Lecture 1 | Modern Physics: Special Relativity (Stanford)

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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Special Relativity. Recorded April 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the third of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu this quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them a wave-like character moving through space and one of the fundamental principles of field theory in fact more broadly nature in general is the principle of relativity the principle the special printless the the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I'm not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know whether it was Galileo or Newton or those who came after them but those early pioneers certainly had the right idea it begins with the idea of an inertial reference frame now inertia reference frame this is something a bit tautological about an inertial reference frame Newton's equations F equals MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what an inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first of all was a reference frame in tale of a reference frame first of all entails a set of coordinate axes in ordinary space X Y & Z and you know how to think about those but it also entails the idea that the coordinate system may be moving or not moving relative to whom relative to whomever we sitting here or you sitting here in this classroom here define a frame of reference we can pick the vertical direction to be the z axis the horizontal direction along my arms here to be the x axis X plus that way X my X is minus in that direction and which one have I left out I've left out the y axis which points toward you from me so there are some coordinate axes for space XY and Z and I didn't this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with a uniform velocity in a definite direction if your frame of reference is an inertial frame of reference in other words if when you throw balls around or juggle or do whatever is supposed to do in an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with uniform velocity relative to you now remember what uniform velocity means it doesn't just mean with uniform speed it means with uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated or if it starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo was often credited with the idea but I never read enough of Galileo to know whether he actually had it or not neither did I read enough of Newtons they both wrote in languages that I don't understand what was I saying oh yes right according to both Newton and anybody else who thought about it very hard the laws of physics are the same in all inertial reference frames laws of physics meaning F equals MA the forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference of and another if you want a kind of pictorial example that I like to use a lot when I'm explaining this to the children or to grownups I like to think about the laws of juggling there are very definite procedures that you train your body to do uh in order to be able to juggle balls correctly now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the x axis and trying to juggle do you have to compensate for the fact that the train is moving and for particular when you throw a ball up into the air that you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame and every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're a good juggler or not but it will not depend on whether you're moving with uniform velocity so the laws of juggling are the same in every inertial reference frame the laws of mechanics are the same in every inertial reference frame the laws Newtonian laws of gravity are the same in every inertial frame according to Newton what about the laws of electrical phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations were the field equations the field theory that governed the electromagnetic field and the way that it propagated and sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth and the fundamental dilemma as you all know I'm sure you all know the fundamental dilemma was both according to well here was the dilemma Maxwell's equations said light moves with a certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down an axis and that velocity comes out to be a certain number out of Maxwell's equations you have two choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case the principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is three times ten to the eighth meters per second which is about what it is if it follows from Maxwell's equations that light moves that fast and if Maxwell's equations are laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because if a light beam is going down that axis and you chase it and run along with it that lets say three-quarters of the speed of light then you want to see that light ray moving much more slowly than three times ten to the eighth meters per second relative to you on the other hand the light ray going in the other direction since you're sort of running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference frame something had to give well the point was of course that they were good laws of nature and that they were the same in every reference frame the thing that had to give is our concepts of velocity space and time and how we measure velocity especially velocities were up which are up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago and there on the Internet I believe relativity and electromagnetism I think that was maybe about three quarters ago I've lost track yeah they're up there they're on the net and they're the lectures on relativity special relativity and electromagnetic theory we're just going to cut through it real fast we're going to cut through the basic ideas of relativity a little more mathematically than I would do if I were teaching it for the first time I teach it the first time I tend to teach it the way Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influence these things instead I'm going to take a more mathematical view of it and think about the properties of various kinds of coordinate transformations coordinates now consists not only of XY and Z but also time T so imagine every event in the world is characterized by just like every particle would be characterized by a position x y&z every event taking place in space-time is characterized by four coordinates X Y Z and T let's suppress for the moment y&z let's just forget I forget them for the moment and concentrate on X and T that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the x axis let's suppose there is no motion along y&z then we can forget y&z for the moment momentarily we'll come back to them and think of motion along X and T and the various reference frames that might be moving along the x axis alright here's here's time vertically is space horizontally physicists always draw space horizontally and time vertically I found out that mathematicians are at least certain computer scientists always draw time going horizontally I didn't know that and I got into an enormous argument with a quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically these are traditions I guess traditions grow up around subjects but time is north and X is east I guess or at least time is upward yeah yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal ok all right now let's in let's imagine a moving observer moving down the x axis with a velocity V let's take his origin of spatial coordinates his origin of spatial coordinates at time T equals zero is just the same let's assume that my I'll be the moving observer I move down the x-axis I am my own origin there's nobody who was your origin that seat is vacant over there so that absent a human over there is the center of the x-coordinates in your frame I'm the X prime coordinates and of course I being very egocentric will take my x-acto is origin to be where I am there I do I move down the x-axis we pass each other our origins pass each other at t equals 0 so that means at T equals 0 your axis and my axes are the same or your origin in my origin is the same but then as I move down the x axis my core my coordinate center moves to the right most of the right that's supposed to be a straight line that's as good as I can do under the circumstances that's a straight line and it's moving with velocity V which means it's X prime equals SR it means x equals VT but it's also that's the way you describe it in terms of your coordinates my centre you described by saying x equals VT how do I describe it I just say X prime my coordinate X prime is 0 X prime equals 0 is the same as x equals VT all right what's the relationship between X Prime and X and T well it's easy to work out if you believe this picture the X prime coordinate is the distance from my origin the x coordinate is the distance from your origin so one of these is X the other is X prime the upper one here is X prime the low and here is X and the relationship between them is that they differ by an amount VT in particular X is equal to X prime minus VT or X prime is equal to X plus VT will have it wrong yes I do X prime is X minus BT and X is X prime plus VT yeah I think I have that's correct now all right what about time itself well according to Newton and according to Galileo and according to everybody who came afterward up until Einstein time is just time is just time is just time there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of God's time God upon his cloud ticking off with his with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time here and so the other equation that went with this is that T prime is equal to T let's forget the top equation here let's just forget it one might say that this was the Newtonian or the Galilean transformation properties between X and T your coordinates and the coordinates that I ascribe to a point in space-time now let's examine a light ray moving down the plus x axis if it starts at the origin here then it moves along a trajectory which is x equals CT C being the speed of light now shortly I'm going to set C equal to 1 we're going to work in units in which C is equal to 1 but not quite yet incidentally once you understand a bit of relativity working in coordinates in which C is not equal to 1 is about as stupid as using different units for x and y are if we used yards for x and feet for y then we will have all kinds of funny factors in our equations which would be conversion factors from X which is measured in feet to Y which is measured in our yards the cycle has its uses log scale has its uses no long skilling long scale well let common interest yep I'm not sure we good but okay I'm just saying it is quite often in practical circumstances that one uses different scales yeah you sometimes you might there might be a good reason I mean um it wouldn't be totally unreasonable for a sailor to use different units for horizontal direction and vertical direction hmm I mean he's used to moving around horizontally he might use what miles miles versus fathoms or something nautical miles versus paddles yeah Persian is relative but um when you talk about a frame of reference you need to specify a period of time because obviously goes that 15 billion years there is no yeah we're ignoring now the fact that the universe began at some time and we're imagining now as Newton did and as the early Einstein did that the universe has just been here forever and ever and ever unchanging totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right but as long as we're interested in time intervals which are not I suspect this is what you're getting at as long as we're interested in time intervals which are not too long in particular time intervals over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything just stays the same as always was is that what you're asking it seems that that this assumption if it is made it needs to what you're describing so well so the question is without imagining to some point as it doesn't lead it doesn't lead to what I'm describing where is this this room for different formulas here this is a formula which is based on an assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you're in a moving frame of reference to different the observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but in any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity C being some numbers in meters per second okay 3 times 10 to the 8th meters per second we will later as I said say C equals 1 let's imagine a light beam moving down the x axis let's describe how X prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out X prime will be X which is CT for that light ray minus VT which is the same as C minus VT all this says is that I see the light moving with a diminished velocity a velocity C minus V why is that because I'm moving along with the light so naturally I see it move slowly the slow compared to what you see it what about the light going in the other direction supposing it was a light beam going in the other direction then how would you describe it you would describe it as x equals minus CT and if I do exactly the same thing I will find that X prime is equal to X that's minus CT – VT which is the same as minus C plus V times T so what this says is that I will see the light moving also in the negative direction that's the minus sign but I'll see it moving with an enhanced velocity C plus V if this were the right story and if these were the right transformation laws for space and time then it could not be the case that Maxwell's equations are laws of physics or laws of nature in the sense that they were true in every reference frame they would have to be corrected in moving frames just like the juggler who had to reach to the right who didn't actually but who thought he had to reach to the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct for motion was the famous Michelson Morley experiment Einstein he just rejected he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate or to the contingent on which reference frame and so he said about to find a framework in which the speed of light would be the same in every reference frame and he basically focused on these equations and after various very very beautiful Gedanken experiments thought experiments about light and about measuring and so forth he came to a set of formulas called the Lorentz transformations I'm going to explain them the Lorentz transformations in a more mathematical way not fancy mathematics but just get we want to get right to the heart of it and not spend the three weeks doing it the best way is to a mathematical problem but before I do let me set up a different mathematical problem which is for most of you you've seen me do this before but nonetheless let's go through it again the problem of rotation of coordinates we're going to do this quickly let's just take spatial coordinates now for the moment two dimensional spatial coordinates let's forget X and T and just concentrate on X&Y two coordinates in space instead of events in space-time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the x and y coordinates just by dropping perpendicular to the x axis in the y axis and we would describe this point as the point at position let's just call it X Y now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician a really nutty one who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between vertical and worth worth worth we should give this direction a name oblique yeah all right the oblique observer the blue observer can blue be seen everybody can see blue okay good ah the blue observer also characterizes points by coordinates which he calls X Prime and Y Prime the X Prime and the Y prime coordinates are found by dropping perpendicular to the X Prime and the Y prime axis so here's X prime is y prime and given a point X Y there's a role it must be a role if you know the value of x and y you should be able to deduce the value of X I'm in y-prime if you know the angle between the two coordinates between the x coordinate and the X prime coordinate and the formulas simple we've used it least in these classes many times I'll just remind you what it is that's X prime is equal to x times cosine of the angle between the two frames between the two coordinate systems minus y times sine of the angle and Y prime is equal to minus plus I think X sine of theta plus y cosine theta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded in two very simple formulas I've used them this signs on these signs of are on the right let's Ella and X prime is bigger than X for small theta since ours here are all so it's Auto Expo Rhine is bigger than it is is it yeah let's see if you rotate it to the next so that y is y prime is zero it's further out X prime rook will have it backward yeah what's your gift I'm not gonna fit nobody so let's say just make sure the links take survive is the little perpendicular there no my life primary so that's y prime y prime is this is why I'm here right right that's why I'm in X prime is bigger than X so there has to be a plus sign on the second you know its prime is bigger than X let's see um yeah X prime is bigger than X yeah X prime is bigger than X looks like that's probably right probably sign but then this one must be man negative yeah okay there's an easy way to correct for it another way to correct for it just call this angle minus theta that would also do the trick because cosine of minus theta is the same as cosine of theta and sine changes sign when you change theta 2 minus theta so if instead of calling this angle theta I called it minus theta then my previous formulas would be right it's true true but the it's an excuse all right what do we know about sine and cosine it's important to understand sine and cosine everything you ever learned about trigonometry can be codified in two very simple formulas if you know about complex numbers the two very simple formulas are that cosine of theta is e to the I theta plus e to the minus I theta over 2 and sine of theta is e to the I theta minus e to the minus I theta over 2i those two formulas contain everything about trigonometry you don't have to know any other formulas other than these for example I will assign you the homework problem of using these two formulas to find cosine of the sum of two angles but the way you would do it is just write the sum of two angles in here and then reexpress the Exponential's in terms of cosine and sine that's easy to do e to the I theta is equal to cosine of theta plus I sine theta and e to the minus I theta is cosine of theta minus I sine theta so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again the other thing to know is that e to the I theta times e to the minus I theta is 1 all right e to the anything times e to the minus the same thing is one those things characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply e to the I theta times e to the minus I theta we will get one on this side but on this side we will get cosine squared of theta plus sine squared of theta naught minus sine squared but plus sine squared cosine squared and then ice minus I squared sine squared that gives us cosine squared plus sine squared cosine squared theta plus sine squared theta so that's equivalent to the fact that e to the I theta times e to the minus I theta is 1 all right now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from XY to X prime Y prime something is left unchanged namely the distance from the origin to the point XY that's something which is you know you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point XY has to be the same independent of which coordinate axes we use well let's take the square of that distance the square of that distance we know what it is let's call it s squared I'm not sure why I use s but s for distance s s for distance s for space I think it must be for space that I'm using it for the spaces for the spatial distance from the origin to the point XY we know what that is it's Pythagoras theorem x squared plus y squared but as I said there's nothing special about the XY axes we also ought to be able to calculate it as X prime squared plus y prime squared well it's not too hard to work out that X prime squared plus y prime squared is x squared plus y squared it's easy to use do X prime squared plus y prime squared will have x squared cosine squared theta it will also have x squared sine squared theta when you add them you'll get x squared plus y squared you know you know the rigmarole so it follows from cosine squared plus sine squared equals 1 that X prime squared plus y prime squared equals also equal is equal to x squared plus y squared work that out make sure that you have this on the control that you understand why from the trigonometry not from the the basic physics of it or the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about sines and cosines if I plot on the blackboard for every angle if I plot sine or cosine along the horizontal axis supposing I plot cosine of theta along the horizontal axis and sine of theta along the vertical axis then if I plot all possible angles they will correspond to a bunch of points that lie on a unit circle Y on a unit circle because sine squared plus cosine squared equals 1 so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating they're convenient for describing unit circles points on unit circles are described in terms of coordinates which are cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions that come in to the transformation the new transformation properties first of all these are wrong and I don't want to use X what's X ya ya now just wrong Newton had it wrong Newton or Galileo however it was postulated who postulated it Einstein modified it now we're going to have to make sure that Einstein's modification doesn't change things in situations where Newton knew where Newton's equations were good approximations the situations where I'm Stan's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the 20th century nobody or nothing had ever moved faster than a hundred miles an hour probably well of course some things did light did but for all practical purposes light didn't travel at all it's just when you turned on the switch the light just went on so light didn't travel nothing and anybody's experienced direct experience traveled faster than 100 or 200 miles an hour and well I should say nothing travels faster than 100 miles an hour and then live to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities newton's formulas must be correct they work they're they're very useful they work Nutan got away with it so there must be good approximations it better be that whatever einstein did to the equations in particular to these two equations here had been a reduced to newton's equations in the appropriate limit okay let's come back now to light light according to the Newton formulas doesn't always move with the speed of light but let's let's try to figure out what it would mean of a better formula of a replacement for this but light always moves with the speed of light first of all let's set the speed of light equal to one that's a choice of units in particular it's a choice of the relation between space units and time units if we work in our light years for spent for a distance and years for time then light moves one light year per year the speed of light is one if we use seconds and light seconds it's also one whatever whatever scale we use for space if we use for time the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of view of very slowly moving things those are odd units but if we were electrons with neutrinos and whizzing around like photons they would be the natural units for us speed of light equals one so let's set the speed of light equal to one as I said it's just the choice of units and then a light ray moving to the right just moves along a trajectory x equals T C is just equal to one a light ray moving to the left is x equals minus T how can we take both of these equations and put them together sorry x equals minus T can I write a single equation which if it's satisfied is a light ray either moving to the left or to the right yes here's an equation x squared equals T squared it has two solutions x equals T and X equals minus T the two square roots or x squared equals T squared is equivalent to either x equals T or x equals minus T in other words this equation here has the necessary and sufficient condition for describing the motion of a light ray either to the right or to the left supposing we found a replacement for this equation which had the following interesting property that whenever let's let's write it this way X square minus T squared equals 0 this is even better for our purposes x squared minus T squared equals 0 that's the necessary and sufficient condition to describe the motion of a light ray supposing we found a new set of rules a new set of transformation properties which which um had the property that if x squared minus T squared is equal to 0 then we will find that X prime squared minus T prime squared is equal to 0 in other words supposing this implied this and vice-versa then it would follow that what the unprimed observer you and your seats see is a light ray the primed observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X is equal to T it does not follow that X prime is equal to the T prime in fact it says something quite different okay so the form of these equations must be wrong let's look for some better equations now at this point let's in fact let's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that when X square minus T squared is equal to zero then X prime squared minus T prime squared is equal to zero let's say something even bolder let's say the relation between XT and X prime T prime is such that x squared minus T squared is equal to X prime squared minus T prime squared in other words pick any X and any T and calculate X square minus T squared then take the same point except reckoned in the primed coordinates in other words we take a certain event a light bulb goes off someplace you say that corresponds to X and T I say it corresponds to X Prime and T Prime but let's require just to try it out see if we can do it let's look for transformations so that X square minus T squared will always be equal to X prime squared minus T's prime squared that would be enough to ensure that everybody will agree about the speed of light why if x squared minus T squared equals X prime minus T prime squared for all X and T and so forth then when X square minus T squared equals zero X prime minus T prime squared will be zero and then if this is a light ray so is this a light ready everybody get the logic ok good so let's assume now that let's ask can we find transformations which have this particular property now it's not so different from looking for transformations which preserve x squared plus y squared equals x prime squared plus y prime squared it's just a little minus sign other than a minus sign here X square minus T squared look of these two is very similar and the mathematics is quite similar here are the transformations which preserve x squared plus y squared what are the transformations which preserve x squared minus T squared well they are the Lorentz transformations they are the fundamental transformations of the special theory of relativity they're not this but they're closely related or perhaps one should say closely analogous to these equations here but we have to substitute for circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions well I didn't want to erase that all right these are the basic rules governing circular functions cosine theta this sine theta is equal to this and the e to the I theta in terms of cosine and sine all right let's see if we have a yeah we do have a blank blackboard here let me write whoops what did I do here I erased something I didn't mean to erase incidentally does everybody see how I got this side from the side you just add and subtract the equations appropriately and you isolate it to the I theta e to the minus R theta that's elementary exercise alright hyperbolic functions what are hyperbolic functions alright those are functions of the form hyperbolic cosine cosh hyperbolic cosine first of all the angle theta is replaced by a variable called Omega which I will call Omega Omega is called a hyperbolic angle it doesn't go from zero to two pi and then wind around on a circle it goes from minus infinity to infinity goes from minus infinity to infinity so it's a variable that just extends over the entire real axis but it's defined in a manner fairly similar to cosine and sine cosh Omega is by definition you're not allowed to ask why this is definition e to the Omega plus e to the minus Omega over 2 all we do is substitute for theta or for Omega theta I theta substitute Omega and that gives you hyperbolic functions likewise or similarly there's the hyperbolic sine and that's given by e to the Omega minus e to the minus Omega over 2 essentially you throw away all eyes out of that formula out of the top formulas just throw away all Sun all eyes the equations on the right-hand side become e to the Omega equals hyperbolic cosh Omega plus sin Chi Omega and e to the minus Omega equals cosh so mega- cinch Omega I think that's right is it right gosh – cinch it is yeah it is right okay now what about the analog of cosine squared plus sine squared equals one that simply came by multiplying this one by this one so let's do the same operation multiplying e to the Omega by each by e to the minus Omega gives one and now that gives cosh squared minus cinch squared you see we're getting a minus what we want we want that minus the minus is important we want the well somewhere is under here was a formula with a minus sign yeah we want to get that – into play here that's cos Omega squared knockouts Prakash squared Omega minus sin squared Omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas these in these and they're more or less definitions but there are the useful definitions now yeah go ahead yeah not only is it worth mentioning I was just about to mention it so I squared minus y squared is what hyperbola yeah right exactly so if I were to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of theta and sine of theta but cosine cosine cosh of that of Omega and since Omega what's in other words on the x-axis now we're going to plot cos Omega and on the y-axis cinch Omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola with asymptotes that are at 45 degrees you can see let me show you why why the asymptotes are at 45 degrees when Omega is very large when Omega is very large then e to the minus Omega is very small right when Omega is very large e to the minus Omega is very small and that means both cosh and cinch are both essentially equal to e to the plus Omega in other words when Omega gets very big cosh and cinch become equal to each other and that's this line here cash equals cinch along this line here so when Omega gets very large the curve asymptotes to to a curve which is a 45 degrees it's not hard to see that in the other direction when Omega is very negative that that it asymptotes to the other asymptotic line here so that's why it's called hyperbolic geometry it the hyperbolic angle the hyperbolic angles the caches the cinches play the same role relative to hyperbolas as sines and cosines do two circles any questions No so cosh Omega equals zero how would you plot that hi purple okay show me hmm Oh cos squared minus sin squared equals zero no that's no no cos squared minus sin squared equals one in the same sense that sine squared plus cosine square it never equals zero I think what I think you want to ask a different question I think oh well since Omega equals zero is the horizontal axis the costume a equals zero is the vertical eyebrows right okay well this is the x-intercept yeah it's it's the vertex I just think here's one point on a minute oh man the x-intercept there is one yeah because Kostroma cost of zero is one to see that just plug one r 0 in here 1 plus 1 divided by 2 is 1 at least it was yesterday yeah stores okay so now we we're sort of starting to cook a little bit we're starting to see something that has that nice minus sign in it but what's it got to do with X and T and X Prime and T prime we're now set up to make let's call it a guess but it's a guess which is based on the extreme similarity between hyperbolas and circles cautions and cosines and so forth he is the guess I'm going to make and then we'll check it we'll see if it does the thing we wanted to do my formula instead of being this has gotten with and we're now going to have instead of x and y we're going to have x and t time and x later on we'll put back y&z we're going to have to put back y&z but they're very easy okay so let's start with X prime X prime is the coordinate given to a point of space-time by the moving observer namely me and I'm going to guess that it's some combination of X and T not too different but not the same as where is it X prime equals X minus VT I'm going to try cosh Omega X let's write X cos Omega minus T sin Omega sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of Omega just as you did here so this let's do it this way X cos Omega minus T sin Omega and T prime going to look similar but without the extra minus sign here this you know the relation between sines cosines and cautious and cinches is one of just leaving out an eye you go from sines and cosines the clashes and cinches by leaving out the I well if you track it through carefully you'll find that this minus sign was really an I squared it's not going to matter much I will just tell you it was really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the guess for the formula connecting X prime T Prime with X and T it equals let's say X since Omega – no – plus T cos Omega in this case there are two minus signs in this case there was only one minus sign okay but but let's check what do we want to check we want to check that X prime squared minus T prime squared is equal to x squared minus T squared your ask you're probably asking yourself what is this Omega what does it have to do with moving reference frames I'll tell you right now what Omega is it's a stand-in for the velocity between the frames we're going to find the relationship between Omega and the relative velocity of the reference frames in a moment there has to be a parameter in the lower end these are the lines in these are the Lorentz transformations connecting two frames of reference in the Lorentz transformations as a parameter it's the velocity the relative velocity that parameter has been replaced by Omega it's a kind of angle relating the two frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here that X prime squared minus T prime squared is equal to zero ah is equal to X square minus T squared I'm getting to that point in the evening where I'm going to make mistakes all right this is easy you just work it out you use all you have to use is that cosine squared minus sine squared is 1 you can work that out by yourself but we can just see little pieces of it here X prime squared will have x squared cos squared Omega t prime squared will have x squared sin squared Omega if I take the difference between them I'll get a term with an x squared times cos squared minus sin squared but cos squared minus sin squared is one fine so we'll find the term with an x squared when we square take the square of the difference between the squares of this and this and likewise will also find the T squared the cross term when you square X Prime you'll have XT cost cinch when you square T Prime you'll have XT costs inch when you subtract them it'll cancel and it's easy to check that's our basically one liner to show that with this transformation here x prime squared minus T's prime squared is x squared minus T squared which is exactly what we're looking for let me remind you why are we looking for it if we find the transformation for which the left-hand side and the right-hand side are equal then if x squared equals T squared in other words if the right-hand side is 0 the left-hand side will also be 0 but x squared but x equals T that's the same as something moving with the speed of light in the X frame of reference if this being 0 is equivalent to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here X prime squared minus T prime squared is equal to x squared minus T squared all right that does follow by a couple of lines using cos squared minus N squared equals 1 but what I want to do let's take another couple of minutes now let's take a break for five minutes and then come back and connect these variables Omega with the velocity of the moving frame of reference somebody asked me a question about the ether and what it was that people were thinking somehow Einstein never got trapped into this mode of thinking um well what were they thinking about when they were thinking about the ether what exactly was the michelson-morley experiment well I'll just spend the minute or two mentioning it certainly Maxwell understood that his equations were not consistent with with Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in a material or water waves propagating on water and of course it is true that if you move relative to the atmosphere or move relative to the substance that sound is propagating in you'll see sound move with different velocities depending on your motion if you're at rest in a gas of material isn't there's a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have zero velocity if you're in that reference frame then first of all light has the same velocity that way as that way number one and it has a velocity that's determined by the properties of the fluid that the sound is moving in okay Maxwell more or less thought that light was the same kind of thing that there was a material and the material had a rest frame and that particular rest frame was the frame in which light would move with the same velocity to the left as to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics but that they would change when you moved from frame to frame just happened by some luck we happen to be more or less at rest relative to the ether to this strange material um of course you could do an experiment with sound if you're moving through the sound you can check that the velocity in different directions is different you do let's not worry exactly how you do that that's what the Michelson Morley experiment was Michelson and Morley I suppose said look the earth is going around in an orbit maybe at one season of the year we just happen to be at rest relative to the ether by accident and some other season six months later we're going to be moving in the opposite direction and we won't well we won't be at rest only at one point in the orbit could we be at rest relative the–this or at any other point in the orbit we wouldn't be so if we measure in November that light moves the same than all possible directions then in what's what's the opposite of November May then in May we should find that light is moving with great with the different velocities in different directions and he tried it and a very fancy and sophisticated way of measuring the relative velocity in different directions but he found that there was no discrepancy that the light traveled the same velocity in every direction at every time of year there were all sorts of ways to try to rescue the ether but none of them worked none of them work and the result was one had to somehow get into the heart of space and time and velocity and mid distance and all those things in a much deeper way in a way that didn't involve the idea of a material at rest in some frame of reference that that propagated the light okay oh where are we I forgotten where we were when we stopped somebody remind me whoo-hah Omega yeah what is Omega forgotten Omega Oh how Omega is really metal speed of light but to the velocity of the moving reference frame here we have two reference frames X T and X Prime and T prime what's the relationship between them well the whole goal here was to understand the relationship between frames of reference moving with relative velocity between them Omega is not exactly the relative velocity but it is closely related to it okay let's say let's see if we can work out the relationship we know enough to do it let's see if we can work out the relationship between Omega and the velocity of the moving frame all right again let's go back to this picture there's a frame of reference moving let's redraw it here's my origin moving along okay what does it mean to say that from your perspective my frame of reference so my origin is moving with velocity V well by definition this is not a law now this is a definition and says that this line here has the equation x equals VT that's the definition of this V here my origin moves relative to your origin it moves with a uniform constant velocity that's an assumption that we're talking about two inertial frames of reference and you in your frame of reference will write x equals VT that's the definition of V if you like what will I call it I will call it X prime equals zero all along there I will say X prime is equal to zero it's my origin of coordinates okay now let's come to this transformation law here and see if we can spot how to identify V well X prime equals zero that's this trajectory moving at an angle with a velocity V X prime equals zero is the same as saying X cos Omega equals T sin Omega X prime equals zero set this side equal to zero and that says that X cos Omega equals T sin Omega all right so whatever the connection between velocity and Omega is it must be such that when X prime is equal to zero X cos Omega equals T sin Omega well let's look at that equation it also says that X is equal to sin CH Omega over cos Omega times T well that's interesting because it's also supposed to be equivalent to x equals VT now I know exactly how to identify what the velocity is as a function of Omega the velocity of the moving transformation the moving coordinate system must just be sin Chi Omega over cos Omega that's the only way these two equations can be the same x equals VT x equals sin Chi Omega over cos Omega times T so now we know it we know what the relationship between velocity and Omega is write it down the velocity of the moving frame now this is not the velocity of light it's just the velocity of the moving frame must just be cinch Omega over cos omega well actually i want to invert this relationship i want to find sin and cos omega in terms of the velocity i want to rewrite these Lorentz transformations where are they i want to rewrite these Lorentz transformations in terms of the velocity that's the familiar form in which you learn about it in in elementary relativity books X prime is equal to something with velocities in it to exhibit that all we have to do is to find Cinch and cosh Omega in terms of the velocity that's not very hard let's let's work it out the first step is to square it and to write V squared is equal to cinch Omega squared over cosh Omega squared that was easy next I'm going to get rid of since Omega squared and substitute where is it I lost it one is equal to cos Omega squared minus cinch Omega squared alright so wherever I see cinch Omega squared I can substitute from here namely cosh squared Omega minus one is equal to sine squared Omega so here we are this is just equal to hash of Omega squared minus one divided by cost of Omega squared or let's multiply by what I want to do is solve for cost Omega in terms of velocity I want to get rid of all these cautions and cinches of Omega and rewrite it in terms of velocity so first x cost Omega squared we have cosh squared Omega times V squared equals cosh squared Omega minus one or it looks to me like this is cosh squared Omega times one minus V squared equals one what I've done is transpose yeah cos squared times V squared minus cos squared itself that gives you cos squared 1 minus V squared equals 1 change the sign can everybody see that the second line follows from the first I'll give you a second yeah yeah yeah it's clear ok finally we get that cos Omega is equal to 1 divided by 1 minus V squared but now I have to take the square root cos Omega / one minus V squared and then take the square root and that gives you cos Omega now we've all seen these square roots of 1 minus V squared in relativity formulas here's where it begins the kayne we begin to see it materializing what about sin Chi Omega let's also write down sin Chi Omega well from here we see that sin Chi Omega is just equal to V times cos Omega this is easy since Omega equals V times cos Omega sorrow sin Chi Omega is V divided by square root of 1 minus V squared let's go back to these Lorentz transformations over here and write them getting rid of the trigonometric functions the hyperbolic trigonometric functions and substituting good old familiar velocities let's get rid of this and substitute the good old ordinary velocities ok so we have here X prime equals x times cos Omega and that's divided by square root of 1 minus V squared then this minus T times sin Omega which is V over the square root of 1 minus V squared or if I put the two of them together and combine them over the same denominator it's just X minus VT divided by square root of 1 minus V squared I think most of you have probably seen that before maybe slightly different let's let's clean it up a little bit X prime equals X minus VT divided by the square root of 1 minus V squared what about T prime T Prime is equal to t minus V X over square root of 1 minus V squared T prime is equal to T times cos cost is just 1 over square root and then x times sin CH that gives us the extra V in other words the formulas are more or less symmetrical and those are all good old Lorentz transformations now what's missing is the speed of light let's put back the speed of light the put back the speed of light is an exercise in dimensional analysis there's only one possible way the speed of light can fit into these equations they have to be modified so that they're dimensionally correct first of all one is dimensionless has no dimensions it's just one velocity is not dimensionless unless of course we use dimensionless notation for it but if velocity is measured in meters per second then it's not dimensionless how do we make V squared dimensionless we divide it by the square of the speed of light in other words this V squared which is here which has been defined in units in which the speed of light is 1 has to be replaced by V squared over C squared likewise over here V squared over C squared now velocity times time does have notice first of all the left hand side has units of length the right hand side this is dimensionless X has units of length but so does velocity times time so this is okay this is dimensionally consistent as it is but over here it's not the left hand side has dimensions of time that's all right 1 minus V squared over C square that's dimensionless this has units of time but what about velocity times X velocity times X does not have units of time in order the given units of time you have to divide it by C square okay let's check that velocity is length all the time times length divided by C squared that's length square R which gets correct but it's correct all right this is probably familiar to most of you who've seen relativity once or twice before these are the equations relating to different moving coordinate systems moving relative to the x axis but you see the deep mathematics or the mathematical structure of it in many ways is best reflected by this kind of hyperbolic geometry here and you know most physicists by now never write down the Lorentz transformations in this form much more likely to write them in this form easier to manipulate easier to use trigonometry or or hyperbolic trigonometry it's a little exercise it's a nice little exercise to use this the hyperbolic trigonometry to compute their to compute the compounding of two Lorentz transformations if frame two is moving relative to frame one with velocity V and frame three Israel moving relative to two with velocity V Prime how is three moving relative to one the answer is very simple in terms of hyperbolic angles you add the hyperbolic angles not the velocities but the hyperbolic angles the hyperbolic angle of three moving relative to one is the hyperbolic angle of three moving relative to two plus two moving relative to one and then you use a bit of trigonometry or hyperbolic trigonometry to figure out how you do the inches and kosh's of the sum of 2 hyperbolic angles very straightforward and I'll leave it as an exercise to see if you can work that out much easier than anything else ok so there there we have the Lorentz transformations yeah oh oh absolutely yes that's that's that's a good point yeah when we that's right if we have frame 1 let's call this x1 and y1 x2 and y2 and finally x3 and y3 well then the angle of – let's call F of 3 relative to 1 let's call it theta 1 3 is just equal to theta 1 2 plus theta 2 3 the angle connecting frame one with frame 3 is just the sum of the angle theta 1 2 plus theta 2 3 so in that respect the Lorentz transformations are much simpler in terms of the Omegas it's the Omegas which combined together to add when you add velocities now how different is omega from the velocity let's work in units in which the speed of light is equal to 1 where is our formula for velocity all right let's take this formula over here what a cinch Omega 4 small Omega let's put the C squared there a let's not put the C square there or not put the C square there since Omega is essentially Omega when Omega is small just like sine is omega where is theta when theta is small the cinch function the cost function looks like like this the cinch function looks like this but it but it crosses the axis with a slope of 1 for small Omega cinch Omega is proportional to Omega for small velocity one minus V squared is very close to 1 if the velocity is a hundredth of the speed of light then this to within one ten-thousandth is just 1 if we're talking about velocities a millionth of the speed of light then this is very close to 1 and so since Omega and velocity are very close to each other it's what's going on here Thanks okay so for small velocities Omega and velocity are the same the actual correct statement is that V over C is like Omega the dimensionless velocity over the speed of light is like Omega for small Omega and small velocity so for small velocity adding velocities and adding omegas are the same things but when the velocities get large the right way to combine them to find relationships between different frames is by adding Omega and not adding velocities when you add Omega like compounding velocities as you've got it there I guess you won't go greater than 45 degrees that guess because that would be faster than light no but Omega no more you see this bit the speed of light is V equals one that corresponds to Omega equals infinity yeah yeah so Omega Omega runs over the whole range from minus infinity to infinity but when it does V goes from minus the speed of light to the speed of light so you can add any omegas and still add any omegas Omega that's right there's no there's no speed limit on Omega is this like we just go on that diagram it looks like it's greater than 45 degrees if here where where I make a and I guess they use the definition of state along the hyperbola yeah that's right sorry where are we right there today I guess that's theta though isn't it this is Theta that's a good oh god yeah right right yeah Omega is the distance along hyperbola that's right distances that's right Omega is a kind of distance along the hyperbola all right now let's let's talk about that a little bit all right now that we've established the basic mathematics structure of the transformations I think we should go back and talk about some simple relativity phenomena and derive them oh one thing which is important which I yeah well let's see we're here are my Lorentz transformations over here I said we should we ought to at the end make sure that our transformations are not too dissimilar from Newton's in particular when the velocities are small they should reduce to Newton that's all we really know that's or at least that's all that Newton really had a right to assume that when the velocities are smaller than something or other that his equations should be good approximations isn't adding velocity good enough isn't velocities adding good enough in fact you're right in fact you're right but let's just look at the transformations themselves all right as long as the velocity is a small percentage of the speed of light an ordinary velocities are what a hundred miles an hour versus 186,000 miles an hour what is that it's small right and it's doubly small when you square it so for typical ordinary velocities even the velocities of the earth around the Sun and so forth fairly large velocities what 60 kilometers per second or something like that 60 kilometers per second is pretty fast that's the that's the orbital earth around the Sun it's pretty fast but it's nowhere near 300,000 kilometers per No yeah looks here on a thousand meters per second we're I'm sorry three times ten to the eighth no three times three hundred thousand kilometers per second right 60 kilometers per second three hundred thousand kilometers per second small fraction and then square it so for ordinary motions this is so close to one that the deviation from one is negligible so let's start with the top equation for the top equation this is negligible and it's just x prime equals X minus VT the bottom equation here you have a C squared in the denominator whenever you have a C squared in the denominator that's a very very large thing in the denominator this is negligible compared to T so here the speed of light is also in the denominator just forget this and it's just T but it's just T prime equals T it's just D prime equals T so in fact Newton's formulas are essentially correct for slow velocities no no significant departure from Newton until the velocities get up to be some some appreciable fraction of the speed of light okay let's talk about proper time proper time and then let's do a couple of relativity examples yeah question the bottom equation when X is very large yes that's right when X is exceedingly large you get a correction but that correction that X has to be very large look let's let's discuss before we do anything else let's let's let's talk about that a little bit X minus VT one minus V squared over C squared yeah let's alright in my drawings I'm going to sitt C equal to one but in the equations you can leave the C there okay this equation we understand apart from this one minus V squared over C squared in the denominator it's just this x equals V T or X minus V X minus X minus VT that's Newton let's look at this one over here okay let's look at the surface T prime equals zero T prime equals zero is the set of points that I in my moving reference frame call T call time equals zero it's what I call the set of points which are all simultaneous with the origin T prime equals zero is just everyplace in space-time which has exactly the same time according to my frame of reference and I will therefore call all those points synchronous at the same time what do you say about them if T prime is equal to zero that says that T is equal to V over C squared X now let's set C equal to one for the purpose of drawing just for the purpose of drawing I don't want this huge number C squared to distort my drawings too much it says the T equals V X what does the surface T equals V X look like it looks like this T equals V X which is also X is equal to 1 over V T so it's just a uniform line like that all of these points are at different times from your reckoning this ones later this ones later this ones later and so forth according to my reckoning all these points are at the same time so we disagree about what's simultaneous this was this was the hang-up incidentally this was the basic hang-up that took so long to overcome that took Einstein to overcome it the idea that simultaneity was the same in every reference frame nobody in fact it was so obvious that nobody even thought to ask a question is simultaneous does it mean the same thing in every reference frame no it doesn't in more in your reference frame the horizontal points are all simultaneous with respect to each other in my reference frame what I call horizontal what I call simultaneous you do not okay so simultaneity had to go let me point out one more thing about these equations I'm not going to solve them for you but I will tell you the solution anyway how do you solve for X and T in terms of X Prime and T Prime well think about it in the case of angles supposing I have a relationship like X prime is equal to X cosine theta what is it plus plus y sine theta and y prime is equal to X minus X sine theta plus or Y cosine theta and supposing I want to solve for x and y in terms of X Prime and Y Prime you know what the solution is just change theta 2 minus theta and write that X is equal to X prime cosine of minus theta but what's cosine of minus theta right cosine theta plus y sine of minus theta what's sine of minus theta minus sine theta times y and likewise for y prime Y prime is equal to minus x times sine of minus theta so that becomes plus X sine theta plus y cosine of minus theta which is cosine theta you don't have to go through the business of solving the equations you know that if one set of axes is related to the other by rotation by angle theta the second one is related to the first one or vice versa the first one is related to the second one by the negative of the angle if to go from one frame to another you rotate by angle theta and to go from the second frame back to the first you rotate by angle minus theta so you just write down exactly the same equations interchange Prime and unprimed and substitute for theta minus theta same thing for the Lorentz transformations exactly the same thing if you want to solve these for X and T write down the same equations replace primed by unprimed and change the sign of omegas to minus the sines of omegas change sinus rgn of all the sign all the cinches okay in other words just send Omega 2 minus Omega and that will solve the equations in the other direction yeah yes it's also the same as changing V 2 minus V yes the way to see that is to go right what was it what do we have cosh Omega yep yeah that's right via sign yes that was correct yeah you just well you change Omega 2 minus Omega it has the action of changing V 2 minus V you can just check that from the equations good alright let's let's talk about proper time a little bit proper time if you're doing ordinary geometry you can measure the length along a curve for example and the way you do it is you take a tape measure and you you know sort of take off you take off equal intervals equal equal little separations you can think of these separations as differential distances DS squared small little differential distances and that differential distance is d x squared plus dy squared with the x squared and the y squared are just the differential increments in x and y DX and dy this is d s alright so that's the way and you add them up you add them up that's the way you compute distances along curves it's quite obvious that if you take two points the distance between those two points depends on what curve not the same for every curve so I'll measure the longer curve you have to know not only the two points but you have to know the curve in order to say what the distance between those points are of course the distance between its longer straight line that's that's well-defined but the distance along a curve depends on the curve in any case D s squared equals the x squared plus dy squared is the basic defining notion of distance between two neighboring points if you know the distance between any two neighboring points in a geometry you basically know that geometry almost essentially completely so given this formula for the distance between two points you can compute if you like the distance along a curve because you've got to take the square root of this and then add them up don't anhedonia the squares add the differential distances all right the important thing is here that square root of DX squared plus dy squared which is the distance between neighboring points doesn't depend on your choice of axes I could choose X Y axes I could choose X prime y prime axes if I take a little differential displacement the X and the y or I just take two points two neighboring points don't even give them labels and measure the distance between them the distance between them should not depend on conventions such as which axes are used and so when I make rotational transformations the X square plus dy squared doesn't change the X and the y may change but the x squared plus dy squared does not change the same thing is true in relativity or the analogous thing we don't measure distances along the paths of particles let's say now that this curve here is the path of a particle moving through space-time there's a particle moving through space-time and we want some notion of the distance along it the notion of distance along it another example would just be a particle standing still as a particle standing still particle standing still is still in some sense moving in time I wouldn't want to say that the distance between these two points and space-time is zero they're not the same point I wouldn't like to say it's zero I would like to say there's some kind of notion of distance between them but it's quite clear that that distance is not measured with a tape measure this point and this point are the same point of space boom here at this point of space and that at a later time boom again at the same point of space two events at the same point of space how do I characterize and some nice way the distance between those two events that occurred in the same place you don't do it with a tape measure all right what do you do with a clock a clock you take a clock and you start it at this point tic tic tic tic tic tic tic a stopwatch you press it at this point tic tic tic tic tic it picks off intervals and then you stop it at that point and you see how much time has evolved that's a notion of distance along a particle trajectory it's not the distance the particle moves in space it's a kind of distance that it's moved through space-time and it's not zero even if the particle is moving standing perfectly still in fact what it is is it's the time along the trajectory what about a moving particle well you can imagine that a moving particle carries a clock with it of course not all particles carry clocks but we can imagine they carry clocks with them as they move and we can start the clock over here and then the clock over here what is the time read off by this moving clock the time read off by a moving clock is much like the distance along a curve measured by a tape measure in particular it should not depend on the choice of coordinates why not this is a question that has nothing to do with coordinates I have a clock made in the standard clock Factory the standard clock Factory and I don't know we're in Switzerland someplace makes a certain kind of clock that clock gets carried along with a particle and we ask how much time evolves or how much time elapses or how much the clock changes between here and here that should not depend on a choice of coordinates it shouldn't depend on a choice of coordinates because it's a physical question that only involves looking at the hands of the clock in fact we can ask it for little intervals along along the trajectory we could ask how much time elapses according to the clock between here and here well the answer again should not depend on what coordinates you use which Lorentz frame you use and there's only one invariant quantity that you can make out of the D X's and DTS describing this point describing these two points there's a little interval DT and there's a little interval DX now we're in space and time not ordinary not ordinary space and the quantity which is invariant there's really only one invariant quantity that you can make out of it it is DT squared minus DX squared it's the same quantity x squared minus T squared for a whole you know for a whole interval the T squared minus DX squared that's the quantity which is invariant it's minus D it's the negative of what I wrote over here x squared minus T squared okay this quantity is equal to the X prime squared minus DT power sorry DT prime squared minus the X prime squared the same algebra goes into this as goes into showing that X prime squared minus T prime squared equals x squared minus T squared incidentally this is the same as saying T prime squared minus X prime squared equals T squared minus x squared doesn't matter which way you write it all right so that suggests that suggests that the time read off the invariant time read off along a trajectory between two points separated by DX and DT is just the square root of DT squared minus DX squared why the square-root incidentally okay you're going to integrate in detail I can integrate DT yeah well alright why not just DT square minus the x squared for the time between here and here is it here's an answer supposing we go to you two intervals exactly the same as the first one we go an interval over here DX and DT and then we go another DX in DT what happens when we double the interval to DT squared minus DX squared it gets multiplied by four because everything is squared well I wouldn't expect a clock when it goes along you know when it goes along a trajectory for twice the the interval here to measure four times the the time I expected to measure twice the time so for that reason the square root is the appropriate thing here okay that's called D tau squared the tau squared the proper time along the trajectory of an object you're right that's just the towel or D tau squared being the x squared minus DT squared the Tau is called the proper time let's go I think we'll let's see the towel is called the proper time and it is the time read by a clock moving along a trajectory it's not just DT that's the important thing it's not just DT the T squared minus the x squared let's do one last thing let's just do the twin paradox in this language I think I think I've had it I'm going to finish you can do the twin paradox in this language all you have to do is to compute the proper time along two trajectories one that goes out with a uniform velocity turns around and comes back with the same uniform velocity versa a trajectory which just goes from one point to the st. the another point along a straight line and it's no more weird it's no weirder really from this perspective than saying the distance from one point to another along two different curves do not have to agree the proper time along two different curves in general will not agree what is a little bit weird is that because of this minus sign the proper time this way is less than the proper time this way that's the consequence of this minus sign here moving with some DX decreases the proper time all right we'll do a little bit more next time but then I want to get to the principles of field theory and and connect some of this with field equations for interesting wave fields the preceding program is copyrighted by Stanford University please visit us at stanford.edu

NEET Physics – Motion in a Plane | IMPORTANT QUESTIONS | AIIMS | 2020 2021 Physics Video Lectures

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25 May, 19 – Motion in a Plane – Live Session and Revision – Abhishek Sir – 6:30 PM – NEET & AIIMS 2020 2021

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Physics – Test Your Knowledge: Energy (24 of 33) Will the Block Slide Down?

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In this video we will determine how far, d=?, a block will travel, and if the block will slide down a 20 degree incline with a frictional-force=15N, and v0=8m/s.

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welcome to our lecture line our next problem deals with a block that is pushed up a hill it's given an initial velocity of 8 meters per second but since its friction and regaining height eventually the block will come to a stop at the end the block will gain a certain amount of height and the velocity at that point will be zero now the question is is the friction force sufficient to keep the block from sliding back down the hill once it's reached its maximum height and reach the farthest distance D so we're going to find the distance D and then we're going to determine if the block will slide back down the hill so to find the distance D will use the energy conservation equation all right that the work put into the system plus the original potential energy plus the original kinetic energy equals the final potential energy plus the final kinetic energy plus any energy loss due to friction and yes indeed the restriction so energy will be lost there's no work put into the system because we're using the original kinetic energy so we don't have to put in a work term so that's zero plus it starts from zero height up the original kinetic energy will be one-half MV squared the final potential energy will be MGH it'll be no kinetic energy at the end because the block is not moving and the energy lost will be forced friction times distance now the height can be written in terms of the distance we can say that the height final will be equal to the hypotenuse which is a distance times the sine of the angle theta because it's the opposite side to the triangle so let's go ahead and plug that in so end up with one half and V squared equals MGH plus the friction force let's see here friction force that would be 15 multiplied times the distance which would be oh wait a minute distance height no we're going to change the distance we're going to write in terms of eight because that way we'll solve for H so let's this was the age divided by sine of theta or maybe we want to change this age to distance let's do that that's probably better let's go ahead and instead of H well write what H is equal to that would be D sine theta the sine theta plus the friction force which is 15 times D and then we can factor out a D install for D so let's see here we have one half and the squared is equal to M G D sine theta and the friction force would be let's see here mm-hmm that would be plus 15 times D what I wanted to do is factor out the D right so plus 15 and the D is factored out that's better okay now we'll solve for D so D is equal to one-half MV squared divided by mg sine theta plus 15 and now let's calculate what the D is because I will come up here we have lots of board space so D equals 1/2 times the mass which is 4 times the initial velocity squared that would be 8 squared divided by mg that would be 4 times 9.8 times a sine of 20 degrees and plus 15 for the friction force all right let's see what that's equal to so we have 4 times 9.8 times the sine of 20 equals at 15 to death move that to the numerator then multiply it times 0.5 times 4 and times 8 square root of 64 equals and that gives us four point 506 let's see here so distance equals four point five zero six meters so that's the distance of the hill now will the blog begin to slide back hmm we know the friction force is 15 Newtons that would be the kinetic friction force which means the static friction force will be a little bit higher but let's say it's the same so how do we determine that notice we have an mg acting downward we have the perpendicular component which is mg cosine theta and we have the parallel component which is mg sine theta and if mg sine theta is larger than the friction force the block will slide so is the question is the M G sine theta greater than 15 Newton's question mark if the answer is yes the block will begin to slide back down the hill so let's figure that out so we have M is 4G is 9.8 and we multiply times the sine of 20 degrees and question mark is that greater than 15 Newtons all right so we have 4 times 9.8 times 20 sine equals that's 13.4 Newton's so 13.4 Newtons is that greater question mark than 15 Newtons and the answer of course is no therefore block will not slide all right and that is how it's done

Physics – Test Your Knowledge: Energy (25 of 33) Where will the Block Stop?

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In this video we will find how many times a block of m=5kg will bounce between 2 spring of different spring constants across a frictionless and friction surface and where will the block come to rest, x2=?

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welcome to our lecture online now here we have a fun problem not that it's not difficult to do but it is a fun problem and what's going on here well we have two Springs one on either end we first push a block that has a mass of five kilograms against the first spring and compress it a distance of 0.15 meters notice the spring constant of both Springs is the same 200 Newtons per meter we let go also notice that for most of the route between the two Springs there's no coefficient of friction it's equal to zero so therefore there's no friction but there's a small patch of distance of 25 centimeters or 0.25 meters where the coefficient of friction is 0.08 so there's going to be some loss of energy but then the block presuming it gets passed this will bump into the spring compress the spring so the first question is how much will that spring be compressed then the block will get pushed back we'll go over the rough patch again will compress the spring over here then it'll go back and forth and back and forth until all the energy is lost on the right rough patch and eventually it will stop somewhere on the rough patch the question is where so we want to answer what is the first compression of spring – how many times does the block go across the rough patch and finally where's the block finally stop so the way we're going to solve this problem is first calculate how much initial energy the block starts with so the initial energy is going to be potential energy so potential energy initial which is the initial energy of the system which is going to be 1/2 K x squared so let's go ahead and calculate that how much energy the system starts with so that means we have 1/2 the spring constant times the distance 0.15 squared and let's see how much energy that is 0.15 squared times 100 which is 2.25 joules now let's calculate how much energy is lost when it goes across the rough patch so energy lost each time it goes across the rough patch is equal to well that would be the work it requires to go across that rough patch which is force times distance it's a friction force times distance and how do we calculate the friction force well when the block is on top of the rough patch and let's put the block over here for a moment then you can see that we'll have the weight mg the normal force n which means that the friction force is going to be equal to the normal force times mu which is equal to M G mu all right so the friction force is going to be equal to M G mu and we have to multiply times the distance the length of that rough patch let's plug in the numbers and see what we get so the mass of the 5 g is 9.8 u is 0.08 and the distance is 0.25 so how much energy do we lose or does the block lose every time it cross the path so it's 49 times point zero eight times 0.25 which is 0.98 joules so let's go ahead and plug this in I may not play them it circles around it so actually we can already answer the second question how many times will the blog go across the rough patch well each time the blog goes across it loses almost 1 joule and the block starts off with a little bit over 2 joules so we lose almost 1 joule going this way almost another jewel that's almost 2 joules going this way and it will not be enough energy to go across it a third time because then the initial energy would require at least or almost 3 joules so the answer for number 2 is 2 times completely but then it will stop on the third pass through from left to right so go from left to right right to left and back left to right but it will not make it all the way across the third time so that the third question is how far will it go before it comes to a complete stop all right but now let's answer the first question what is the compression of the spring on the other side and for that we can use our energy equation we could say that work put into the system plus the original potential energy plus the initial initial kinetic energy equals potential energy final plus kinetic energy final plus the energy lost to overcoming friction so in this case we have zero work put in the initial potential energy was 2.25 joules plus the kinetic energy initial is zero because it starts at rest potential energy final is going to be one-half K x2 squared plus zero kinetic energy because when spring 2 is completely compressors no motion and energy loss will be 0.98 joules but other words when I subtract point 9a from both sides I get one point two seven joules is equal to one-half KX 2 squared which means if we come over here and finish that up we can say that X 2 squared is equal to two times one point two seven divided by K or X sub two is equal to the square root of two times one point two seven divided by 200 which means the amount of compression of the second spring one point two seven times two divided by two hundred and take the square root of that would be zero point zero zero point one one three meter so x2 is equal to zero point one one three meters so notice it start out with compressing X 1.15 meters then it goes over there it loses some of the energy then the compression will be point one one three meters it will come back across a patch then back on to the patch so here we could say our first answer is X up two equals zero point 1 1 3 meters and finally 3 where does it come to a complete stop so notice how much energy is left after it's gone through the second time and now it's trying to get through the patch the third time so on the third pass we start with the initial energy energy initial which is 2.25 joules – 0.98 for the first pass and – 0.98 joules for the second pass so that means that the energy left is this minus that that would be 0.29 joules that's how much energy is left and then how much are that then how far does that get us onto that rough patch so we could say that energy initial equals energy lost and of course energy initial will then be equal to the friction force times distance or distance equals energy initial divided by friction force and energy initial is zero point two nine jewels and the friction force is where do we find the friction force we calculated it it will be mg mu so that would be mg mu it so that makes the distance is equal to zero point two nine divided by the mass which is five kilograms times 9.8 all right let's see here time time point eight times mu which is zero point zero eight and therefore how far will go across the rough patch point two nine divided by 5 divided by nine point eight divided by point zero eight equals and there it is it is zero point zero seven four meters or seven point four centimeters so that's how far the block will go on the third pass not make it all the way across and the block will stop at that point and that is how it's done

Physics: Fluid Dynamics: Bernoulli's & Flow in Pipes (16 of 38) Pumps in Bernoulli's Equation

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In this video I will explain how power from a pump (pumps deliver energy to a system) is introduced into Bernoulli’s equation.

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welcome to electron line in the previous video we calculated the amount of power required to push fluid through a pipe but how do we introduce that into bundlers equation so here we have the equation and notice we added an extra term on the left side of the equation we called it P sub P and this is a term that it's representative of pressure now we know that pumps add energy to a system and that is why we place it on the left side of the equation because we're adding energy to the system but why is it in terms of power or in this case in terms of pressure I should say instead of in terms of power that's because Bernoulli's equation every term is representative in the units of pressure and pressures Pascal's that's Newton's per square meter so how do we relate that how to relate the power input of the of the pump to a term that's expressed in terms of Pascal's so what we're trying to do is we're trying to relate the power delivered by the pump to the pressure term piece of P so it's the pressure delivered by the pump now notice that we have an equation here with the power delivered by the pump is equal to the efficiency of the pump times the power produced and so maybe we should put some parentheses around it like that so it's the efficiency times the power produced and so yes he is in the efficiency so if the pump gives you five horsepower but it's only eighty percent efficient that means that you only get force four horse power delivered to the system that's our cat again alright continuing so how do we do that well first of all we start with the definition of power now notice we have P for pressure and P for power so to give you the difference between the two I put a little notation here that this is P for power and by definition it's work divided by time and work by definition it's four times distance so force times distance divided by time we can look at distance divided by time which is really philosophy so the power input to the system by the pump is going to be the force applied times the velocity of the fluid inside the pipe and then since pressure can be defined as force divided by area force can also be expressed as pressure times area so instead of force we're going to replace force by pressure times area so now we have the power of the pump is equal to the pressure and we'll call it the pressure the pump times area times velocity so the force is now replaced by pressure times area and of course we're talking about the pressure of the pump now we also know that the product of cross sectional area times velocity can also be thought of as the amount of volume of fluid going through the system per unit time so eight times V can be expressed as the amount of volume per unit time so the amount of fluid flow through the pipe so we can replace those two now we have the power produced by the or delivered to the system by the by the pump is equal to the pressure to provided by the pump times the amount of fluid flowing through the pipe per unit time now if we solve this equation for the pressure the pump it is now equal to the power delivered which is the power produced times the efficiency divided by the amount of fluid flow or if you like it is said writing at Delta V delta T we can write as the cross section area times velocity either way this is how that term that we add on the left side equation to indicate the pressure provided by the pump can be expressed in terms of this or expressed like that and so now we know where the term came from why it's there and how to calculate it that is how it's done

Physics: Fluid Dynamics: Bernoulli's & Flow in Pipes (6 of 38) The Moody Diagram

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In this video I will explain the Moody Diagram, which is used to find the friction factor=f=? in the frictional head loss equation when we have turbulent flow which is dependent on the Reynold’s number and the relative roughness of the pipe.

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welcome to electoral line let's go back and review what we've done so far we've figured out that there's something called the frictional head loss in fluid flow in a pipe when there's internal friction because of the pipe the construction of the pipe and because what happens inside the molecular forces are the fluid flowing through the pipe to calculate the frictional head loss we found this equation right here which included what we call the friction factor and when the flow is laminar we could calculate the friction factor by taking the number 64 and dividing it by the Reynolds number note that we can find the Reynolds number by taking the density of fluid and there should be a small V for the velocity of the fluid times the diameter of the pipe divided by the internal viscosity of the dynamic viscosity of the fluid now this is good if we have laminar flow but what if we have turbulent flow well for that we need what we call the Moody diagram the purpose of the Moody diagram is to find the friction factor that very same constant right here which is unitless for turbulent flow instead of laminar flow and it's a lot more complicated for turbulent flow because it depends not only on the reynolds number but it also depends on something we call the relative pipe roughness and we can calculate the relative roughness of the pipe by taking epsilon divided by the diameter the pipe will explain in next video what that really means all we need to do in this video is try and understand how to obtain the friction factor if the fluid flow is turbulent instead of laminar so on this diagram on the left side we have the friction factor on the right side we have the relative pipe roughness on the bottom we have the relative the Reynolds number notice that we start by calculating the Reynolds number so that would be no different but let's say that the Reynolds number is greater than 2000 below 2000 we have laminar flow above 2000 we first have what we call the transition region and then we have turbulent flow Motors that not until the Reynolds number gets well above the 2000 range we have what we call complete turbulence between between that between 2000 and some value we have what we call transition and it depends upon the relative pipe roughness if we have greater roughness the turbulence happens quicker if you have less roughness on the inside of the pipe to complete turbulence happens at a much greater Reynolds number so it's not as simple and straightforward as we thought before it does depend on several things but let's say for example we calculate the wrote the Reynolds number and we found it to be 10 to the fifth or 100,000 but then you realize a hundred thousand that has to be turbulent well it again depends upon the relative roughness of the pipe but under normal circumstances the relative reference of the pipe is high enough so let's say the relative roughness is point zero three so we come up here until we meet the line that represents point zero three then we come over here and that represents a friction factor of about point O five two point O six now these numbers since I have them hand-drawn are not exact so don't take them as exact numbers but they're approximate numbers but it gives you the picture of how it's done using a real Moody diagram notice that if the relative roughness of the pipe is greater then yes we'd expect a greater friction factor if the relative roughness of the pipe is not as big then we accept then we expect a smaller friction factor and of course a smaller friction factor gives you a smaller head loss a greater friction factor gives you a greater head loss and it all has to do with the reynolds number and the relative pipe roughness and on the Moody diagram were then able to find the corresponding friction factor necessary to calculate the head loss and that's why we need a Moody diagram but how that it'd be pretty tough

Lecture 1 | The Fourier Transforms and its Applications

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Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood provides an overview of the course, then begins lecturing on Fourier series.

The Fourier transform is a tool for solving physical problems. In this course the emphasis is on relating the theoretical principles to solving practical engineering and science problems.

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this presentation is delivered by the Stanford center for professional development we are on the air okay welcome one at all and as I said on the TV when you were walking in but just to make sure everybody knows this is e 261 the Fourier transform and it's applications Fourier transforms at all Fourier and my name is Brad Osgood circulating around are two documents that give you information about the class there is a general description of the class course information how we're going to proceed some basic bookkeeping items I'll tell you a bit more about that in just a second and also a syllabus in a schedule and I'll also say a bit more about that in just a second let me introduce our partners in crime in this course we have three courses fins thomas john thomas one stand up where's Thomas there we go Rajiv Agarwal I smell that right very good Reggie going to stand up is Rajiv and Michael medias okay so far am gonna correct that okay that's like a Metis everybody thank you all right so now and we will be setting up times for the review sessions and so on all right you know so that'll be that will be forthcoming we have a web page for the course some of you may have already visited that but let me give you the and it's the addresses on the one the sheet one of the sheets that's being passed around but let me write that up now so you can be sure to visit it and register for the class because it is on the web page that you will find course handouts course information I will email people via the web page alright so you have to be registered I five to send an announcement to the class post an announcement and send out an email then that'll be done through the web page and you have to be registered on the Paige in order to get those emails I won't be doing it through access all right so it is at like many of the other classes HTTP slash slash however you do those where the colons go or is it here EE class Sanford et you you can find it very easily edu slash e 261 okay go there if you have not already and and register yourself for the class all right now let me say a little bit about the information that you have I want to say a little bit more about the mechanics I'll talk more about the content in just a second the let me say a bit about the syllabus and schedule and the course reader the syllabus is as I said on the on the top an outline of what we're going to be doing I hope a fairly accurate outline of what we're going to be doing but it's not a contract alright so there will be a natural ebb and flow of the course as things go along and when we get to particular material or what we cover in what order this is more or less I say accurate but it is not written in stone what you should use it for however is to plan your reading so I things will be much better for all of us if you read along with with the material as the syllabus has the schedule basically outlines all right because there's there times what I'm going to want to want to skip around a little bit there are times and I'm going to derive things there are times when I'm not going to derive things and you'll get much more out of the lectures our time together if you've read the material thoroughly before you come to class so that's one thing I ask you to do we have two exams scheduled we have a midterm exam and a final exam I'm going to schedule the midterm exam midterm exam is is already actually on here at least tentatively sort of toward the end of October we'll have it outside of class that is it'll be a sit down regular exam but I want to do it for 90 minutes rather than 15 min 15 minutes is just too short at a time for a class like for material like this so it'll be a 90 minute exam and we'll schedule it several sessions outside of class this is the way I've usually done it hasn't been any problem it's worked out alright for everybody so we'll have you know alternate times and so on and the final exam is scheduled by the char's office do not come to me right before the final exam saying oh I scheduled a trip out of town I hope that's not a problem right you know what the dates are ahead of time we'll also have regular problem sets none of these things that I'm saying should be new to you you've been through the drill many times the first problem the problem sets are going to be I had a starling innovation last time I taught the course where I handed out the problem sets on Monday and had them do the following Wednesday so you actually had you know like a week and a half to do the problem set so there was overlap between the two and people thought that was just a brilliant idea so we're going to do that again this year except for the first problem set and I decided it was not such good policy to hand at the very first problem set on the very first day of class so I'll hand that out on Wednesday and I'll post that also or at least I'll post it on that sure I'll hand it out it'll be available on Wednesday and it'll be due the following Wednesday and again these sorts of things are pretty routine for you I'm sure even through many times it will be practice although again not necessarily every time without fail to have MATLAB problems on the homework one or two MATLAB problems on the homework so I'm going into the assumption that people have some experience with using MATLAB you don't have to be terribly advanced and also access to using MATLAB so if you do not have experience using MATLAB and you do not have access to MATLAB get some experience and get some access won't be hard okay now let me say us a little bit about some of the things this is the course reader for the course it's available at the bookstore and also available on the course website all right doesn't have the problems in it but it has the material that were going to be covering in class now this is a basically a stitched together set of lecture notes that I've been using for a number of years in the class and I sort of tinker with it every time I teach the class but because it is a stitched together set of lecture notes they're the organization is sometimes a little bit odd like you have an appendix in the middle of the chapter and what that means is it was used to be an appendix to a set to a particular lecture that went on that particular day and it never got moved to anywhere else all right so the organization can be a little bit funny you can help on this all right that is if you find typos if you find errors if you find things that are less than clear in their in their wording if you want if you if you have some other ideas for you apples or other explanations please tell me I am working on this I have to say that the I'm hope because these these were written as a set of lecture notes these are meant to be a good and I hope helpful companion for the class that is they're meant to be read and they're meant to be used so you can help as generations of students in the past have helped try to refine these and turn them into something that's really a good accompaniment to to the class as we go on okay one other thing that's special this quarter is the class is as always taped and the lecture notes the lectures are going to be available to everybody but this time for the first time the lecture is going to be available to the world all right Stanford is decide on an experimental basis we're sort of competing with MIT here I think to try to make some classes some of the materials for some classes available to the world all right so the Elektra notes are going to be everything's going to be done through the website but instead of needing a Stanford ID to view the tape lectures I think anybody in the world can view these lectures was a little bit daunting I have to watch my language to try to dress well all right so we'll see what we'll see what goes with that I will however issue a warning I will not answer the world's email all right I will answer email from the class but I will not answer and I think I speak for the TAS here the TAS neither will answer the world's email on this all right how we're going to keep the world out of our inboxes I'm not sure exactly whether this is going to be a problem or not but at any rate that's what's happening okay all right any questions about that any questions about the mechanics of the course or what your expectations should be what my expectations of you are okay all right now I always like to take an informal poll actually when we start this class that's what it's a number of times now and it's always been a mixed crowd and I think that's one of the things that's attractive about this class so let me ask who are the E's in this class who are an electrical engineering your undergraduate or graduate all right so that's a pretty strong show of hands but let me also ask who are the non E's in this class all right that's also a pretty strong show of hands the EES are as is typical the majority of the students in the class but there's also a pretty strong group of students in this class who are not elect engineers by training by desire by anything all right and they usually come from all over the place I was looking at the web I was looking at the classes before I got the class and I think there's some people from chemistry somebody from chemistry anybody from chemists I thought there were somebody up see back there all right and other some people from Earth Sciences uh somebody from somebody is talking actually from Earth Sciences this morning somebody from Earth Science okay where else I think there was an Emmy couple of Emmys maybe yeah all right now that's important to know I think the course is very rich in material all right rich in applications rich in content and it appeals to many people for many different reasons okay for the ease and who are taking the class you have probably seen a certain amount of this material I don't want to say most of the material but you probably seen a fair amount of this material scattered over many different classes but it's been my experience that one of the advantages of this class for electrical engineering students either undergraduate or graduate students is to see it all in one piece all right to put it all in your head at one time at least once all right because the subject does have a great amount of coherence it really does hang together beautifully for all the different and varied applications there are core ideas and core methods of the class that it is very helpful to see all at once alright so if you have seen the material before that's fine I mean that is I mean that you can you can draw on that and draw on your experience but don't deny yourself the pleasure of trying to synthesize the ideas as we go along I mean there's nothing so pleasurable as thinking about something you already know trying to think about it from a new perspective try think about it from a new point of view trying to try to fold it into some of the newer things you'll be learning so I have I've heard this from electrical engineering students many times in the past that it's a it's a pleasure for them to see the material all together at once it may seem like a fair amount of review and in some cases it will be but not in all cases and even if it is a review they're often slightly different twists or slightly new takes on things that you may not have seen before I may not have thought of quite in quite that way so so so that is my advice to the electoral engineering students for the students who have not seen this material before they're coming out of from a field and maybe only heard you know secret tales of the Fourier transform and its uses well I hope you enjoy the ride because it's going to be a hell of a ride a heck of a ride as we go along alright now for everyone I sort of feel like I have to issue I don't know if I call this a warning or just sort of a statement a principal or whatever this is a very mathematical class this is one of the sort of Holy Trinity of classes in the Information Systems lab in electrical engineering the electril engineering is a very broad department and split up into a number of laboratories along research lines I am in the Information Systems lab which is sort of the mathematical part of the subject there's a lot of signal processing coding Theory imaging and so on and this course has been for a number of years taught by faculty sort of thought of as a cornerstone in the signal processing although it has a lot of different applications to a lot of different areas the other courses in that Holy Trinity are 263 dynamically near dynamical systems and 270 a statistical signal processing who's taken to say whose likes let me ask you so because this is also very common who's taken to 63 in the class also a strong majority and who's taken to 78 yeah ok so there's a fair a little little bit less but still number of people we will actually see not so much with 270 oh well actually with both classes with 263 in 278 you'll actually see some overlap that I also hope you find interesting the language will be slightly different the perspective will be slightly different but you see this material in this class melding over into the other classes and vice-versa and again I think it's something that you can really draw on and I hope you enjoy all right so it is those those classes and the perspective that we take the faculty your teaching those classes is a pretty mathematical one but it's not a class in theorems and proofs you can breathe a heavy sigh of relief now all right I can do that but I won't all right I will derive things I'll derive a lot of a number of formulas I'll derive it and I'll go through those derivations or I'll hope that you go through the derivations in the book when I hope and I think that they will be helpful all right and when in some case that is there's an important technique or there's an important idea that you'll see not only in the tick Euler instance but over all that you'll see the same sort of derivation the same sort of ideas be applied not only for one formula but for other sorts of formulas and also in some cases to my mind as twisted as that may be I sometimes think of the derivation of a formula almost as identical with a formula I mean to use the formula effectively almost as to know the derivation because it's to know where it applies and to know how it applies and where to expect to use it all right so that's why I will go through those things for the purpose of teaching a certain amount of technique and for the purposes of sort of having those techniques really at your fingertips so that you can apply them again in a situation that may not be quite identical with with what we did but will be similar enough so that the simp so that the ideas may apply in this situation that's that's very important we will also do plenty of different sorts of applications but again because the field the subject is so varied and because the clientele because the students in the class are also varied will try to take applications from different areas will have applications from electrical engineering but will also have applications from physics and from other areas i i've also done in the past and will see if i get to this some applications from Earth Sciences for example and we'll just see how they go so we all have to cut each other a little bit of slack and if an application or particular area is not exactly to your liking well chances are it might be to somebody's liking to your right or left so you say cut everybody should cut each other a little slack and just enjoy the ride I should also say that many of the more specialized applications are found in more specialized courses all right so we will touch on a lot of things and I will use the words that are used in a lot of different courses and a lot of different subjects but we won't always do see an application to its bitter end so to speak or we won't do every pot we certainly won't do every possible application because there are just so many of them so you will find you will not run out of ways of using the Fourier transform and Fourier analysis techniques in any classes here they go it goes on and on and on but we'll only be able to see a certain amount of a certain amount of that all right and actually that leads to a very important point release of the start of the class that is where do we start all right that is this subject which is so rich and so diverse forces you forces me forces all of us to make hard choices in some ways about where what we're going to cover where we're going to start what direction we're going to go and all the different choices are defensible you will find books out there that take very different taps toward the subject they take different starting points they have different emphases they go off in different directions and you can make a good argument for any one of those choices but you have to make a choice so for us we are going to choose I have chosen not we me I have chosen to start the class with a brief discussion of Fourier series and go from there to the Fourier transform all right whereas it is also very common choice to forget about Fourier series and maybe pick them up a little bit along Angier or pick them pick them up a little bit on the edges or assuming that everybody seen Fourier series then go right into the fray transform I don't want to do that because I think that the subject of Fourier series is interesting enough in it we're not going to do very much with it but it's interesting enough in itself again it's something you may have seen in different context but it provides a natural transition to the study of the Fourier transform and it is historically actually the way the subject developed okay so that's how we're going to that's how we're going to do things will start with Fourier series and use them as a transition to Fourier transform now first of all what is this concerned with overall I it may be a little bit too strong a statement but for our purposes I want to identify the idea of Fourier series as almost identified with the study of periodic phenomena alright so for us it's identified most strongly with a mathematical analysis of periodic phenomena now it certainly shouldn't be necessary for me to justify periodic phenomena as an important class of phenomena you have been studying these things for your entire life pretty much ever since the first physics course you ever took where they do the harmonic oscillator and then the second physics course you took where they did the harmonic oscillator and then the third physics course you took rhythm they did the harmonic oscillator you have been studying periodic phenomena alright so that shouldn't be a controversial choice Fourier series goes much beyond that but it is first and foremost for us associated with a study of periodic phenomena the Fourier transform in although again it doesn't maybe doesn't do it's just justice completely is can be viewed as a limiting case of Fourier series it has to do with a study of the mathematical analysis on phenomena so if you want to contrast Fourier series and Fourier transforms then that's not a bad rough-and-ready way of doing it doesn't it say it doesn't capture everything but it captures something so Fourier transform as a limiting case and in a meeting that I'll make more precise later is limiting case of Fourier series Fourier series of free series techniques is identified with or has to do with is concerned with how about that for weaseling way out of it is concerned with the analysis of non periodic phenomena so again it doesn't say everything but it says something and one of the things that I hope you get out of this course especially for those of you who have had some of this material before are these sort of broad categorizations that help you sort organize your knowledge all right it's a very rich subject you've got to organize it somehow otherwise you'll get lost in the details all right you want to have certain markers along the way that tell you how to think about it how to organize it what what what a particular formula what cat it what general category it fits under okay now it's interesting is that the ideas are sometimes similar and sometimes quite different and sometimes it's the situation is simpler for periodic phenomena sometimes the situation is more complicated for periodic phenomena so it's not as though there's sort of a one-to-one correspondence of ideas but that's one of the things that we'll see and one of the reasons why I'm starting with Fourier series is to see how the ideas carry over from one to the other see where they work and see where they don't work alright some ideas carry easily back and forth between the two some phenomena some ideas some techniques some don't and it's interesting to know when they do and when they don't sometimes the things are similar and sometimes they're not now in both cases there are really to kind of inverse problems there's a question of analysis and there's the question of synthesis two words that you've used before but it's worthwhile reminding what they mean in this context the analysis part of Fourier analysis is has to do with breaking a signal or a function I'll use the term signal and function pretty much interchangeably alright I'm a mathematician by training so I tend to think in terms of functions but electrical engineers tend to think in terms of signals and they mean the same thing all right so analysis has to do with taking a signal or a function and breaking it up into its constituent parts and you hope the constituent parts are simpler somehow then the complicated signal that as it comes to you so you want to break up a signal into simpler constituent parts I mean if you don't talk in just in terms of signals here or you don't use exactly that language that's the meaning of the word analysis I think close enough whereas synthesis has to do with reassembling a signal or reassembling a function from its constituent parts a signal from its constituent parts kind of stitch one alright and the two things go together all right you don't want one without the other you don't want to you don't want to break something up into its constituent parts and then just let it sit there all these little parts sitting on the table with nothing to do you want to be able to take those parts maybe modify those parts maybe see which parts are more important than other parts and then you want to put them back together to get that to get either the original signal or a new signal and the process of doing those things are the two aspects of Fourier analysis I use I use the word analysis they're sort of in a more generic sense now the other thing to realize about both of these procedures analysis and synthesis is that they are accomplished by linear operations series and integrals are always involved here both analysis and synthesis free analysis analysis and synthesis are accomplished by linear operations this is one of the reasons why the subject is so I don't know powerful because there is such a body of knowledge on and such a deep and advanced understanding of linear operations linearity will make this a little bit more explicit as I go as we go on further but I wanted to point it out now because I won't always point it out all right because when I say linear operations when I'm thinking of here integrals in series all right eg ie integrals and series both of which are linear operations the integral of a sum is the sum of the integrals the integral of a of a constant times a function is a constant interval the function and so and similarly with sums alright because of this one often says or one often thinks that Fourier analysis is part of the study of linear systems alright in engineering there's there's a there's their courses called linear systems and so on and sometimes Fourier analysis is thought to be a part of that because the operations involved in it are linear I don't think of it that way I mean I think it's somehow important enough on its own not to think of it necessarily as subsumed in a larger subject but nevertheless the fact that the operations are linear does put it in a certain context in some in some ways in some cases more general context that turns out to be important for many ideas alright so often so you see you often hear that Fourier analysis Fourier analysis is a part of the subject of linear systems the study of linear systems so I don't think that really does complete justice to Fourier analysis because of because of the particular special things that are involved in it but nevertheless you will you'll hear that okay now let's get launched alright let's start with with the actual subject of Fourier series and the analysis of periodic phenomenon a periodic phenomena and Fourier series as I said it certainly shouldn't be necessary for me to sell the importance of periodic phenomena as something worth studying you see it everywhere all right the study of periodic phenomena is for us the mathematics and engineering or mathematics and science and engineering of regularly repeating phenomena that's what's always involved there's some pattern that repeats and it repeats regularly right so it's the mathematics and engineering so this is an engineering course I'll put that before science or maybe I won't even mention science mathematics and engineering of regularly repeating patterns I'm relieving a couple of terms here I'm leave all these terms somewhat vague what does it mean to be regular what does it mean to repeating what is a pattern in the first place but you know what you know what I mean you know it when you see it and the fact you can mathematically analyze it is what makes the subject so useful now I think although again it's not ironclad trouble is this subject is so rich that every time I make a statement I feel like I have to qualify it well it's often true but it's not completely true and sometimes it's not really true at all but most of the time it's true that it's helpful but not always helpful but most of the time helpful occasionally helpful to classify periodicity as either periodicity in time or periodicity in space all right you often see periodic phenomena as one type or the other type although they can overlap so you often periodic phenomena often are either periodicity in time a pattern repeats in time over and over again you wait long enough and happens again so for example harmonic motion so eg harmonic motion a pendulum I think bobbing on a string G harmonic motion or periodicity in base periodicity in space the city in space alright now what I mean he is there is often a physical quantity that you are measuring that is living on some object in space one dimension two dimensions whatever that has a certain amount of symmetry alright and the periodicity of the phone on is a consequence of the symmetry of the object so it's often the cow giving example just a second so here you have say some some physical quantity physical not always but often you know physical quantity distributed over a region with symmetry the region itself repeats all right the region itself as a repeating pattern all right so the periodicity of the phenomenon the periodicity of the physical quantity that you're measuring is a consequence of the fact that it's distributed on on over some region that itself has some symmetry so the periodicity arises from the symmetry for periodicity here of the object of the of the physical quantity that you're measuring arises because the periodicity of the are the symmetry of the object where tributed where it lives I'll give you an example there from the symmetry matter of fact I'll give you the example the example that really started the subject and we'll study this is the distribution of heat on a circular ring so eg the distribution of heat on a circular ring alright so the object the the physical quantity that you're interested in is the temperature but it's a temperature associated with a certain region and the region is a ring all right the ring has circular symmetry it's around okay so you're measuring the temperature at points on the ring and that's periodic because if you go once around you're at the same place so the temperature is periodic as a function of the spatial variable that describes where you are on the ring time is not involved here position is involved all right it's periodic in space not periodic in time periodic in a spatial variable that gives you the position and the periodicity arises because the object itself is symmetric because the object repeats that's why this sort of example is why one often sees and this actually turns out to be very far-reaching and quite deep that free analysis is often associated with questions of symmetry in a sort of most mathematical form you often find for a series developed in and in this context and Fourier transform is developed in the context of symmetry so you often see so you see Fourier analysis let me just say free analysis analysis is often associated with problems or just not off with with analysis of questions that have to do with that have some sort of relying symmetry so let me say often associated with problems with symmetry just leave it very general this is the very first of all that for the problem of distribution of heat on a ring we're going to solve that problem that was the problem that Fourier himself considered alright they introduced some of the methods into the into the whole subject let's launch everything all right so again it's not periodicity in time its periodicity in space and for those of you who have had or may have courses in this that the mathematical framework for this very general way of looking for a analysis is group theory because the theory of groups in mathematics is a way of mathematize the ADEA of symmetry and then one extends the ideas for elseís into to take into account of groups that is to say to take into account the symmetry of certain problems that you're saying and it really stays very quite it's quite far-reaching we're not going to do it we'll actually have a few occasions to to go to go into this but but with a light touch all right I'm just telling you I'm just giving you some indication of where the subject goes all right now what are the mathematical descriptors of periodicity well nothing I've said so far I'm sure it is new to you at all you just have to trust me that at some point before you know it some things I say to you will be new I hope but one of the mathematical descriptions of periodicity again that in the two different categories say the numbers the quantities that you associate with either either a phenomena that's periodic and timer function or a phenomenon that's periodic in space for periodic and time for periodicity in time you often use the frequency all right frequency is the word that you hear most often associated with a phenomena that is periodic in time you use frequency the number of repetitions the number of cycles in a second say if a pattern is repeating whatever the pattern is again if I leave that term sort of undefined or sort of vague it's the number of repetitions of the pattern in one second or over time all right that's the most common descriptor mathematical descriptor of a phenomenon is periodic that's periodic in time for a function for a phenomenon is periodic in space you actually use the period that's the only word that's really in use in general for the particulate well one thing a time so for periodicity in space you use the period all right that is sort of the physical measurement of how long the long the pattern is before repeats somehow all right the measurement of how whether its length or some other quantity measurement of how let me just say how big the pattern is that repeats they're not the same all right they have a different feel they rise off from from different sorts of problems that's probably too strong a statement but I think I think it's fair to say that mathematicians tend to think in terms of mostly in periodic they tend to think in terms of the period of a function or the period is the description of periodic behavior whereas engineers and scientists tend to think of systems evolving in time so they tend to think in terms of frequency they tend to think of how often a pattern repeats over a certain period of time all right that's like everything else is that statement has to be qualified but I get tired of qualifying every statement so I'll just leave it at that now of course the two phenomena are not completely separate or not always completely separate they come together periodicity and time and periodicity in space come together in for example wave motion all right that is traveling disturbance a travelling periodic disturbance so the two notions of periodicity come together two notions here periodicity and time periodicity in space come together in EEG wave motion understood very generally here as a periodic as a regularly repeating pattern that changes in time that moves because more jumps up a little bit I think of their skipping so a regular a moving a subset regularly moving disturbance you know a group of freshmen through the quad you know just they're everywhere mostly regular mostly moving all right now there again the two descriptors come in the frequency and the wavelength so again you have frequency and wavelength you have frequency nu and wavelength usually associated usually denoted by this is for periodicity in space and for periodicity and time frequency nu for periodicity in time that's the number of times and repeats in one second this is cycles per second the number of times that the pattern repeats in one second so for example you fix yourself at a fix your position in spate both time and space are involved so you fix yourself at a point in space and the phenomenon washes over you like a water wave all right and you count the number of times you're hit by the wave in a second and that's the frequency that's the number of times that the phenomenon comes to you for periodic for periodicity and time the function the phenomenon comes to you for periodicity in space you come to the phenomenon so to speak all right so I fixed myself at a point in time the wave washes over me at a certain characteristic frequency over and over again regularly repeating it comes to me new times per second the wavelength you fix the time and allow the platen and see what the phenomena looks like to distribute it over space so for periodicity in space fix the time and see how the phenomena is distribute to see the pattern distributed over space distributed my writing is getting worse distributed then the length of one of those a complete to speak is the period or the wavelength length is a term that's associated with the periodicity in space for a traveling traveling phenomena for a wavelet wave for wave motion so the length the length of the disturbance I say one complete disturbance if I can say that one complete pattern is the wavelength now like I say ever since you were a kid you've studied these things and especially don't know the number by lambda but I bring it up here because of the one important relationship between frequency and wavelength which we are going to see in a myriad of forms throughout the quarter that is there's a relate in the case of wave motion there is a relationship between the frequency in the wave length determined by the velocity and there could be two different phenomena all right periodicity in time and periodicity in space may not have anything to do with each other but if you have a wave traveling if you have a regularly repeating pattern over time then they do have something with to do with each other and they're governed by the formula distance equals rate times time which is the only formula that governs motion all right so there's a relationship between frequency and wavelength that is distance equals rate times time I love writing this in a graduate course because it's the up the equation in calculus actually in all of calculus I think this is pretty much the only equation used in very clever ways but the only equation and in our case if the rate is the velocity of the wave then this translate V is the velocity the rate of the wave of the motion and the equation becomes as I'm sure you know many times lambda that's the distance that this this the the wave travels in one cycle it traveling it's traveling at a speed V if it goes nu cycles in one second then it goes one cycle in 1 over nu seconds let me say that it going to make sure I got that right if it goes nu cycles in one second if it just passed you nu times in one second then in 1 over nu seconds it rushes past you once rushing past you once means you've gone through one wavelength so distance equals rate times time the time it takes to go one wave length is 1 over nu seconds so I have lambda equals V times 1 over nu or lambda nu equals V again a formula européenne many times now why did I say this if you've seen it many times because I never have the confidence that I can talk my way through that formula for one thing so I always have to do it secondly it exhibits a reciprocal relationship to quantities all right there's a reciprocal relationship you can see it more clearly over here where the constant of proportionality or inverse proportionality is the velocity all right lambda is proportional to the reciprocal of the frequency or the restore the frequency is proportional to the reciprocal of the wavelength at any rate or the or expressed this way lambda times nu is equal to V so there's a reciprocal relationship between the frequency and the wavelength all right this is the first instance when you talk about periodicity of such reciprocal relationships we are going to see this everywhere all right it's one of the characteristics of the subject hard to state as a general principle but but they're plain to see that in the prop in in in the analysis and the synthesis of signals using methods from Fourier series or Fourier analysis there will be a reciprocal relationship between the two between the quantities involved all right I'm sorry for being so general and but you'll see this play out in case after case after case and it is something you should be attuned to all right all right so you may never have thought about this in these types of simple enough formula you've used millions of times all right you may not have thought about it somehow in those terms but I'm asking you to think about stuff use you once saw in very simple context and how those simple ideas sort of cast shadow into much more involved situations all right the reciprocal relationship between as well as we'll learn to call it the reciprocal relationship between the two domains of Fourier analysis the time domain in the frequency domain or the tie or the store the time domain and the spatial domain or the spatial domain in the frequency domain and so on is something that we will see constantly alright and I will point that out but if I don't point it out you should point out to yourself you should be attuned to it because you will see it and it's one of those things that helps you organize your understanding of the material because sometimes when you're called upon to apply these ideas in some context that you haven't quite seen you have to ask yourself it's at least the good starting place is to ask yourself questions like well should I expect a reciprocal relationship here you might lead you to guess what the formulas should be or guess what the relationship should be so you say well somehow I want to use for a analysis to do this problem so I'm sure I should be looking for some sort of reciprocal relationship the quantities that I'm interested in somehow should be related in some kind of reciprocal way and what that might mean might be more or less involved depending on the particular kind of problem but you'll see it trust me you'll see it okay right now we're almost done for today why does mathematics come into this in the first place I mean periodicity is evidently sort of a very physical type property why is it allow any kind of mathematical description well it does because there are very simple maybe not so simple mathematical functions that exhibit periodic behavior and so can be used to model periodic phenomena so math comes in because there are simple mathematical functions that model that are periodic that repeat and so can be used to model periodic phenomena I am speaking of course of our friends the sine and cosine now you may think again we've only talked about elementary things in very elementary contexts but you know I have a PhD in this subject and I get excited talking about sines and cosines I mean you know and it's not just creeping old age I mean I think there you know there's a lot there's a lot to reflect on here and sometimes the miraculous nature of these things cosine of I'll use I'll use T is the variable cosine of T and sine of T our periodic of period two pi that is cosine of T plus two pi is equal to cosine of T for all values of T and sine of two pi + t + 2 pi is equal to sine of T why dead silence because the sine and cosine are item don't tell me I want to do it because this I'll do it over here because the sine and the cosine are associated with periodicity in space because the sine of the cosine are associated with an object that regular repeats the simplest object that the regularly repeats does circle you didn't meet sine and cosine that way first you met sine and cosine in terms of ratios of psiy lengths of sides in triangles that's fine but that's an incomplete definition the real way of understorey way but the but them but the more sophisticated way the ultimately more far-reaching way of understanding sine and cosine is as associated with the unit circle where the cosine of t is the x-coordinate and the sine of t is the y-coordinate and T is Radian measure I'm not going to go through this in too much detail but the point is that the sine of the cosine are each associated with the phenomenon of periodicity in space they are periodic because if you go once around the circle that is to say T goes from T to T plus 2 pi you're back where you started from all right that's why it's periodicity in space all right that's the definition of sine and cosine that exhibits their their periodic phenomena not the definition in terms of right triangles it's not the definition it's not that the definition in terms of right triangles is wrong it just doesn't go far enough it's incomplete all right it doesn't reveal that fundamental link between the trigonometric functions and periodicity and it is fundamental if not for that mathematics could not be brought to bear on the study of periodic phenomena and furthermore this clear and will quit in just a second that is not just 2pi but any multiple of 2pi positive or negative I can go clockwise or I can go counterclockwise I can say the cosine of t plus 2pi n is the same thing as cosine of 2t and the sine of 2pi t plus 2pi n is the sine of T for n any integer n 0 plus or minus 1 plus or minus 2 and so on and so on the interpretation is that when n is positive I'm going count and it is just an interpretation is just a convention when n is positive I'm going counter clockwise around the circle when n is negative I'm going clockwise around the circle but it's only when you make the connection between periodicity and space and the sign of the cosine that you see this fundamental property all right now all right I think we made it out of junior high today that's that was my goal all right what is what is most amazing and what and what was what we'll see you next time is that such simple functions can be used to model the most complex periodic behavior all right the simple from such simple things some simple acorns mighty oaks grow or whatever you excuse me whatever whatever stuff you learn out there that the simple these simple functions that associated with such a simple phenomena can be used to model the most complex really the most complex periodic phenomena and that is the fundamental discovery of Fourier series all right and is the basis of Fourier analysis and we will pick that up next time thank you very much see you then

How To Solve Any Projectile Motion Problem (The Toolbox Method)

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Introducing the “Toolbox” method of solving projectile motion problems! Here we use kinematic equations and modify with initial conditions to generate a “toolbox” of equations with which to solve a classic three-part projectile motion problem.

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hello everyone I'm Jesse Mason and in this installment of teach me we're going to take a look at a classic projectile motion problem involving an angled launch from uneven ground so this problem we're given the initial position and velocity of the projectile and our task is to determine the projectiles range peak height and velocity at some point in time along its trajectory the first step as always is to draw a picture so here's the cliff from which our projectile is launched and here's the level ground below will draw a cannon to fire the projectile and also to mark its initial position and there's our trajectory Dead Gazette and we're done with the picture not much to this one yet so what are our knowns and unknowns to start we're given the height of the cliff which is to say the initial vertical displacement and that's 100 meters of course we've already implied our coordinate system but we need to be explicit and included on our drawing next we're given the magnitude of the launch velocity the initial speed of our projectile 30 meters per second we're also given the direction of the launch velocity so we'll label the angle between the velocity vector and the horizontal as theta and theta is equal to 20 degrees so there are some of our knowns there are a few more that are implied and we'll address them shortly the first unknown were tasked with determining is the projectiles horizontal displacement upon impact it's range we'll label this X and by doing so we've implied the location of the origin so let's indicate that on our picture now it should be clear to see that our projectiles initial position along the x axis is 0 meters we'll label the maximum vertical displacement that is to say our peak height as Y max lastly we're looking for the projectiles velocity 2 seconds after launch we're going to guess at the projectile a cannonball in this case would be somewhere around here at that time accuracy with our guest is unimportant because the equations will tell us the rest of the story so at this point in the trajectory the instantaneous velocity looks something like this we're after the magnitude and the direction so we'll label the angle between the velocity in the horizontal is fee fie fie fie fie foe fum whatever it's all Greek to me and let's not forget to jot down the given time for this part since we'll be looking for each component of the projectiles velocity separately we'll break V into V sub X and V sub y and while we're at it let's resolve our initial velocity into its constituent X&Y components as well so here's our given launch velocity V sub o in the vertical direction we have V sub o sub y and in the horizontal we have V sub o sub X and here's the given launch angle theta since will eventually be writing these components in terms of V sub o and theta will write their trigonometric relationship right here let's see so the Y component of V sub o is opposite theta so it's equal to V sub o sine theta and the X component of V sub O's adjacent to theta so it's equal to V sub o cosine theta by the way we've got an implied known that should be listed the acceleration due to gravity what value should we use for G or more to the point is the magnitude of G positive or negative well it all depends on our coordinate system so we need to compare the acceleration vector to our chosen coordinate system what's the direction of the acceleration vector downward of course since we align the positive Y axis with the upward direction and the acceleration vector is downward that means that the magnitude of G for this problem will be negative 9.8 meters per second squared okay we're finished with identifying the knowns and unknowns now we're ready to move on to the next step selecting the appropriate equations for our problem since we're dealing with projectiles and ignoring the effects of air resistance the kinematic equations namely the displacement equation and the velocity equation are entirely sufficient to analyze the motion of our projectile of course our projectile is moving through two dimensions so with an out of appreciation to Galileo we'll apply these equations to each direction separately so in the x-direction starting with the displacement equation instead of s we have x equals 1/2 a sub X that's the acceleration in the X Direction times T squared plus V sub o sub X the X component of the initial velocity times T x sub o the initial horizontal position before moving on we're going to simplify this equation a projectiles horizontal motion is unaccelerated so we can set a sub X to 0 we've defined V sub o sub X as v-sub-oh cosine theta and X sub o is 0 so the displacement equation in the X Direction simplifies to x equals v-sub-oh cosine theta t we'll label this equation 1 and set it aside for now now we apply the velocity equation to the x-direction and we get V sub X equals V sub o sub X plus a sub x times T we'll set V sub o sub X to v-sub-oh cosine theta and again the horizontal acceleration of a projectile is zero so the velocity equation in the X Direction simplifies to V sub X equals v-sub-oh cosine theta we'll label this equation two and set it aside okay so now we'll apply the kinematic equations to the y-direction the displacement equation becomes y equals one-half a sub y t squared plus V sub o sub y t plus y sub o first a sub y the acceleration in the Y Direction is the gravitational acceleration of a projectile so we can set that to g and we've defined V sub o sub y the Y component of our projectiles initial velocity as V sub o sine theta so the displacement equation in the Y direction simplifies to become y equals one-half GT squared plus V sub o sine theta T plus y sub o label that equation three and we're almost done the velocity equation in the Y Direction is V sub y equals V sub o sub y plus a sub y times T again V sub o sub y was defined as v-sub-oh sine theta and a sub y is just G so our fourth and final equation is V sub y equals v-sub-oh sine theta plus GT these four equations now constitute what we call our toolbox every handyman knows you got to have the right tools for the right the same goes for projectile motion simplifying the displacement and velocity equations for each direction is selecting the right tools and with these tools we can determine everything we want to know about our projectiles motion and trajectory first up determining the projectiles horizontal displacement upon impact ie its range where do we start well equation 1 would give us the horizontal displacement but we don't have T the time of impact to get T we're going to need to use equation 3 along with a key implication at the moment of impact the projectiles vertical displacement is 0 now we can solve equation 3 for time so we'll start by writing equation 3 we'll set Y to 0 and rewrite for clarity sake does this equation ring any algebraic bells perhaps if we recall it's more general form in solution if 0 equals 8 e squared plus BT plus C where a B and C are all constants then T equals negative B plus or minus the square root of b squared minus 4ac all over 2a yep that's how I remember the quadratic formula so our 1/2 G here will play the role of a our v-sub-oh sine theta will be b and our y sub o is C so T equals negative v-sub-oh sine theta plus or minus the square root of quantity v-sub-oh sine theta and quantity squared minus 4 times 1/2 g y sub o all divided by 2 times 1/2 g thank you thank you very much simplifying just a few terms and then inserting our values we find that T equals negative three point five nine seconds or positive five point six eight seconds the negative value is not useful for this problem negative time what does it mean so we'll lovingly discard it now we proceed with equation 1 write it out here and insert the appropriate values 30 meters per second cosine 20 degrees and for time we use five point six eight seconds from Equation three and we get a horizontal displacement of 160 point one meters all things considered not an unreasonable range okay now let's determine the maximum vertical displacement of our projectile that is the peak height of its trajectory we'll use equation three to determine this value but we first need to obtain the time which corresponds to this moment to obtain the time we need to recognize that the moment our projectile crests its trajectory the vertical component of its velocity is zero using this value along with equation four will give us the time at which our projectile reaches its peak height okay solving equation four for time we get T equals negative v-sub-oh sine theta divided by G inserting our values we find that at one point zero five seconds our projectile levels out and begins its descent so to speak now we'll insert this time into equation three to determine the maximum vertical displacement of our projectile and when we assign our values to the variables using one point zero five seconds the time coincident with y max for T we find that our projectile reaches a peak height of one hundred five point four meters not very impressive but very reasonable given our shallow launch angle lastly we'll determine the projectiles velocity two seconds after launch we'll need to determine the x and y components separately so starting with V sub X we'll use equation two V sub X equals v-sub-oh cosine theta using the after mention known values we get twenty eight point one nine m/s for the horizontal component we'll use equation four to determine the vertical component inserting our known values including the time in question two seconds we get a value of negative nine point three four m/s for V sub y what's up with that negative sign that negative sign implies motion in the negative y direction that is downward now at this point we can write our solution in vector notation where the velocity vector is equal to the magnitude of the velocities X component times the unit vector I hat plus the magnitude of the velocities y-component times unit vector J hat that will give us the solution v equals 28 point one nine m/s I hat plus negative nine point three four m/s J hat which is fine and dandy but let's go ahead and determine the magnitude and direction of this velocity vector so we've got 28 point one nine meters per second in the x-direction and nine point three four meters per second in the negative y-direction will recombine these components to determine the magnitude of V using Euclid's xlvii proposition better known as Pythagoras's theorem the square of the hypotenuse is equal to the sum of the squares of the legs taking the square root of both sides and inserting our values and dropping the plus or minus sign we get a magnitude of 29 point seven zero meters per second physically reasonable check are we done not yet we still need the direction of its motion that is to say the angle between the velocity vector and the horizontal to get fiy we use trigonometry so the tangent of V equals opposite nine point three four meters per second over adjacent twenty-eight point one nine meters per second take an inverse tangent and we get a value of eighteen point three degrees for fee so two seconds after launch our projectile is moving at twenty nine point seven zero meters per second at 18 point three degrees below the horizontal I'm Jesse Mason I hope you found this video helpful and that I've convinced you of the utility nay the necessity of creating a toolbox to solve projectile motion problems if you have any suggestions for future teach me videos or just want to say hello from your part of the world please do so in the comment section below and as always happy learning you

Physics Fluid Flow (1 of 7) Bernoulli's Equation

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In this video I will show you how to use Bernoulli’s equation to find the pressure of a fluid in a pipe.

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welcome to AI electron line and today continuing with physics in terms of fluid dynamics we're now going to talk about a topic which is very interesting very practical and sometimes very confusing it's called Bernoulli's equation it has to do with fluid flow and typically fluid flow through a pipe but it can also be applied to other examples and because there's so many different applications I'm going to show you a set of examples I have planned seven examples of how we can look at Bernoulli's equation so you may say what is Bernoulli's equation now we're not going to derive the equation but let me say what Bernoulli came up with he took the pressure in a fluid the velocity no fluid and the height of the fluid relative to some reference point into an equation so that we could relate those three concepts to each other so pressure velocity and height of the fluid all related to one another and he put it together into what he called Bernoulli's equation but starting out we're going to say that the pressure at any point inside the pipe plus Rho gh of the fluid now Rho would be the density of fluid G is of course acceleration due to gravity and height is the height above some reference point plus one-half Rho V squared again rose the density of the fluid and V is the velocity of the fluid through the pipe and again it doesn't have to be through a pipe but for simplicity let's start there and he said that those three terms combined are always constant so which means that if one increases something else has to decrease for example when the pipe gains height so that this term becomes bigger something else has to become smaller now in this particular case since the pipe doesn't change in diameter the velocity at point 1 must equal the velocity at point 2 so we know that that must be the same no matter what and since this will increase the only other option that we have is that the pressure then will decrease over here because of its increased height and that's how Bernoulli was able to manipulate what happens in fluid flow now let's make let's write down the equation so at point one we can say pressure at point one plus Rho gh one plus one half Rho V at one squared must equal since it's constant to these three terms combined at point two so we can say pressure at two plus Rho hoop yeah Rho G H at two plus one half Rho V 2 squared now in this particular case since the diameter the pipe doesn't change and we know that DV DT which is equal to 8 times V must also be constant and therefore we can say that a 1 V 1 must equal a 2 V 2 and if a doesn't change then V cannot change so that shows that the velocity must be the same over here and over here so that means this term and this term are constant they don't change in this particular example of course which means that if the height increases that means that here Rho gh 2 has increased relative to Rho gh 1 that means in order for the equation to balance that the pressure at 2 must be less than the pressure 1 and that's how you have to read that equation that's a good way to take a look at renewals equation when only the height changes and nothing else changes and of course because of that the pressure has to change as well all right now let's do an example let's say that this height is equal to 5 meters above the reference point and this height is equal to 10 meters above the reference point let's say that the fluid is water h2o and let's say that the velocity in the pipe is equal to 2 meters per second and the question that would be given the change in the height given the velocity here and of course knowing that the velocity there also must be 2 meters per second what is the pressure at the second point all right let's go ahead and do that well first of all we have an equation it's balanced that left side equals the right side and since this is constant that's constant it doesn't change we can simply get rid of that part in our equation now solving that for p2 that means I have to take Rho gh 2 and move to the other side and of course I can then flip the equation around which means that the pressure at point two equals the pressure one plus Rho G H one minus when I bring this across that would be Rho gh two simplifying that equation a little bit by factoring out a GNA IG in a row so that becomes p2 is equal to pressure one plus Rho G times h1 minus h2 now notice I did not give you pressure one well I can put something in there just lets say that pressure one p1 is equal to two times atmospheric pressure and of course atmospheric pressure hmm that would be two times 1.01 three times 10 to the fifth Newton's per square meter all right so let's say that pressure one was two atmospheres how much will the pressure have changed by allowing the pipe to go up an additional five meters from where it was before all right so let's plug these numbers in so this is equal to I'll leave this at two atmospheres for now plus the density of water and of course the density of water is a thousand kilograms per square meter so one thousand kilograms not per square but per cubic meter of course because it's per volume g is 9.8 meters per second square then we multiply that times h1 minus h2 now h1 is 5 meters h2 is 10 meters and so very quickly you can see that that would be a negative number which means that the pressure will be less at point two compared to 0.1 now how much less are L let's find out so that would be at 1,000 times 9.8 that would be 9800 times 5 minus 10 which is minus 5 so times 5 equals that's 49,000 so this is equal to 2 atmospheres minus forty-nine thousand Newton's per square meter now there's a unit for that we call Pascal so you could say Pascal's or Newton's per square meter now if one atmosphere is one hundred and one thousand three hundred Newton's per square meter what is 49 thousand new tunes per square meter into of atmospheres so let's do that real quick here if I have 49,000 Newton's per square meter and we convert that to atmospheres so one atmosphere is equal to one hundred and one thousand three hundred Newton's per square meter so we can see if we take down divided by 101 three hundred one too many equals and that would be a zero point four eight atmospheres all right so this can then be written as two atmospheres minus zero point four eight atmospheres so we can say this is equal to one point five two atmospheres the way there we go now quickly looking at again Renu's equation we have three terms the pressure Rho gh and one-half Rho v1 squared notice that all three of course are terms of pressure if the velocity goes up then something else has to come down if height goes up something else that has to come down so in this case the pipe gained height this became bigger since the equation has to stay constant or the left side and the right side stay constant this goes up that has to come down and we just calculate it by how much in our next example I believe has something where the velocity changes and see how that affects the pressure in the pipe

Physics – Test Your Knowledge: Energy (3 of 30) Force and Power

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In this video I will find v(t)=?, P(t)=?, and P(ave)=? of a force=F(t)=24kt acting horizontally on a 2kg box across a horizontal surface.

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welcome to our lecture online our next example deals with force in power and ultimately they want us to find velocity as a function of time the power is a function of time and the average power for the time period from time equals zero to time equals four seconds and let's say that the initial velocity V initial is equal to zero so what is the problem well we have an object with a mass of two kilograms and a force acting on the object and the force is a variable force it's equal to 24 times T so as time increases the force increases and so they're trying to figure out or they want us to figure out the velocity is a function of time the power applied to the object as a function of time in the average power for the four first four seconds all right how do we do this well let's start out with F equals MA that's usually a good place to start F equals MA and realizing that the force is a variable force a can then be written as the change in velocity as a function of time so in other words we could say that F is equal to m times dv/dt because after all we're trying to find the velocity which means that we can write that M DV is equal to F times DT and then if we replace F by what F is equal to and we divide both sides by M we can say that the V is equal to 24 T DT divided by M and then if we integrate both sides we can then say that the integral of DV is equal to 24 over m but M would be equal to 2 because M is 2 kilograms so divide by 2 times the integral of T times DT and then if we integrate both sides we get V is equal to 24 divided by 2 which is 12 T squared divided by 2 plus of integration but remember when t is equal to zero velocity is equal to zero so the constant integration drops off so excited that this is equal to 6t squared so we have our first answer in the velocity as a function of time so V as a function of time is equal to 6t squared all right now we need to find power now power well let's see here we know that power by definition is equal to work over time and work is equal to Force Times distance over time and this is over time its velocity so this would be force times velocity now notice that the force is a function of time and velocity is a function of time and we're trying to find power as a function of time which means that this is the right equation so we can say that power as a function of time is equal to the force which is equal to 24 T multiplied times the velocity which is 6t squared so in other words the power as a function of time is equal to 24 times 6 that's 120 plus 24 that would be 144 T cubed there we go there is our second answer but now we need the final answer the average power for the first four seconds so how do we find the average of anything well what we can do is so we can find the area need to curve and divided by the width but other words if I draw a graph and on the vertical axis I have power and on the horizontal axis I have time that curve this curve right here would look something like this and let's say that this would be the power after 4 seconds so if we come down here so this would be 4 seconds this would be 0 so that would be in terms of seconds and of course power would be in terms of watts so to find the average power for the first 4 seconds what I need to do is I need to find the area need to curve and divided by the width so we can say that power average is equal to the integral of the power from 0 to 4 divided by the time 4-0 that would be the width of the horizontal width of that graph so that means that this is equal to the integral from 0 to 4 of 144 T cubed DT and of course I need to multiply this times DT all divided by 4 so this is equal to 144 T to the 4th divided by 4 times 4 because we divide by the new exponent times 4 so this is equal to oh we have to evaluate from 0 to 4 so this is equal to 144 divided by 1616 that would be 9 that would be 9 times T to the 4th power actually that would be 4 to the 4th power minus 0 so we could plug in the lower limit we get nothing so 16 goes into 144 9 times we have T to the fourth power evaluated from 0 to 4 sets forth to the fourth power which is 16 times 16 which is 256 so this is equal to 9 multiplied times 256 that would be 25 16 minus 256 well let's get a calculator that makes it easier so 256 times 9 23 hundred and four watts so this is equal to two thousand three hundred and four watts that would be the average power over the first four seconds and so there's the answers wasn't that easy it was it we start out with an object may have two kilograms being pushed by variable force of 24 T we first want to find the velocity so we use ethical sigh made but it can be written as DV DT and F of course is 24 T so we solve that to integration then we need to find the instantaneous power which is a function of time and it simply work overtime work is Force Times distance and this isn't overtime is velocity so it's the force times velocity the force 2014 velocity is 60 squared and finally to find the average we take the area need to curve of this graph so we integrate 1 over 144 T cubed and we divided by the width that gives us the average power so this would be average power and simply the area underneath the curve divided by the width so divided by 4 and that gives us the average power for the first 4 seconds of 20 304 watts that is how it's done

Physics – Mechanics: Rotational Motion (1 of 6) An Introduction

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In this video I will develop the three basic angular rotational equations and show their equivalence to the linear motions equations.

welcome to AI electron line and today we're going to talk about rotational motion another topic in physics very fundamentals of physics and I wanted to give you a little bit of an introduction to this particular topic so what is rotational motion well think of like a disk rotating around so let's take a nozzle disk let's say that it's rotating in a clockwise direction and let's take a particular point on the desk like a little corner a little edge or a particular dot on the edge of the disk and so you can imagine things does now going to be rotating around and if we use the horizontal x-axis as a reference and this disk is rotating around a clockwise direction you can see that sometime later the dot will be in a different position so we'll have moved from there to there and you can see that it has moved through it a certain what we call angular distance and the angular distance can be denoted by the angle let's say theta so theta is the distance or angular distance that it covered so theta represents angular distance all right now how fast is a disc rotating how fast is it spinning and so the way to express that is how big of an angle per unit time are you covering and the variable you use for that is angular velocity and we use the letter Omega for that the Greek letter Omega and it represents angular velocity and you can see that the word angular keeps on being repeated which means that this is a measure of how much an angle we cover in a certain amount of time or how much an angular we cover over a certain amount of time all right then another unit that we need to think about is one of the disk rotates increases its rotational speed or decreases the rotational speed that we call acceleration and in this case we use the letter alpha to express the the term angular acceleration so those are the three units of three terms that we use to describe rotational motion and the definitions of them is that Omega by definition is the change in angle over time and if you want to do that in a differential equation we write that we can write that as d theta/dt angular acceleration is defined as the change in the rotational speed or rotational velocity as a function of time or we can write this as the Omega DT so those are the definitions and you can see that there's a lot of parallel between these units or these ways of writing motion compared to the linear units of motion and let me write the equation kinematics in both versions so for linear motion we could write the three equations you can the max as follows we could say that X is equal to X at naught plus V sub naught times T plus one-half 87th naught plus 80 or you could write that V squared is equal to V initial squared plus 2ei X okay those are the three equations of linear motion called ik equations of kinematics and we can write the equivalent equations for rotational motion and where the equivalence comes from is that theta is the replacement for the variable X so rotational distance is the parallel to the linear distance Omega is the parallel to the linear velocity and alpha is the parallel to the linear acceleration so if we take the three equation kinematics and replace the variables X V and a a by theta Omega and alpha we get the three new equations that theta is equal to theta sub naught plus Omega sub naught times T plus one-half alpha t squared get omega is equal to omega0 plus alpha t and we get omega squared is equal to Omega initial squared plus 2 alpha theta and so those are the three equivalent equations to rotational motion which will do just this exactly the same thing for us as the equation linear motion have for linear motion so now that we have this concept of what rotational motion is and how it there's a lot of parallelism between rotational motion or linear motion let's do a few examples now to see how we use those equations I'll leave these equations up on the board so we can come back to them later as a reference and we'll go ahead and show you some interesting examples to help you understand these concepts all right so onto the next video

Physics – Test Your Knowledge: Energy (2 of 30) The Circus Act

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In this video I will find v0=?, theta=?, and horizontal-distance=x=? of a trapeze artist shot out of cannon, catching a swing, and landing on a platform 20m above the ground.

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welcome to our lecture online the next problem is kind of an interesting problem it has to do with a circus act and they're asking you to figure out various things about the act the objective is to shoot a person out of a cannon at some initial velocity have the person fly to the air catch the bottom of this ring that at the bottom of a 2 meter long rope which is attached attached to the ceiling and then the momentum of the person that was shot out the cannon should be sufficient to then swing the person to the top of the ramp right here which is even with the top of the rope at a height of 20 meters above the ground now they're asking you to figure out what the initial velocity should be of the person being shot on the cannon but the angle of the cannon should be and what the distance should be from where the can is placed to where the ramp is at hmm how do we do that well think about it in terms of energy the person will have a velocity in the Y direction so let's call that velocity initial in the Y direction and the person will have an initial velocity in the X Direction V initial in the X direction and when the person reaches this point the energy the person had in the Y Direction will be completely converted from kinetic energy to potential energy because when the person reaches the ring here the you can see the motion will only be in the horizontal direction so the kinetic energy in the Y Direction is completely converted from the initial kinetic energy that it had at the wind when the person was shot from the cannon and that was then converted to potential energy over here so this here would then be of course 18 meters and that would then be the potential energy the person's gained in the Y direction but in the X direction the person will still be moving when it clamps onto the ring and it brings them to the very top that means that the kinetic energy the person has in the X direction will be used to gain an additional two meters so this here would be an again of an additional two meters so what we can say is for the vertical direction in the my direction the kinetic energy initial equals potential energy final and we can say the same in the x-direction we can say that the kinetic energy initial in the x direction this would be of course in the y direction will be converted into potential energy final at the very top in other words we can say that one-half the mass times the velocity initial in the Y Direction squared will be equal to the potential energy gained which would be MGH would be M G times the height of 18 meters and then in the X direction we can say that one-half M V initial in the X Direction squared that would be the kinetic energy the person has in the X direction should equal the height gain which would be M G times two meters because it only gives them an additional two meters both equations we can go ahead and get rid of the mass and we can then solve for V initial in the Y direction so V initial in the Y Direction is equal to the square root of 2g times 18 and here we can say that V initial in the X direction would be equal to the square root of 2g times two in other words this is equal to the square root of 36 G and this would be equal to the square root of 4 G so this will give the this will give us the initial of loss in the Y direction and the initial velocity in the X direction so the initial velocity in the Y Direction is equal to the square root of 36 G which is equal to the square root of well actually I don't even need to do that we know what G is yes 9.8 so 36 times 9.8 OOP 36 times 9.8 take the square root of that which is 18 point 7 8 3 that would be 8 ten point seven eight three meters and the initial velocity in the X Direction is equal to the square root of four G which is equal to right so four times 9.8 take the square root is six point two six one six point two six one meters per second of course not meters but meters per second all right so we have the two initial velocities in the X and in the Y Direction using the energy equivalence now we can find the angle because we could say that that the tangent of the angle theta would be equal to the velocity in the Y Direction initial in the Y Direction divided by the velocity initial in the X direction because it would be the opposite side over the adjacent side so this is equal to the square root of 36 G divided by the square root of four G and of course the G's cancel out the square root of 36 would be six the square root of 30 the square root of 4 would be 2 which is equal to three but other words theta is equal to the inverse tangent of three which is equal to and that will give us the second piece of information we need three take the inverse tangent which is seventy one point five seven degrees 70 71 71 0.57 degrees so there's the angle that the cannon needs to be directed at the initial velocity can be found by doing this V initial is going to be the square root of V initial in the Y Direction squared plus V initial in the X Direction squared that's equal to the square root of we have eighteen point seven eight three squared plus six point two six one squared and now let's find out the initial velocity that the person needs to be shot at out of the cannon and cannon 18 point seven eight three squared plus six point two six one squared equals take the square root and I get 19.8 meters per second so the initial must be equal to nineteen point eight meters per second okay so there's the second piece of information there's the first piece of information what else do we need X being the distance all right how do we find X well that looks like we probably need to use the equation kinematics I'm going to find time in the air time in the air based upon the Y motion so we can say that y equals y sub naught plus V sub naught in the Y Direction times time plus one-half GT squared so the final height would be a height of 18 meters initial height would be zero initial velocity in the Y direction would be eighteen point seven eight 3t minus 4.9 T squared so putting that in a quadratic equation we can say that four point nine T squared minus eighteen point seven eight 3t plus 18 equals zero and now we can find the time the time is equal to minus B that would be 18 point seven eight three plus or minus the square root of 18 point seven eight three squared minus four times a times C four point nine all divided by 2a which is nine point eight so the time is equal to let's take a look so here we have 18 point seven eight three squared minus four times 18 times four point nine equals take the square root mmm let me see it well that's equal to zero so this is equal to zero which means it's 18 point seven eight three divided by nine point eight eighteen point seven eight three divided by nine point eight equals one point nine one six six seconds one point nine one six six seconds I'm keeping a few extra in significant figures just to eliminate to run off air so now we know how long the person is in the air by the time the person reaches this distance of course X needs to be an additional two meters so now we can say that X is equal to V initial in the X Direction times time plus the additional two meters gained with the swing at the very end so this is equal to six point two six one meters per second times one point nine one six six seconds plus two which is equal to so times six point two six one which is twelve so this will be 12 plus 2 which is 14 meters and so the total displacement in the horizontal direction is four meters so if you're going to help the circus you need to tell them that the angle needs to be seventy one point five seven degrees that the initial velocity of the person shot at the cannon is nineteen point eight meters per second which is quite fast and finally that the horizontal distance from the cannon to the platform needs to be forty meters and then you can tell them rest assured everything will just go just fine and the person will reach the platform without any problems we hope and that is how it's done

Mathematics Gives You Wings

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October 23, 2010 – Professor Margot Gerritsen illustrates how mathematics and computer modeling influence the design of modern airplanes, yachts, trucks and cars. This lecture is offered as part of the Classes Without Quizzes series at Stanford’s 2010 Reunion Homecoming.

Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.

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Stanford University thanks very much I'm a very tickle that you're here because I know I'm competing with a football game the start of a baseball game so I'm keeping a very close eye on the time because I know the first pitch is at 457 as I want to be out before then and I'm also competing with the rain and then the first thing is I'm talking about mathematics which generally is something that people want to escape from after college not revisit at a homecoming weekend but my purpose for this afternoon is really sharing with you a bit of the passion that I have for computational mathematics and as Adam was saying I'm directing quite a large Institute now on campus is the Institute of computational mathematical engineering and you may not have heard of that because we haven't actually been in existence all that long there was a precursor to this institute which was a program in computer science and I actually did my PhD in that program so I'm an alumni myself from 96 and I was fortunate enough to come back here in 2001 as a faculty member after getting my PhD in computer science in energy resources don't ask me how that happened but they allowed me back in and since for weeks I'm director of this Institute but we have 140 graduate students 70 masters and 70 PhDs we have no undergraduate students but we're teaching 20 courses in the School of Engineering and for our Sciences for Applied Mathematics and so on in the whole for the whole University 4,000 student units and so most of the harassing the mathematical harassing of undergraduate and graduate students is led by my Institute and so that's a wonderful feeling to be able to control that for some people very painful experience but I love it and I've always really loved mathematics and I want to show you a little bit today why and one of the reasons why I absolutely love this is on the board right now and that's a whole bunch of complicated Asians and those are the equations we're going to be talking about today but my idea is that I make this much simpler for you because really when you look at it very carefully all of the equations that govern fluid flow processes be it climate models where the models optimization of SIL design for competitive yacht races one of the things I've done another thing I've done is optimizing wings for pterosaurs of course they don't exist but we just made up some for a National Geographic I've also done fluid flow in the oil and gas reservoirs aquifers groundwater models all of these processes that may seem completely different are all governed by these equations there's all the same stuff and then they look very complex right and this is of course what we what we like we like to use Greek symbols we like to use long mathematical expressions so we can impress people at Friday beers and so on now sometimes we even put it on t-shirts and have something like that and now we understand this and you don't but really when you start looking at it and breaking it down it's all relatively simple and it's really quite wonderful when you see how all of these fields know coastal oceans wind turbine optimization you name it is all connected in exactly the same way and this is one of the reasons why I love computational mathematics and why in the 25 years or so that I've been doing this professionally I've worked on 10 12 13 different projects all sorts of different fluid flow problems and sometimes when people look at my CV or my publication say you're crazy you know you're all over the place they know I'm always doing just this and so that's what I'd like to share with you so by the end of this hour you're either going to tell yourself that you will never ever do a math course classes without quizzes again at a homecoming weekend or you're going to be so excited that you'll come back next year and apply to do a master's in my program all right it's normally either-or okay so I'm hoping to see you all back next year for masters it actually happened once about five years ago I gave a talk at Homecoming weekend and there was somebody in the audience who was 84 years old and decided at the end of the talk that he wants to do another PhD at the PG already from Stanford in physics from 1954 I think it was and he came back and he started it but unfortunately it was a little bit too fast for him but he did take some of my courses which is absolutely fantastic anyway so let's go and and look at these equations so so here they are and they like I said these equations govern fluid flow no matter where the fluid flow is no sometimes the equations look slightly different but in concept they're all the same so I wanted to tell you a little bit about it and I won't raise the screen right now I'm just just going to write here because you know I am just right so let me start here um there is a couple of things in these equations a couple of terms when we understand these terms that you put one other down here you've understood all these equations and then I can start talking about how to solve these on a computer because ultimately that's what we'll do okay so and then we'll move between blackboard and screen for a bit because you know this is a map or so obviously I have to use this blackboard okay now what are all these things here you is stuff that we're interested in knowing now when you think about fluid flow what could you be you is just a name for something we're interested in when we're looking at fluid flow something that describes ocean flow wind what could you be up for example energy or velocity volume we're leaving pressure if we had all of that right if we had we're assuming that density does not change so we're not taking that into account right now but if we knew how the velocity changed as I'm traveling through space or as time changes if I know how the pressure behaves in the system and if I knew how energy behaves or what the temperature was like then the main regulus thing is I can really describe this whole process and moreover I can predict it because think about for example air flow like this in the room ok so we open some windows and there was a nice breeze outside we could maybe feel the air move in this room what drives that air now what is causing it to move difference in pressure now you can say well it's the wind outside right but ultimately that wind is driven by pressure differences low pressure high pressure what else ultimately by the summer we're not going to go that far I want to keep the computations a little bit less than than the solar system because then maybe we have a chance to actually simulate it sure size of the opening so configuration right now how things are moving is dependent on the configuration of my room and other words on the boundaries that I have in in that flow domain you know if you understand that lingo well how about gravity no you can say gravity doesn't really play a big difference if I have an air molecule flowing around it's not going to just drop down right but gradually gravity will also influence things so things move because there is a force acting on them right and as a result of this force things change velocity changes when things start to heat up somewhere can maybe have a density change I can have energy change because things can start flowing faster so kinetic energy changes and so on so all I really need to understand of any fluid system is in that system what's going to be moving or changing that I'm interested in and most of the time that is just that's my you that is things like pressure velocity and most of the time we don't think just about velocity but we actually think about something called momentum which is nothing but a mass times the velocity but if my mass is constant but if I say our air is always the same density weighs the same God so a certain percentage of nitrogen certain percentage of oxygen all these sort of ways the same per volume then I can just think about velocity and I can maybe write something down for energy but that's coupled to pressure so very often these things are taken together so I think about momentum I think about energy ok and then if I need to I think about mass that's all so that my you and these things are changing because of forces and most of fluids processes have very few forces acting on them there's there is a pressure gradient right pressure difference pressure differences there is gravity sometimes there are forces at boundaries and you think of a force at the boundary of something that may influence a flow friction right or sometimes we like to call that skin friction because it's friction that happens at the boundary for example on an airplane wing the actual airplane surface is holding back air flow you don't realize that when you're sitting in the airplane when you're sitting in in an airplane and you're looking outside over the wind and you know that you're going 500 miles per hour the funny thing is that at the surface of the wing the air is actually not moving at all it's just changing very very quick from 500 miles per hour to zero in a very thin layer over that wing which we call a boundary layer like a very very strong pressure or velocity change in that very thin layer there's a lot of force on that wing as a result of it hey that wing is really stopping that flow and you can just imagine if you're trying to stop a runaway train and there's a lot of force on you it's the same on that wing right but luckily we know how to engineer these things so we have maybe some boundary forces okay and all of these things here they determine how you changed so let me just write on the determines how you changes they'll be it pressure momentum energy whatever as a function of time right so how it changes in time how fast it changes how slow it changes know how long does it take for it maybe doesn't change at all and also as a function of space because not everywhere in my domain I may have the same velocity or the same pressure right so if I look at whether we have nice low pressure here now and somewhere else maybe high pressure so it changes also as a function of space now this is really all we need to do and all we need to know we not just need to set up some equations that describe this that says equations that say if I have a strong pressure gradient so my pressure somewhere is high and very close to it it's low what do you think will happen to the air velocity it's going to be high right because I have a strong force pushing that air from high pressure to low pressure to have a strong wind so I need an equation that says this high pressure gradient gives me a strong high velocity I need to have things like if gravity is strong then my downward velocity my vertical velocity will grow know things will start to fall if I have skin friction at the boundary then near that boundary my flow will need to slow down that's all we do it's just a bunch of rules the govern the behavior very natural intuitive behavior of this fluid flow so we look at the fluid problem we write down the sort of things that we want to understand about this flow we write down all of the forces that we can think of that may drive that flow sometimes we throw some out because not all of them may be important right for example in this room if I did air flow in this room I would leave out gravity sure it has some influence but it's so tiny compared to any drafts or other winds that I may feel in this room that I can throw that out so we look at it we simplify and then we decide how this changes and the ratio you know these rules that say if my pressure gradient my pressure force is this big then it starts to flow this fast or that fast well that I just know from experience that comes from experimentation because that depends on things like viscosity of a fluid and so on but really all of these rules the general behavior is very intuitive pressure gradient higher flow stronger but how much stronger what that ratio is that is something we just get from experimentation there's no rocket science involved with that ok so we look at tables and other things and we just write down an equation now let's look at this a bit it determines how you changes as a function of time or function of space bad stuff function of time function of space is in symbols like this so if I see D u DT what it says is how much does you change with respect to a change in time the little Delta is just the change in so if I translate this in words this just says change in u in a certain change in time that's all right what would this say then yeah so if X my X Direction you know we are in three dimensions so I have an X Direction I've got a y direction of God the Z Direction this is says if I walk in this direction so I'm changing X this is how much you is going to change now right now here nothing much is changing but it could be that if I'm walking here all of a sudden and I'm picking up a draft from one of the open windows on that side and I start feeling a breeze and then obviously the velocity of the air in this direction changes right so that would then be you so here it's zero and here start picking up the breeze and say oh the udx must be positive because as I'm changing X all of a sudden that velocity starts to grow so that's the UDS the u dy is the same in the Y direction I can do this for pressure I can do this for velocity I can do this for mass and that's it so as a physicist and mathematician we just set these things up looking at these physical rules and they're come the equations so that's the first step it's not so hard can we load the screen go back to the presentation yep be Thanks so now let's look at these equations that are a little bit fuzzy which I said before is some by choice because that's how we sometimes feel about them no it's just because copy and paste didn't work very well from leh Tech I don't know if you can see this but do you see all these DTS the X's device and what do you see you see V's you see little peak that's what that stands for pressure what do you think little you would be that's a velocity but it is one of the real how many velocities do I need to describe flow three right an X and y in the Z direction so we're so incredibly creative as mathematicians we generally call them uvw okay so we have X Y Z uvw so you recognize it the u DX this is the change in velocity in the X direction it's multiplied by something that's the fist causa T that makes a lot of sense if I start to push on something whether or not that goes real fast or not well that depends on whether I'm pushing air or whether and pushing peanut butter well what's the difference between them well one is a lot runnier than the other no one is very sticky we call that very viscous so obviously that viscosity needs to come in somewhere well there it is now let's see if I can find some forces right because these are all equations that tell me how velocity changes or momentum changes and they can only change as a result of applying a force okay so where are my forces they're hidden a little bit but you can see some forces here's a little guy this is G G is the gravitational constant it's multiplied by the density this is just gravity mg now so there's a gravity force and then I had one other force that I talked about there was the pressure gradient so what would be a pressure gradient that tells me how much a pressure changes over space right so how what would that look like what sort of term would the pressure gradient the pdy dpdx a DP DX and the P DX and the U DX are in the same equation no makes perfect sense okay so that's really it so now we have the equations and then the only thing that we need to think about is how do we model these now you know from algebra that if you have an equation like x squared is 9 you can solve for x I can you still do that plus or minus 3 right or a more complicated one to X is 16 X is 8 right so here in this case though the equations are a little bit more complicated first of all we've got these DD X's DDT's all put together then as you can see we have a bunch of equations because how many unknowns do we have here well we have P we need to know you we need to know V we need to know W right so these are already four equations if I have four unknowns how many equations or four unknowns how many equations do I need to solve for four unknowns for okay that was the easy one about algebra it's always the same okay and so I have four equations and are these equations independent can I take one and just solve it and then I have the solution and solve the next one now they're all related to each other and that makes a lot of sense because I can start moving things in this direction in the air without impacting flow in that direction of why would that be suppose I was god it's always like the fantasize about so this is my room right it's filled with air and I point my finger there and they tell these air molecules there to start moving in this direction what do you think will happen with the rest of the room will I just see that flow only in this direction I would have a vacuum there then right that would not be sustainable so what what happen things would rush in from this side so if I flow in this direction I'm immediately causing a flow in that direction so I cannot solve an equation for this direction only it will all be coupled together right and so all of these systems are coupled the very complex and I cannot just find a formula that gives me the solution at any point in space at any point in time that's impossible in other words there is no analytic solution to this I cannot do what you did in algebra ok so when I first saw this I thought for all these years I have been taught algebra and now I finally come to grad school and the first thing they tell me it's I can't use it I have to go do something else so what do we do well let me go on and show you first a couple of pictures of things that I've worked on these equations that you just saw can do things like this okay so this is a vertical takeoff and landing aircraft and you see the jets coming down now what these streams are here these are for DISA T or velocity lines but you can sort of imagine Highfill ASSA T coming down okay so this would be W in my equation will be very high here right this is the vertical velocity and what is driving that well a jet and in this jet I'm really just creating huge pressure difference that pushes that air out right it's just a pressure gradient that is doing this so if I were to put the pressure here then I see really really high pressure somewhere there and immediately a lower pressure as that flow is pushed down and then here it hits the boundary and then there is a boundary force right that I talked about skin friction and other things and the flow is forced away in this direction and then becomes turbulent and all of these things are governed by these equations okay I don't need anything more for turbulence or all of that it's all just in those equations that I showed you with the DD exes the DD Weiss the disease it's all the same okay but obviously I didn't do this algebraically this is another example one of my colleagues it's just air flow past a race car and then here you see pressure so these these are streamlines to indicate how the air is actually moving and how the particles move over this over this car and here you see the pressure build up at the surface some areas at the surface experience a much larger force than than others as this air is hitting it that depends on how aerodynamic this this car is but again all of this is just governed by these equations that I showed you and they're all solved in exactly the same way in fact these two could be solved by the same software this is another thing that I've done it's looking at flow 4-cyl design I worked for T New Zealand for a while for the America's Cup if you know who knows about the America's Cup okay good some sailors in the audience it was wonderful to work with them I worked on developing the gennaker and if you remember well in 2000 that was the only thing that didn't break right but that may be because we didn't design it very well so that we overloaded all the other systems you don't know that either right but here in in seal flow and I just show you this again exactly the same equations that we just had but I show this because the behavior of the flow is very different depending on this boundary configuration all right so these equations are all exactly the same but what you see in observe can change a lot no way the vertical takeoff and landing aircraft we had all the turbulence coming up but sometimes flows are very smooth so for example if I'm sailing upwind very close to the wind with my with my sails my sails just act like a wing of an airplane it's the same sort of thing but you can see here with smoke this is all smoke so you can visualize how the airflow okay so these are just how the smoke particles are moving and you see that especially on the on the jig there the the headsail it's very very smooth but there's nothing turbulent there is everything is smooth almost as if it is attached here you see a little bit of a wake a little bit of a turbulent wake where the flow is coming off but on the downwind leg when I have my gennaker when the flow hits that I see big eddies and turbulent flow again forming very very different behavior but exactly the same equations and you would think the same configuration because I have a boundary like a seal in both of these cases it's just that the uncoming flow is from a different direction and in one case it's from a direction that is very directly aligned with the seal and then the flow stays very smooth is not really him that so much by that seals so we call this attached flow and then the other case it's more like a parachute coming down where the flow just hits the seal that on is going around it and then the seal with its friction on the sill surface cannot keep that flow close to it it's almost as if you're on your bike your motorbike and you're trying to go around a sharp bend and you're going too fast and you're flying out of the bend what is causing that well you don't have enough centripetal force right not enough friction to in other words to balance the centripetal force that's the same here airflow hits this sail needs to go around the bend goes really really fast and the seal just doesn't have enough skin friction now it's a very simplistic way of looking at it to keep that flow attached to the seal and flies off it detaches and it starts twirling around okay but the same sail the same equations just different behavior and it I found it amazing that with that set of equations I can do all of these different things I'll show you a little simulation now this is just a if it works yeah this is just a simple did you see it move isn't it amazing this is my my little very simple little simulation but here I've have a downwind sill like a like a gennaker the wind is coming in goes around and here are the touches and I can just on my little laptop this just plays on this little thing do these type of simulations with these equations and then I could do more complicated things like going into three dimensions and actually doing both of these sails but now this particular simulation here that I that I ran for Team New Zealand would take like a week to run it's not very good right so these sometimes these equations they describe everything but they're very very expensive to simulate so we need to do something about it and this is why when we're doing things like silh design we generally just look at two dimensional cross-sections and when they do wing design they generally look at two dimensional cross-sections of a win yeah did my work change the look of the seal not really see when you are working on something like this for competitive team what they're looking for with Team New Zealand is looking for shaving off two or three seconds of a lag of the sailing race that maybe not 20 minutes long they just want to shave off a few seconds that's all and so when you think about this the changes that are looking for really are very subtle changes and I can design a beautiful sail but if the crew is not behaving now then they won't see the two or three seconds the errors that are making because I'm using approximations right I'm ignoring some stuff even when I write these equations down here I simulate things in windy conditions while wind is gusty and I can predict what it's going to be over 20 min a times so I make some approximations to the win they're probably off by more than the two or three seconds so here we always say these simulations give me bread and butter right on the shelf in my cupboard but they don't help because I'm working in the error margin if you'd like but the nice thing about doing these calculations is that with a simulator like this I can look at some really strange shapes and display now this is the way I see it this is not to give them the final design that is exactly a little bit better and it's the same in the aircraft industry you don't use these computers nowadays anymore to get this final design that is just right you always do that with Windtunnel tests and then you have to go and actually fly these things to see how it really behaves right ultimately but what you can do in this it's like a virtual laboratory right sort of at skill and I can put in all sorts of funky designs be creative and see if they do anything I can explore in other words much more than I can if I have to build a skilled model and put it in the wind tunnel test which is much more expensive so what we do with a lot of this stuff is just play trying out new ideas and with the ginika for Team New Zealand we did that we tried out some new ideas and then the other there's a really interesting thing because now I can come up with a new Jenica design what does that mean I can come up with a new shape saying if you had this shape of sail you could seal a little bit faster then I give it to the SIL designer and say I want this shape of civil and they say are you kidding me how can I build a sail that retains that shape when I put it in the wind because of course it's a lot more complicated when you actually need to build this right and so there's all sorts of stuff going on but see this more as a virtual laboratory where you can play okay all right now I will let's just leave this up but I'm going to scribble a little bit here on the side because we had those equations and I told you we cannot compute this with an algebraic equation right so and I've shown you results so obviously we must be doing something to be able to compute this but it's not a formula and so what do you do well the very first thing that you do as an engineer when you have an equation that is way too complex to solve exactly right or algebra Utley is what you compromise right I mean I'm call myself a computational engineer I'm not really an applied mathematician as I look at these equations to say I cannot find one formula I'm not interested and simplifying the physics so much that I can do this analytically I could do that could look a very simple domain very simple flow maybe just two dimensions or even just one dimension they' people do that one dimensional flow that's and funny right and so that they can find real algebraic equations that model this but I don't want to do that I want to have the three-dimensional feeling of this flow but I can't find it exactly and I think who cares if I can't find it Klee I'll find it approximately I don't need to know exactly that it is one point three seven four five six nine meters per second I just want to know it's around one that's probably enough right or maybe I want to get a few floating points in there but I don't need this completely accurately so here is the general idea of my field they say okay suppose that I want to simulate something and it's on this domain we're looking down on the Pacific Ocean okay and I want to simulate wind here maybe the wind sort of goes like this okay and all of these equations that are just showed with all the DD X's and DDT's simulate model this and I'm going to say to myself I don't need to know the solution everywhere that is the trick right so I want to know things only approximately and the way I'm going to compromise is to say I don't need to know the solution at every single XYZ and that every time T I'm going to be happy if I know the solution in some set of points okay and this was a fantastic idea in the whole field of computational fluid dynamics that I'm talking about is based on it so what they do is they divide this domain into a grid okay and the idea is that at each of these intersections of these grid lines are called grid points you find an approximate solution so in one dimension but would that look like so one dimension do something like this now say we have a domain between 0 & 1 I don't find a solution everywhere but I just find it add a bunch of points all of these points here and if I was looking for you of X but now I'm saying no I'm not going to look for U of X as a function but I'm going to find u 0 u 1 u 2 all of these points this is X 0 this is X 1 this is X 2 through two well maybe this is X capital n and all I'm doing is finding those solutions UN the UN minus one I just find n values that's all somehow we still need to talk about how but somehow I find them and then I start to think hey this the solution here or maybe the solution here in this Pacific Ocean through the equations that I have of course depends on the solutions they on there right for example the velocity here will depend on the pressure I have there and the pressure I have there but it also depends on the velocity I have there right because as I said earlier all of these things are coupled together so when I start translating these complex equations to relationships between the use at each of these points I will probably get a large set of coupled equations out of it right but it's now just a system of equations for the use in each of these grid points and even if there are million grid points and every great point says five unknowns I don't care it's only five million unknowns with five million equations my computers are big enough that I can solve this no problem right on the computer so the first step is that we do this that we say okay instead of wanting the solution everywhere we just wanted on the grid and then my question to you is what if I now one they have the solution right here because this specifically maybe Hawaii you know I'm interested in the wind on the Big Island and it's not in the grid point what would I do I could take a finer grid but I don't have enough computer power for that maybe I could shift the grid I heard that so that the grid point overlaps with that exactly but that may be hard because then the next thing I know somebody in Tahiti wants to know it and has just shifted my grid and Tahiti is no longer on it okay so I keep shifting don't want to do that either but I could just interpolate right so if I have a solution here here here and here I can probably find a pretty decent approximation to something in the middle and take some sort of average of them interpolated in other words if I have the solution at the bunch of points I can probably find a solution that is okay and that's all we're looking for in between those points if I want my accurate solution I got a computed for more points is a simple thing no such thing as a free lunch right you want accuracy you got to do it for more for more points now this opens up fantastic area several one is there are graders that look at the domain and create a nice grid for that domain and there's all sorts of ways you can do this I'm a very simple grid or I look at the domain and I want to put in straight lines like this because I like working with these types of grits because computationally they lead to simpler things but there are people that like using triangular ization so instead of having these straight lines they grid something like this anybody from near puts itself so you recognize this so this is simulation we they don't push it sound on for tidal flow modeling and everywhere did there is a vortice vortex of a triangle a point in a triangle is one of these grid points they're not as organized as nicely but these points is where we compute a solution and guess what the closer we get through the land to boundaries where things are changing faster the more points we want okay out here not much is happening really sort of steady flow we sort of know what it is it doesn't change very rapidly it just changes only with the tides I don't have all sorts of small disturbances or turbulence or small eddies of a river coming in all of these things happening at small skill don't have any of that so I can take what we call a coarse grid here there's only a couple of points in which you need to know the solution and then I know pretty much everywhere but here where there's a lot of stuff happening with the river coming in and all sorts of you know Jets and Eddie's here I need to have a lot of points so much in fact that you can't see them anymore a lot of points are together so this actually took one of my students about two months to create this grid to get it just right it seems very simple to do this triangular ization but there all sorts of things you have to be aware of for example if I do a grid here and the triangle is completely skewed almost squashed together like this then my equations don't behave so well this is just too skewed and and when I actually look at the equations and then how these equations solve themselves on the computer I get problems now I may have too big of an error now and so there's a whole science behind is that when you start translating these equations on these grid points how they behave okay so but that if they look very fancy I think we like looking at these things we've done one for Elkhorn Slough here in Monterey Bay right this is the grid and I gave you the answer but I always ask what's this thing when I show that to people that's actually the train track it's a little bit higher so we there's no flow going there it's just flow here in this little area here under the Train okay so that's Alchemist loop again takes a long time to think of these types of grids years agreed on the airplane now obviously this is not just the airplane itself that we're grading but also the flow around it but I won't show you that because it's so hard to see these things grow in three dimensions but in three dimensions just little tetrahedra so not my angles on the surface but tetrahedra volumes in three dimensions and you can see something's here near this edge here the leading edge of the airplane wing very very dense grits because a lot of stuff is happening there see huge differences in pressure and so on you can actually see the pressure distribution on the wing here blue is very low pressure I will be the top of an airplane wing have very low pressure well the better if it wasn't we'd be in real trouble because it's the pressure difference between the top of the wing in the bottom of the wing that keeps this plane up and so we like seeing very very low pressure here and these low pressure areas to true low pressure areas where most of this lift is is actually near the front of the airplane wing okay so if the back of the airplane wing damages a little bit don't worry too much when you're looking out and you see that fall off or break but if the front falls off you better get out okay now you can do other things like this is just a little simulator very simple thing but as you're moving something this is gas going through a reservoir you see as the gas is moving you see what the grid is doing can you see that very hard to see you but as this gas is moving through if you looked at this really carefully you can see the grid change so locally we're putting in more grid points because it's much harder to calculate some of this this gas flow in those regions where the gas and the oil come together so we can we do what we call adaptive grading so do all sorts of wonderful tricks that you can play to get good solutions the question I still haven't entered and I'm going to take three minutes for this and that's the end of your math class is how do I do the translation from these equations on to the grid okay and they're whole courses at Stanford called numerical analysis courses finding different sources final volume courses they give you ways to do it and there's all sorts of different methods but they're basically all the same thing so that's what we're going to talk about and it's something like this but I'd like to use the board so if we can just raise this close this and let's just give you a quick idea of how this is done and then you can probably use your your creativity to understand that this is a little bit more complicated in multiple dimensions but it doesn't matter it's all the same thing really nice let me just use this B I think yeah okay so here we are in one dimension okay just one dimension and we have a bunch of points in which we want to compute a solution okay and this point I want to compute you and that point is point number three or four number four point number five but I call it point number I okay so I give it a little number I and what point is this then I plus one see you get an A for this class and this is UI minus 1 okay now maybe you look something like this I don't know I don't know what that solution is right but suppose that this is my real solution now in my equations what do I have floating around I have things like D DX right the U DX right that is the change in U as X is changing what is that Billy it's a derivative remember that from way past so in this on at this position here now I'm interested in the U DX in other words I'd like to replace the U DX in all of these equations that I had with an approximation right that involves only Solutions at these points right because I cannot compute the true the udx because I don't have all that information about you the only information that I want to compute the only thing I want to know about the solution is these guys just in those points okay so it's this one this solution and that solution so now my task is find the udx an approximation to it in this point using only discrete values like this again I don't know how big the U is and the UI minus once and so forth are but it doesn't matter because I'm going to compute it all I want to find is how to replace this in my equation by an approximation and we call this a discrete approximation because I'm only using the discrete points yeah doesn't make sense so how do I do that I want to find an approximation to the slope in this point right the slope of this line the tangent line using only those points what would be really nice approximation well the slope is nothing but the change in U corresponding to some change in X so how about if I take a change you this minus this for example right so you I plus 1 minus UI minus 1 divided by this distance well I like these grids that have the same distance between no grid points ok because this would then simply be that so now imagine all these complex equations I had in the beginning and every D DX and every D dy and every D DZ is simply replaced by an approximation like that what would I get I would get a whole bunch of equations with use phase WS and PS in I plus 1 I minus 1 all of these grid points that I have and and for every grid point I would get a bunch of those equations so altogether if I have a million grid points and five unknowns foregrip one to five equation together a system of five million equations that all contain the solution values UI minus 1 UI plus 1 and so on and I can solve them all together but it's a large coupled system of equations how many of you have solved systems of coupled equations ok so it just takes a big computer and the smart algorithm I may maybe some of you who have heard about a Gaussian elimination to solve a matrix vector equation this is just a make big huge matrix vector equation that you get but for those of you know about it it's very large a fully coupled often nonlinear so I don't know if you notice this but sometimes we don't have UD UD X but we have something like u times 2 u DX it comes up in the momentum equation for example when I derive a momentum equation and so when I start discretizing that I get something like UI times UI plus 1 minus UI minus 1 divided by 2 H and that is part of my equation so now I have a product a multiplication of two unknowns and becomes nonlinear maybe it's quadratic okay but I have to solve a large system of nonlinear equations ultimately I get a huge matrix vector equation and I need to solve this on my computer but that's all ok so if you know how to do this so you take a couple of my courses next year I think you need with rusty calculus you need to and then you know everything there is to know about this so two quarters worth of investment and you can go and use commercial packages and do your simulations it tricks and the reason why I get money for consulting the the hard parts of this is to have good boundary conditions that are realistic know what parts of the physics you can ignore so that means that you need to know a little bit about the underlying physics I do need to understand it well enough to be to know what parts to throw out you need to be a reasonable gritter you need to be able to create nice grits and that's actually more of an art than a science this takes a long time to play with this especially if you have complex domains like put it sound or Monterey Bay but sometimes for a room like this would be very very simple because you kind of square very easy to put a good grid in this and then once you have all of these equations you really need to know a little bit about matrix calculations to do this fast okay so that's maybe another course so let's say three courses right – numerical analysis one matrix computation and you can solve almost any fluid flow problem okay and you can make the bit money you're looking for another career not as much as if you do search engine design which is also just a matrix calculation okay anyway that's my job and that's what I love and hopefully you learned a little bit for more please visit us at stanford.edu

Classical Mechanics | Lecture 1

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(September 26, 2011) Leonard Susskind gives a brief introduction to the mathematics behind physics including the addition and multiplication of vectors as well as velocity and acceleration in terms of particles.

This course is the beginning of a six course sequence that explores the theoretical foundations of modern physics. Topics in the series include classical mechanics, quantum mechanics, theories of relativity, electromagnetism, cosmology, and black holes.

Stanford University

Stanford Continuing Studies

Stanford University Channel on YouTube:

Stanford University classical mechanics is basically a set of rules about what those laws of motion look like it's a set of rules and in fact there are two varieties of questions the first question is I'm not sure about the order of them but let's take them first the first kind of question is what are the specific laws for particular kinds of system particular system could be a planet moving in the field of a heavy mass that has its own particular laws those laws are different than an electrically charged particle moving in the field of a magnet they're different then you know there are those specific specific laws of nature then there's a more general framework what are the rules for the allowable laws are there rules for the allowable laws what's the grand framework in which all of the various different specific laws are framed in and the second it's really the second question which we're really interested in the first question will provide us with illustrations illustrations of the principles that govern what the allowable laws of physics are okay so I like to start and I always do in these classes with a very simple set of illustrations who can guess what my illustrations are going to be coins right right of course that's because you've been here before going's let's take the very simplest system that I can think of a coin I don't have a coin I so when I say a coin I mean a abstract coin laid down on an abstract table the only thing about the coin which is relevant to us is whether it's heads or tails the coin has two states two states of being two configurations are two values however we want to call it heads or tails so there are two States this is our dynamical system it's one coin and it has two states heads or tails that's all there is to the world to this very simplified world question of course there's an even simpler world we can even go back to a simpler world it's a coin with only one side then it's even simpler all it can be is that one state right that's a little too simple nothing can happen nothing ever happens so this is our world what is an initial condition an initial condition is it's either heads or tails now what about laws of motion now we're going to make up some laws of motion the laws of motion of course for this coin that just sits here the law of motion is very very simple let's say let's put two circles here one stands for heads and one stands for tails what is the law of motion of this particular coin sitting on this particular table the answer is it stays the same so if we imagine breaking up time into little successive intervals the stroboscopic world of the squeak point like that then the only thing that happens is nothing happens heads goes – heads tails goes – tails or we can draw a picture heads goes – heads starting with heads we go back – heads tails goes – tails that's really a very boring law of physics but it's very powerful law of physics and that it tells you however you start you know where you will be arbic terally down the line and time alright if you start with heads the history of the world will be heads heads heads heads dot dot dot forever and ever so that's this law may be very boring but it's very powerful or if you start with tails it's going to be tails tails tails tails as far as the eye can see so that's an example of a dynamical system with a law of motion where the law of motion is just an updating but it's in this case very trivial there's only really one of the law of motion that you can imagine for this system and what is it it is that whatever it is at one instant at the next instant of the stroboscopic light it's the opposite in the next instance the opposite we can write it this way heads goes to tails tails goes to heads and we can make a picture of it we can make we could picture this law of motion by again arrows an arrow from head to tails which says that if you started hits the tail end of the arrow always means what you start with the head end of the arrow means where you'll end up and that would be this law of physics heads goes to tails tails because the heads heads goes to tails and the history now of the world would be if you start with heads you then go to tails and you go to heads and you go to tails not thought but if you start with tails you go to heads tails heads still pretty boring but a little more interesting than the original first law ah we could write we could write some mathematics for this in fact we can write an equation of motion let's invent a symbol a variable which takes two values let me call it Sigma why do I call it Sigma it's because Sigma is a traditional variable associated with two unison physics it has to do with the spin of the electron but about them or spin of particles but we don't need to know about that now Sigma is a variable it is either equal to one let's say or minus one one for heads – one for tails alright so we can instead of calling this heads and tails we can call it Sigma equals 1 and Sigma equals minus 1 alright we now have the idea of a configuration space which is labeled by the two possible values of a certain variable it's not as rich as the variables we will use for example positions of particles values of fields or all sorts of other things but nevertheless it's a mathematical symbol whose value tells you which of the configurations you're in alright let's take this first law of physics the first law of physics not the first law of physics the first law that I wrote down the previous law now as heads goes to heads and tails goes to tails let me write that in the mathematical form oh let's call time T and in our stroboscopic world T is an integer T is 0 1 2 3 it can also be minus 1 minus 2 minus 3 so in the scrubmites copic world time is discrete and it takes on integer values so let's see if we can write an equation for the board the most boring law in the world what it says is that whatever Sigma is at a given time let's put that over here Sigma at time T at the next instant T plus 1 it will be equal to whatever it was at the instant T this is just a law that says that the spin or that the coin doesn't flip it just stays the same whatever it is at one time it will be the same at the neighboring time at the next time ok so that's the simple law heads heads heads heads heads or tails tails tails tails tails what about the other law the one which says that the spin the coin does flip between each between each successive flash of the of the stroboscope that's easy that's just says Sigma of T plus 1 is equal to minus Sigma of T if it starts at 1 then at the next instant it's minus 1 if it starts at 1 the next min simmer it it's the opposite so there we've written down equations of motion for a very very simple system notice that it is completely predictive completely deterministic there's no ambiguity about what happens arbitrarily fire far down the future now of course I could come in and grab the coin and do something with it do something else with it that would disturb it in our language that we set up now that would be because the system was not closed I intervened and I was not part of the system so for a closed system laws of physics are completely deterministic in classical mechanics ok now let's go let's think about a more complicated system more complicated system not much more complicated instead of a coin let's take what's the next case friends a died about half of a half of a pair of dice you know I never knew until I started teaching this that as the singular of dice was die I really didn't somebody corrected me once I called it a I guess I called it a dice somebody said oh you mean a die and I guess I did ok so we have a die a die has six states 1 2 3 4 5 6 that's the only significance for our purposes today about the fact that it's a by is that it has six states again we could label the six states one through six let's put them 1 2 3 4 5 six and these stand for the six different configurations of laying the dice down the died down on the table that's an initial condition well sorry an initial condition is a choice of one of these six configurations what about a law of physics what about a law of motion an equation not an equation I'm not going to write an equation is a little bit too complicated but what's a possible law of physics for this simple system well a very simple law would be or give you a very simple law a simple laws nothing happens nothing happens however you start you stay the same way to graph it we would just graph it like this whatever you start with it's what you get at the end that's about as boring as as the as the simple coin okay a more interesting law would be that you cycle around this collection of configurations for example a possible law for zexis would be 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes to 6 6 goes back to 1 I'll leave it to you to try to write an equation for this an equation of motion it's not hard to do I'll just leave it to you do ok but you can see what happens the history of the world would now be well give me the starting point for after that the history of the world is 4 5 6 1 2 3 4 5 6 1 2 3 endlessly cycling around and that would be the theory of this particular die with a particular equation or a particular law of evolution now we can write down other laws another law and the simplest way to write them down is to graph them Oh again notice that it's completely deterministic another law would be to cycle in the opposite direction 2 goes to 1 1 goes 6 6 goes to 5 or we could do more complicated things they're not really more complicated it's just more complicated to draw 1 goes to 2 2 goes to 5 5 goes to 3 3 goes to 4 4 goes to 6 and 6 goes to 1 now this law is logically equivalent to the one which just cycled around it's just a relabeling it's a relabeling of the states but it's again 1 cycle 1 to 2 2 to 5 5 to 3 3 to 4 4 to 6 6 back to 1 if I were to just rearrange them redraw them then it would just look exactly the same as the original cycle so this law would be what should we call logically equivalent to one of the others okay but let's think about some laws which would not be logically equivalent here's a law that's not logically equivalent again 6 6 points I won't bother labeling them with numbers 1 goes to 2 2 goes to 3 3 goes back to 1 4 goes to 5 5 goes sorry 6 goes to 5 5 goes to 4 4 goes back to 6 it's again completely deterministic wherever you begin the future is laid out for you completely if you start with 3 you go to 1 – 2 – 3 – 1 – 2 – 3 on the other hand if you start with 4 you go – whoops if you go to for you so you go to 6 6 2 5 5 back to 4 and so forth so it's again completely deterministic but it's not logically equivalent to any of the previous ones it now has two cycles and if you're on one of the cycles you will never get to I'm giving them a name now cycles okay if you're on one of the cycles you will never get to the other one so it's not that there are two systems is one diary only one die but with this particular law of physics you're trapped on one cycle or the other cycle there's a name for this kind of behavior it's called having a conserved quantity a conserved quantity is something non-trivial that you can label the system with which doesn't change with time for example we could label this cycle up here cycle number one and assign it a value one this cycle down here we could label with a value 2 and then we would say that if you start with 1 the quantity 1 or the kind of conserved quantity 1 you stay with 1 forever if you stay with to use if you start with – you stay with 2 forever this is called a conserved quantity in other words a quantitative quantity all quantities are quantitative but a quantitative quantity which just doesn't change it's conserved are let's call the conservation law so this cycle over here the single cycle system doesn't have a non-trivial conserved quantity you pass through every one of the states this has a conserved quantity to the 1 what did I say 1 or 0 I don't remember what I assign them but I assign them to different numbers this one in this one oh we can make up more complicated or not necessarily more complicated but other examples all right and this one one goes to warn if the die is it 1 the rule is it stays at 1 if it's at 2 it goes to 3 and if it's a 3 it goes back to 2 and if it's a 4 it goes too far five if it's a five it goes to six if it's six it goes back to one I've changed my color coding sorry about that all right again if you there are three cycles now there are three cycles and if you get on to any one of them if you start on any one of them you stay on it you could describe this by saying the first cycle over here corresponds to the value zero let's say corresponds to the conserved quantity having value 0 it could be 1 over here and it could be 2 over here if you start with a conserved quantity being 1 then it stays 1 if you start with one less sorry zero if I start with a conserved quantity being 0 it stays 0 if I start with it being one that doesn't tell me exactly where I start either here or here but it tells me I start with one of these two and I stay with that value I don't change I change the state but I don't change the value of the conserved quantity likewise over here so we can consider a variety of different logically different evolutions what is it that distinguishes them well really it's just a number of cycles the number of distinct cycles this is a 3 I'll know um it's a little more than that it's a little more than that but but you can see you can you can play around with this and investigate it and you'll very quickly get the point if you haven't already gotten it any questions up to now yes sir hhhh to go to write so those I mean I can imagine her the sequences of agency T yeah but if I were to try to construct one of those I think it would have to depend on more than just the preceding state in several state factors not for that stuff for those laws HT th that's a different right right right you can right that's right you could invent laws where to know what happens next you have to know not only the previous but the second previous state right that's correct yes yes it will become important it will become important but under those circumstances you would say that the state of the system is not characterized by just whether it's heads or tails it's characterized by the configuration and the previous configuration and so you will have in that case four configurations the four configurations would be heads preceded by tails heads preceded by heads tails proceed and then you were in and then you would do exactly the same thing except you would say there were four possible states right so and that of course is them I know exactly where you want me to go but I'm not going to go there yet but the right not yet so it's what I said before the state of a system or the configuration consists of all the things that you need to know to predict the future so in the case that you described what you need to do what you need to know the initial condition consists not only of what the coin is doing but what it was last doing good so that's a good point and we will come back to it all right so we have the idea of configurations we have the idea of isolated systems we have the idea of conservation laws let's just point out that there's nothing to prevent us from considering systems that have infinite number of states it doesn't have to have an infinite number of objects to have an infinite number of states it just needs to have one object which can be rearranged in infinite number of ways for example if we had an infinite line and on that infinite line we marked off the integers and we said as a particle what does a particle mean a particle now just means a thing which occupies one of the places the places now consists only of the integers what's in-between doesn't matter doesn't count all right so a particle is simply a object which sits at one of the integers you could have an infinite array like this and then you would say the particle has an infinite number of states namely the infinite number of possibilities of where it can be we can label them by an integer in and then we could invent laws such as wherever you are move to the right one unit wherever you are move to the right one unit and that would be a picture that would look like this it doesn't cycle nevertheless we're going to call such a thing a cycle but but you can see it doesn't cycle it just goes on and on forever and ever we could have another law the other law could be wherever you are jump two units ahead wherever you are jump two units ahead you know I think I'll do it this way if you're over here jump two units ahead if you're over here jump two units ahead if you're over here if you're over here and so forth all right so we make a picture then of a kind of law of physics which tells us to jump two units ahead again make up an equation for that make up an equation for that it's easy to do in the first case there was no interesting conservation law there was only one cycle if you like wherever you are you will always either get to any other point or if you go back into the past if you imagine running it back into the past you will have come from that point so on so there cannot be any interesting conserved quantity because you'll pass through every single one of these integers and so they can't be distinguished by a conserved quantity in the second case there is a conserved quantity and you can call it the oddness or evenness of the position of the particle you could give all odd numbers are represent them by a value of a quantity which is zero and all even that's that's weird isn't it let's make the odd numbers have integer value one and the even numbers be labeled by integer values zero and then you would say there's a conserved quantity which can either be 0 or 1 if it starts at one of them it stays there it starts at the other it stays there all right so having an infinite number of configurations doesn't change the picture very much it does open the possibility that there's a kind of endless evolution which never repeats itself so in that sense it gets a little more interesting and you can think of all sorts of generalizations of this all right so I laid out some I'll call them allowable laws of physics allowable rules let me talk now about some rules which are not allowable no not allowable by whom by me but of course I'm simply reflecting the way classical real classical physics works when I say allowable and not allowable I give you an let me draw a picture of a non allowable or unallowable M in a stick it completely predicts the future but there's something different about it then the laws that I've drawn over here I'm going to do it by drawing the picture for it it's got something we can make up many many examples but for this particular example it's got three states so this is a three sided coin a three sided coin it's got heads tails and no sides heads till I was going to think something a little more risque but the tails tails us about as rescales we're going to get ya an edge sorry edge that's pretty edgy heads tails and edges okay all right here's a law of a kind which represents something that we will not allow in classical physics heads goes to tails tails goes to edges and edges goes back to tails all right now wherever you start wherever you start the history is completely predictive if I start with tails i go to i go to edges if a tails edges edge tail circles tails edge tails edge tails edge tails edge you just follow the lines if you start at heads you go to tails then edge then tails then edge then tail so here's very hit some histories first of all starting with heads it's heads tails edges tails edges tails popped on top if I start a tails I get tails edges tails edges pop I thought if I start on edges now what is it it's odd about this law what's on about this law is that it is completely predictive into the future but it is not predictive into the past so to speak um if you know that you're a tails you don't know where you came from did you come from edges well maybe if but you could have also come from heads so while you can predict the future you can't read icked what's the opposite of pre bit richer dick that's the word I'm looking for you can't retro dick the past from this law of motion one configuration or several configurations run into the same configuration and so you can't tell if you're over here whether you came from here or here the word for this is that it's not reversible here's the way to think about it reverse every arrow now you have an unpredicted situation if you're a tails you don't know whether to go to heads or whether to go to edges so it's a predictive situation one way but not retroactive the other way the word for this is not reversible I won't call it irreversible that's a little too definite it's not reversible it can't you can go one way but not the other way this is the kind of law that is not allowed in classical physics yes sir well ma'am I can't see well okay that of course depends on what you're trying to represent um classical physics doesn't allow probability so I think I could escape from the question by just saying we don't do that in classical physics but we could we can imagine because we can imagine anything we want in fact we can imagine this and we can study its properties the point is in one way or another it conflicts with the rules of classical mechanics probability or let's call non deterministic laws also conflict with the rules of classical mechanics so it's a very good question and it's something we want to come back to whether when when quantum mechanics for example which is not deterministic does quantum mechanics have a analog of this reversibility idea even though it's not deterministic and the answer is yes but not tonight well classical statistical mechanics also has keep in mind that the rules of classical statistical mechanics begin with the laws of mechanics okay we begin by assuming alright let's talk a little bit about the limits on predictability for a moment if we have a perfectly predictive system of equations it won't allow us to be completely predictive if we don't know the initial conditions exactly so we need to know two things to predict the future one is what the rules are laws and one is what the what the initial conditions are now in these very very simple systems it's easy to imagine that we could know exactly what the initial conditions are because we may not we may know that we again with either not this one but this one we may know that we begin either with edges or heads we don't know which and then there would be some ambiguity in what happens not ambiguity because the equations have ambiguity in them but because our knowledge of the initial state is imperfect ok this is easy to understand for these simple discrete systems it's easy to imagine that we can do enough experiment and very quickly just look at a coin and know the initial condition perfectly in the real world where we're faced with degrees of freedom which are continuous meaning to say they can be any number on the continuous real axis any number of numbers the positions and velocities of all the particles in the world you can never know them perfectly no matter how many decimal points you may have that you have may have measured decimal places you may have measured about the location of a particle you still don't know it exactly so there is always a degree of ambiguity in your knowledge of the initial conditions that degree of ambiguity may or may not sort of blow up in your face that small changes in what the initial conditions are may or may not give rise to large changes in what happens in the future so the right way to say it is if you knew well we are we have to be quantitative but if you know the initial conditions perfectly then you could predict the future forever and ever in a true classical mechanical system and you could also predict the past perfectly if you have imperfect knowledge of the initial conditions you want to quantify that how imperfect is it and if you can quantify it it may allow you to answer the question how long into the future can you predict things so if you knew enough about the initial conditions of the atmosphere you might be able to predict the weather for 3 days but as your knowledge is limited and because the atmosphere is one of these systems where little errors build up they called chaotic because the atmosphere is chaotic no matter how well you know the initial conditions it is always true that if you wait long enough you won't be able to predict the future on the other hand if you say I want to be able to predict the future for X number of years it should be possible to say how precisely you have to know the initial conditions okay so when we come to the real world this idea of predictability becomes a little more complicated but I started on purpose with very simple discrete systems ok so laws that are allowed laws that are not allowed the laws of our classical mechanics are not only deterministic into the future but they deterministic into the past which makes them reversible now how do you look at one of these pictures and decide whether it's a legal law or not well it's very simple if every state has one outgoing and one incoming arrow that means that you know where it came from and where it's going so when you draw one of these pictures if you want to know whether it is an allowable legal law in the sense that I've defined I'm using legal now just as a term for for reversible systems if you want to know whether it's reversible look at the arrows and if each state has one incoming and only one incoming and one outgoing only one outgoing then it is both deterministic and reversible there are analogs of all the things I'm telling you now about more complex systems and more interesting systems such as particles moving around okay so that's a that's sort of warm up preliminary about what classical mechanics is about it's about predicting the future or using the predictability the fact that in principle you could predict the future in other ways such as mystical ways to to limit what will happen all right now we want to move on to a more realistic world and in fact the world of particles we're going to be interested in the world of particles moving and a particle for our purposes can be thought of as a point particle if we want to make a complicated system we'll make systems of particles points particles but we'll consider point particles moving around in space so what do we have to know what are the configurations of a point particle what do we need to know well ok before we do so I think we should do a little bit of mathematical preliminary I want to remind you for those who don't know or who know knew but don't remember or remember but just barely what vectors are and what coordinate frames are a coordinate system is just a way of describing space quantitatively and incidentally for our purposes today and largely in general we will of course assume that spaces three-dimensional and so a point of space will be labeled by three coordinates but we're perfectly free to think about systems which are higher dimensional or lower dimensional and we will do so since we're interested in formulating the basic principles we don't have to restrict ourselves to very very specific examples a particle could move in one direct or one dimension it could move in five dimensions and and we will be interested in all the possibilities but for the moment let's just think of particles as things which move in three dimensions so in order to be quantitative about the location of particles we introduce a coordinate system coordinate system Cartesian coordinates will be the usual things we'll introduce later on we'll introduce other ways of describing locations of particles but for the moment we take space we identify an origin the origin is up to us where we want to put it I can put it over here I can put it over here it should be that the important questions that were interested in should not depend on the convention of where we put the origin but it's useful to fix it once and for all and say the origin of coordinates is located in Palo Alto wherever wherever we want to put it here it is then introduced axes the axes are taken to be mutually perpendicular you can check that the perpendicular by by with a t-square or whatever it is that you use to align axes and you label them we can label them XY and Z or X 1 X 2 and X 3 we'll use both kinds of notations but there's also an ambiguity about the orientation of the axis given that they're mutually perpendicular we still have to decide you know I think you know what I mean which directions they go in and so that's like fixing the origin of coordinates we also have to fix an orientation for the X and y axis once we fix the orientation for the x and y axis the third one is fixed it's perpendicular to it incidentally there's a convention and the convention is called a right-handed coordinate system the convention is when you've picked x and y you still need to know one discrete piece of information is Z pointing out of the blackboard or is it pointing into the blackboard and we settle on that by a rule it's a convention it's arbitrary the right-hand rule if we take our thumb and our index finger thumb along X index finger along Y then Z is the middle finger the direction of the middle finger right so that's the right-hand rule and it selects out this coordinate system from the other one where Z goes in the other direction okay so that's the idea of a right-handed coordinate system we also have to mark off distances along here so distances are marked off equal distances with a roller and of course in saying that we mark off distances we're also assuming a set of units the units could be meters it could be inches it could be feet it could be lightyears so again another ambiguity but another convention is to choose our units but once we've chosen I chosen our units we can lay off distances along here and then every point in space can be labeled by the value of XY and Z let's see how to redraw this it has a height Y it has an X X and it has a Z Z if you like you can think about how do you get to this point from the origin you go a certain number of steps along X you go a certain number of steps along Y and then you scroll a certain number of steps along Z and those quantities XY and z are the coordinates of the point so a point is labeled by a set XY and z now I know you all know this but let's spell it out anyway okay so that's the way we describe a point that's the way we describe a particle and of course if we have a system of particles many particles then of course we just have to put in one such point for each particle okay vectors what is a vector a vector is an object which has both length and direction for example a very simple vector is the position of this point relative to the origin is the origin and the position of this point relative to the origin has a magnitude which is the distance from the origin and it has a direction namely just the direction of the of the arrow connecting the point with here think of that vector as an object which has a length in a direction but don't think of it as being located anywheres think of it as being the same no matter where I draw it in space in fact I don't even have to think about drawing it in space it is what it is it's a magnitude and a direction in space that's called a vector and from now on we will label vectors by putting a little bar on top of them if I'm really conscientious I'll put a little arrow on top of that if I forget the top of the arrow or I get bored writing arrows I'll just put a little bar on top and here and there I may even forget to put anything on top but you'll remember you'll remind me okay so a vector has a magnitude the magnitude is its length it does not have to necessarily be a relative position it could be a velocity it could be an acceleration something there are other things it could be an electric field with all sorts of things the criteria for it to be a vector is that it has a length and a direction okay so that's that's the notion of vector every vector can be described or has associated with it a length let's call the length of it in fact for the length of it I don't have to put a vector sign two bars on either side of it the absolute value of it are called its length so this is a way of writing its length okay so that's the length of the vector and the vector it's always positive or zero it could be zero if the vector has no length at all in other words at this point or right at the origin would have no length at all it's always are the positive or zero so every vector has a length and it has a direction which is not so easy to write down okay it can also be described by components a components the components of the vector are exactly what I said before if you wanted to go from the origin to this point over here you would go a certain number of units of X a certain number of units of Y and a certain number of units of Z the other way to define it is to drop a perpendicular from the point to each one of the axes I don't draw this very well let's see I think how are we going to do this I want to get this vector to be out of the blackboard how can I get it to be out of the blackboard but draw it on the blackboard anytime I draw it in the blackboard always looks like it's on the blackboard okay from here to here okay we drop a perpendicular from the tip of the arrow to each one of the axes and that lays off for us a distance along those axes which are the three components of a vector we'll call them V X V Y and VZ when I'm writing the components I have no need to put the arrow on top of them the components themselves are numbers what is the length of this vector whose components are VX v y and VZ ya the square of the length is VX square plus V Y square plus VC squared Pythagoras is theorem except in three dimensions I'll assume it we won't prove it that the length of V the magnitude of V is the square root of the sums of the components that's the notion of the magnitude of vector and the the direction of the vector is encoded in the ratios of the different components for example if the X component is much bigger than the Y component then the vector is pointed more or less along the x axis and so forth the components can be positive or negative if they're negative they're pointing along the negative axis and so forth all right but as I I want to emphasize that now again that the notion of a vector is not necessarily tied to the location of anything in space it is what it is it's a length or magnitude and a direction and if you move it around it doesn't change the vector that's just a mathematical definition of the way you think about vectors you don't think of them as being tied to a point in space you can I mean there were circumstances where you may want to tie a vector to a point of space for example you might want to ask what is the electric field at a particular point of space then that vector is tied to that particular point in space but the notion of the vector transcends that doesn't matter where you put it all right now we have to talk about the algebra of vectors I really feel sorry for all of you people who've sat through this endless number of times before but it's funny there are some things like a book which no matter how good the book is pretty much except for some extremely special cases you really only want to read it once you know you may say you want to read that book you could you love that book so much you could read it endlessly but it's not really true you read it once there are other things like a good piece of music which you want to hear over and over and it doesn't matter how many times you've heard it it just is always good to hear I assume my lectures are like that okay alright so let's talk about the algebra of vectors the algebra of vectors are first of all you can add them you can subtract them you can multiply them by numbers so let's talk about that a little bit given a vector let's call the vector a again it's a length and it's a direction you can multiply it by a number an ordinary number not an integer but a real number let's multiply it by the number see what's the definition of that it depends on whether C is positive or negative if C is positive it's a vector along the same direction except its length is multiplied by C so twice a vector means a vector of exactly the same direction but twice as long and so forth and so on we can also define the negative of a vector in other words we could let C be minus one I can either put here minus 1 or just minus the vector and of course the negative of the vector is exactly the same vector except with a direction in the opposite direction that's the definition of multiplying a vector by a number by an ordinary number now you can also add vectors let's just remind ourselves about the rules for adding vectors there are three ways of thinking about adding vectors in the first way let's say a plus B we take a and we lay it out is there is one end of it tail end of it here's the arrow end of it then we take B and we put the tail of B at the head of a we draw a triangle the triangle has its third leg C which is the sum of a plus B it might be a degenerate triangle it might be that B is along the same direction is a in which case C will also be along the same direction it'll look something like this in which case it's a kind of a degenerated triangle but still we just think of it as a triangle we can add vectors by this role we can multiply by vectors by numbers oh before I do that let's just talk about the two other ways of adding vectors they're trivial in according to this rule it looks like it might depend on which one I laid down first but of course it doesn't you can choose a rule which is symmetric between the two of them if this is a and this is B you put the tails of a and B down together you draw a parallelogram and you draw the diagonal of the parallelogram and that's called C and in this form it's completely manifestly clear that it doesn't matter whether you put a down first or B down first so a plus B is the same as B plus a ok so that's that's vector addition you can multiply vectors by numbers either positive or negative let's call this a we know exactly what that means we can multiply B by a number B B and we can add them we know how to multiply by numbers we know how to add and that means we know how to construct the vector a times vector a plus B times vector B we know exactly what that means okay the third way of adding vectors is to use the components so if we have two vectors with components a X a Y and a Z and B X B Y and B Z then CX is just equal to ax plus BX and likewise C Y and cz we could summarize this all by saying C sub I where I could be X Y or Z or 1 2 or 3 is equal to AI plus bi and that's an equivalent way useful way to add vectors since often we specify the vectors by the by their components all right if we can add vectors and arm subtract vectors and multiply them by numbers the natural question is can we multiply vectors can we divide vectors well now we're not going to divide vectors the idea of vector division is not a well-defined idea but there are vector products there are two kinds of vector products two distinct concepts of multiplying vectors in one of them when we multiply two vectors we don't get another vector we get a number the thing that's called a scalar sometimes the number is just called a scalar so one definition of the product of two vectors is called the dot product let's talk about the dot product of two vectors a dot B a dot B is defined in the following way you take a you take B and now you think of the component of a along the axis of B what does that mean that means you drop a perpendicular from the end of a to the axis not to the x-axis not to the y-axis not to the z-axis but to the axis defined by B so you drop a perpendicular and now you take the length of let's call this here a sub B the component of a along the B axis you take the component of a along the B axis and you multiply it by the component of B along the B axis what's the component of B along the B axis it's the magnitude of B all right so you take the component of a along the B axis you multiply it by the component of B along the B axis alright and that's called the dot product well I'll write it out in a minute more definitely that's called a dot product but the way they defined it it looks like a dot B might not be the same as B dot a II after all the rule for B dot a would beat the drop of perpendicular from B onto the axis defined by a which is a different thing and then multiply them together is it obvious that a dot B and B dot a are the same thing well if we can write them in a manner which is symmetric between the two of them then we'll know they're the same thing okay so let's call this angle here theta the magnitude of a is the length of a the magnitude of B is the length of B what is the component of a along the B axis a sub B is equal to the magnitude of a time's the cosine of the angle between them right let's call it theta a B theta is an angle now and it's the angle between a and B that's the component of a along the B axis we now multiply that by the magnitude of B to get the dot product all right so dot product is a dot B and it's equal to a B cosine theta I won't bother writing theta a B it's a along the component of a along the B axis that's this times the component to be along the B axis which is just a magnitude of B it's a B magnitude of a B cosine of theta now in this form it's completely clear that it's completely symmetric between a and B and it doesn't matter in which order you multiply them together the cosine of the angle is just the cosine of the angle is a dot B always positive and under what circumstances if not under what circumstances would it be negative if the cosine is negative when is the cosine negative the cosine is negative if the angle is bigger than 90 degrees so that would be the situation for example where a was pointing like that then the component of a in the B direction would be negative cosine of theta would be negative for an angle bigger than 90 degrees the cosine is negative what about two perpendicular vectors zero cosine of theta is zero for 90 degrees so a diagnostic for deciding whether two vectors are perpendicular or not is the calculate the dot product now that can be useful and the reason it can be useful is because we can express the dot product in component form if we express the dot product and confirm the component form I won't prove this this is not hard to prove that in terms of the components of the vectors this is equal to ax BX plus a y be y plus AZ BZ in other words you multiply the components in your ad that can be easily proved a little bit of a little bit of trigonometry not much that that this is equal to this over here now that's nice let me show you how you can use that well let's see do we really need that well if I just what to give you the components of a three numbers and the components of B three numbers and asked you to compute the angle between a and B your first reaction might be to throw up your hands and say well I don't know get me a protractor and I'll try to I'll try to measure it but here's a tool now you calculate the dot product between them that gives you a dot B you can also calculate what is the magnitude of a it is the dot product of a with itself let's write eight oh sorry that's not quite right a dot a what is that that's just a square of the magnitude of a so that's a squared likewise for B so you could calculate the magnitude of a or the square of the square of it is ax squared plus say Y square plus AZ squared likewise for the magnitude of B and then you would calculate the angle between them by calculating the dot product of the two of the vectors so from the magnitude of each one of them and the dot product of two of them you would compute the cosine of the angle between them all right so that's a good trick if you want to know the angle between two vectors let's let's prove the law of cosines let's prove the law of cosines there's a simple elementary thing supposing we have two vectors a and B and we want think of it as two sides of a triangle and we want to compute the length of the other side of the triangle so let's call that C C is equal to a plus B how do we compute the length of C that's just C is equal to a plus B squared or a plus B dotted with a plus B the square of the length of a vector is the dot product of a vector with itself so the vector happens to be C this is C dot C D that's the square of the length of C let's get but I'm a chemist a my sorry a minus B a minus B thank you a minus B C is a minus B it's not a plus B a plus B would be over here we could do it that way couldn't we but let's do it that's a minus B and you can work out very quickly that this is a minus B do it yourself okay so it's a minus B squared what is this equal to this is equal to a dot a plus B dot B minus two a dot B right just multiplying them together and this is just the square of a this is a magnitude of a squared plus the magnitude of B squared and this is minus twice the magnitude of a time's the magnitude of B times cosine of the angle between them that's called the law of cosines that the size of the third leg of a triangle is given by the sums of the squares minus twice the product of the lengths times the cosine of the angle between them okay so that's a that's a vector addition volta vector multiplication vector subtraction even vector multiplication of that product will we'll worry about the cross product another time not today all right now one not more important but more interestingly we have not only a algebra of vectors which means adding subtracting and so forth Oh as I said there's no notion of division of two vectors and notice that for the dot product the product of two vectors is not a vector it's a number the cross product is another matter that we'll come to another time okay now let's talk first of all about a particular vector which characterizes the position of a particle here's the position of a particle now think of it as a particle in fact we're going to allow it to move around in a little while and the origin is a particular special point that we've picked out we do have to worry about what happens if we change the origin but we're not going to do that today and therefore the position of the particle defines a vector that vector is usually called little R or I suppose stands for radius and in fact the magnitude of the vector R is the radial distance from the origin to the particle that is true or presumably I don't know where the notation came from I think it was for radius but for now it stands for the position of the particle what are the components of our that just the x y&z of the particle they're just the coordinates of the particle x y&z so R has coordinates XY and Z or let me write it that way R sub X is equal to X the position X and so forth and so on that's the notion the simple notion of the location of a particle described as a vector now in general we are interested in the motion of particles the motion of particles is what classical mechanics is about how the motion goes from one instant to the next in other words how it's updated from one instant to the next and so and should think about R as a function of time in general it moves around however it moves around we'll assume it moves around continuously differentiable smoothly at the same time its components if R is a function of time then R sub X is also a function of time likewise for y sub y of T and Z of T and so the motion of the particle is summarized in this case by three functions of time x y and z what about the velocity of a particle the velocity of a particle is also a vector it has a direction it's not necessarily the direction of the position of the particle from the origin for example the particle might be over here but moving this way moving out of the blackboard in that case its velocity would be this way but its position would be this way and so they're two separate distinct vectors the position could be any vector and the velocity any other vector but how do we describe the velocity the velocity is the time derivative of the position alright I'm not going to spend a lot of time explaining that fact I think you all probably know if not you're probably in a little over your heads but I will assume you know this that the velocity of a particle is the time derivative of the position of the particle so we can write that in a number of ways we can first of all say that the components of the velocity the velocity along the x axis is the derivative with respect to time of the position of the X component of the position or we can write it either as R sub X or we can just write it as DX DT unlike Y's for the other two components of velocity so the velocity is also a three thing with three components it's also a thing with a length the magnitude of the velocity which is called the speed the magnitude of the velocity and it has a direction and the components of velocity are just the derivatives of the components of the position so there's a velocity vector and it's easiest to specify by specifying its components the X well I won't write equals it it has components XY and Z which are the X DT dy DT and the comer and DZ DT that's the notion of the velocity vector and I will frequently in almost always use a notation which many of you know but I will introduce it here anyway differentiating anything with respect to time calculating its rate of change as time proceeds the a traditional notation for it some of you know it some of you don't I'll tell you right now the derivative of anything let's call it any function of time with respect to time is just labeled this is just 1/2 this is just in order to keep from having to write D by DT over and over is just to put a little dot on top of it dot definition of a dot on top of a function is its derivative with respect to time it doesn't mean the general you wouldn't use it for derivatives with respect to space you wouldn't use it for derivatives with respect to anything else dot means derivative with respect to time okay so the we could rewrite this saying that the components of velocity let's write it in one formula V sub I where I could be X Y or Z is equal first of all to the X sub I by DT which is the same as X sub I dot that's the notation for velocity velocities are important let's say I work out an example well know before we work out an example let's talk about acceleration what is acceleration it's the rate at which the velocity is changing acceleration is zero if velocity is not changing whenever the velocity changes this acceleration acceleration is also a vector it's not just 30 miles an hour per hour or 30 miles 3 feet per second per second it's also got a direction thing can accelerate that way it can accelerate that way if an object is moving slowly along the plus so that's your plus x-axis if it's moving that way slowly and then it speeds up you would say the X component of acceleration is positive if it's moving fast and it slows down we would ordinarily call that deceleration but mathematically it's negative acceleration which means that the X component of the acceleration is negative it could be exactly the same acceleration as if you started from rest and accelerated along the neck of x axis in both cases a change of the velocity along the x axis is called acceleration and it's simply the time derivative of the velocity all right so we write acceleration the components of acceleration are the derivatives of the velocity with respect to time or we could write it as V of course VDOT re but that makes it the second derivative second derivative with respect to time of the position the second derivative is usually labeled with two dots on top of it two dots means second derivative one that means first derivative no dots mean don't differentiate it at all three dots means a third dot time derivative and so forth so acceleration is the second time derivative of position with respect left second time derivative position period we can also write this in vector form we could write that the velocity as a vector is equal to the time derivative of the possessor time derivative of the position vector or just this gets a little annoying r dot r with an arrow to indicate it's a vector dot to indicate the time derivative likewise acceleration acceleration is the second derivative R double dot okay now that we have these concepts let's work out an example or to two examples two specific examples of position velocity and acceleration are we going first example motion along the line particle has a position X this is the x-axis a particle is labeled as having a position X of T along one axis it's hardly worth thinking of it as a vector we just think of it as X of T it is a one-dimensional vector but one dimensional vectors are too trivial to even call vectors just X of T and let's as an example let's write down a particular formula for X of T this is X of T which we're going to assume is a constant some number plus B times T plus C times T squared now there's nothing to prevent us from going on but let's suppose that is the formula which tells us where the particle is in any given time in particular at the start let's take the start to be at T equals 0 at T equals 0 the particle is at a let's calculate the velocity and the acceleration you all know how to do this to calculate the velocity we write X dot and that's the first time derivative the derivative of a is 0 this is the constant derivative b times t b and the derivative of c t squared plus 2 CT right alright so this tells us now that the velocity at time 0 is B to start with but then the velocity starts increasing of course it depends on whether C is positive or negative or whether increases one way or the other but whatever it is it starts at B and then as time goes on it either increases or decreases depending on C linearly with time what about the acceleration we just differentiate again what's the derivative of B and the derivative of 2 of 2 C T – C okay so this is the acceleration it's twice C it's constant with time it's constant with time it doesn't change so this is a uniformly accelerated particle okay that's a uniformly accelerated particle that has acceleration to see all right that's the kind of motion you have of an object falling in a gravitational field with a constant acceleration but here it is and here's the the reason why what I want to do before we finish is one more example which is more interesting and it's circular motion motion in a circle this will teach us something interesting and new let's think of a particle moving around in a circle it starts at time T equals 0 along the x axis so here's X and here is y x and y are the components of its position for us now we're going to ignore the third Direction Z plays no role it's a particle moving on a plane it has to coma coordinates x and y and at any given time it's on the circle the angle increases linearly the angle just continues to increase and increase of course when it goes all the ways around the angle goes from 2pi back to zero but we don't have to say zero we can we can keep letting it go wind up higher and higher the angle is that so here's the angle theta and the angle increases with time according to the rule fada is equal to some constant called Omega times T it's not a w it's an Omega a Greek letter Omega how long does it take to go all the ways around the circle well to answer that we just say how long does it take for it to go from theta equals 0 to theta equals 2 pi 2 pi is 360 degrees in radians or working in radians of course so we simply solve the equation 2 pi is equal to Omega times T or 2 pi Omega is the amount of time that it takes to go all the ways around and that's called the period the period of the motion is 2 pi over over Omega and then of course determines what Omega is if we know the period we solve this equation for Omega if we want a particle it moves around in a circle and 1/10 of a second we put in for the period a tenth of a second then we calculate Omega all right so that's what Omega is it's called the angular frequency but let's now consider what are the X components and what are what if the X and the y components of the position of the particle and that's easy oh let's have it moving around on the unit circle for simplicity let's having it move around on the unit circle which means radius equal to 1 okay what is the X component of a position cosine theta right cosine theta where is the y-component sine theta okay so now we know the components of position as functions of time X of T is equal to cosine of Omega T y of T is equal to sine of Omega T and now we can start calculating the velocity and the acceleration and so let's calculate the velocity and the acceleration I assume that you know how to differentiate a function like cosine Omega T so we just have to compute the first time derivative to calculate V sub X and the time derivative of cosine Omega T is what minus Omega times the sine of Omega T okay everybody know that anybody not know that when you're in the notes I suspect this there's a whole firm primer on calculus and one of the things that I think goes discussed is derivatives of trigonometric functions so to differentiate cosine the derivative of cosine is minus the sine but because of the omega here is an Omega here so this much I expect that you will be able to either either you know you recognize it or you'll be able to go home and figure out why this is true what about V sub y Omega times cosine Omega T okay now I have an interesting question for you what's the angle between the velocity and the position the position is over here what's the angle between the position and the and the velocity it's pretty obvious that it's 90 degrees you can see that just by saying well we know the velocity is going to be moving along there but how can we check it we can check it by checking the dot product here's x and y are the components of a unit vector V X V and Y are the components of the velocity vector what's the dot product between the position vector let's call R and the velocity vector V well it's the product of the X component of position times the X component of the velocity plus the product of the Y component of position and Y components of velocity so it's cosine Omega T times minus sine Omega T plus sine Omega T times cosine Omega T they cancel this product here has a minus sign in front of it this product here has no sign both of them have cosine times sine in them so there's a direct calculation of the dot product of position and velocity and it's zero what does it tell you that the dot product of two vectors is zero it tells you that they're perpendicular so without you know intuitively it's obvious that the that the velocity and position are perpendicular to each other but here's the calculation that proves it okay let's go another step and calculate the acceleration the acceleration is another derivative all right so what's the Dare what's the derivative of minus Omega sine Omega T well it's minus Omega times the derivative of the sine which is another factor of Omega and a cosine so this would give us minus Omega squared cosine Omega T and what about a sub y derivative of Omega we first of all have a factor of 2 factors of Omega and the derivative of cosine equal minus sine well let's compare a with the position itself the position is the vector cosine sine the acceleration is minus Omega squared times the same vector okay so apart from let's the Omega squared is important it tells us how fast the velocity is but what's important here apart from the Omega squared is the fact that a lies in the same direction is R itself except not quite it up it lies in the opposite direction there's a minus sign here so that tells us that when the particle is over here its acceleration R is pointing outward the acceleration is pointing inward the acceleration is pointing inward toward the origin we all know that that the acceleration of a particle moving in a circle is toward the origin it's a centrifugal acceleration but here's the mathematics that demonstrates it that the velocity is perpendicular to the variant of them to the position of the particle and the acceleration is parallel but in the opposite direction how about the magnitude the magnitude of the position is one I put it on the unit circle what about the magnitude of the velocity velocity this is Omega right just this the x squared plus V y squared is equal to the square of the magnitude sine squared plus cosine squared equal at the one so we're just left with Omega squared for the for the square the velocity the velocity of this particle is just Omega then that makes sense the larger omega is the faster this thing circles around here and in fact the velocity is just proportional to omega how about the acceleration the magnitude of the acceleration to make a squared okay so the bigger omega is the bigger the velocity but even more so for the acceleration but acceleration opposite to the direction of the position of the particle alright i think i think that's all we wanted to do for today we've covered some material i wanted to go through the simple elements of acceleration velocity circular motion motion on the line and and the next time we'll start to talk about what is what are the initial conditions for a particle what is the configure what is the space of initial conditions and what are the laws of motion what are the things which tell us how the particle moves from one instant of time to the next okay

Physics – Test Your Knowledge: Vectors (11 of 30) Proof (A+B).(A+B)=A62+B^2+2AB

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In this video I will proof vector (A+B)(dot)(A+B)=A^2+B^2+2A(dot)B.

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welcome to a lecture line now let's have a bit of fun take a look at this we do have the two vectors a and B like we did in the previous example but let's ignore those for a moment let's say that we're trying to prove that a plus B dotted with a plus B the dot product of the two sums of a and B should be equal to the magnitude of a squared plus the magnitude of B squared plus twice the dot product of a and B and we're going to do this in the general fashion first and then we'll also show that yes indeed we can get the solution by taking two vectors like in this example so let's first do it for any two arbitrary vectors which means that if I'm going to add a plus B together I can do the following I can say that the vector A plus the vector B can be written as ax plus BX in the hi direction plus a y plus B Y in the J direction plus AZ plus BZ in the K direction so we know that this is the sum of two vectors in an arbitrary sense so Ned Lots let's now find the dot product between two sums like this so for now going to write the following a plus B dotted with a plus B we should get the following result so here we're going to multiply then the X the X components of the two vectors so that means that we're going to get ax plus BX multiplied times ax plus BX plus a y plus B y a Z plus B Z so what we did here we simply use the definition of the dot product to move to multiply via the dot product these two sums so that would be the X component of the first times the X component of the second the Y component of the first times the Y component of a second plus the Z component of the first times the Z component of the second so now let's just go ahead and multiply all these out so this is equal to a x squared plus b x squared plus two times ax BX we do the same over here plus a y squared plus B y squared plus two times a YB y plus a Z squared plus B Z squared plus two a Z BZ alright now we're going to collect terms in a certain way so what we're going to do here is we're going to collect all the X terms and not all d squared XY and Z terms of a so we write this as ax squared plus ay Y squared plus a Z squared so we got this term here this term here and this term here now we do the same for the B vector so we have plus B sub X squared plus B so Y squared plus B sub Z squared and notice that this term does this term and does this term and now what we can do is add all the remaining terms together so that would be two plus two times because I can factor out a two that would be ax BX plus ay Y B Y plus AZ BZ and now we're going to take a look at this I realize that this here is equal to the magnitude of the a vector squared so this can then be written as a squared plus this here we realize now that this is equal to the magnitude of B squared so this is squared and plus two times notice that this is the definition of the dot product of a and B so this would be two times a dot B and that's exactly what we wanted to find when we did that particular dot product so that means that yes indeed the sum of a plus B dotted with the sum of a plus B is indeed the magnitude of a squared plus the magnitude of B squared plus twice the dot product of a and B and so that is how we prove that

Physics – Test Your Knowledge: Vectors (10 of 30) Find the Angle with Respect to x-Axis

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In this video I will find the angles with-respect-to x-axis alpha(A)=? and alpha(B)=? given A=3i+4j-5k and B=-i+2j+6k.

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welcome to electron line in this problem we're given two vectors vectors a and vector B in three dimensions and we're supposed to find the angle between each of those two vectors and the x-axis so how do we do that well Direction cosines comes to mind for example if we have something in three dimensions and let's say we have some vector in some arbitrary direction let's call it vector let's see hmm how about D vector D alright because I want it to be different from what we have up here so how do we find the angle between that particular vector in X so let's call that angle alpha so what we could do is we could find the direction cosine we can find the cosine of alpha and that is defined by taking the x component of the vector and dividing it by the magnitude of the vector so I can maybe write the magnitude like that and that'll give us the direction cosine the next to find the angle we take the inverse cosine of that so D sub x over D and that's how we find the vector alpha for a particular vector for any vector for that matter and so using the same principle we're going to do that over here so what we need to do is first find the magnitudes of the two vectors so the magnitude of a is going to be equal to the square root of the individual components squared a sub y squared in sub X square plus a sub y squared plus a sub Z squared add them all up take the square root that gives us the magnitude of the vector so in this case is this equal to the square root of three squared plus four squared plus a negative five squared and so that is equal to 16 plus 9 is 25 plus 25 that would be the square root of 50 we do the same for vector B that's equal to the square root of B sub X squared plus B sub y squared plus B sub Z squared and so this becomes equal to negative 1 squared plus 2 squared plus 6 squared and so that's 36 plus 4 that's 40 plus 1 that's 41 that's equal to the square root of 41 so now we have the magnitudes of our two vectors now we find the direction cosines so we can say that the direction cosine for vector a is equal to the X component of a divided by the magnitude so in this case the X component would be 3 and the magnitude is the square root of 50 do the same for vector B the direct and cosine for vector B is equal to that would be a sub now B sub X divided by the magnitude B B sub X is equal to negative 1 and divided by the magnitude which is the square root of 41 and now we're ready to find the individual angles so alpha sub a is equal to the inverse cosine of 3 divided by this square root of 50 let's find out what that's equal to so we have 3 divided by take the square root of 50 take the inverse cosine of that and we get 64 point nine degrees we do the same for the B vector inverse cosine of would be negative 1 over the square root of 41 let's see what that is equal to so we have 1 negative 1 divided by take the square root of 41 inverse cosine and that would be an angle of ninety eight point nine eight make it 99 point zero ninety-nine point zero degrees and so here are the two angles with respect to the x-axis for both of these vectors and that is how it's done

1. Course Introduction and Newtonian Mechanics

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For more information about Professor Shankar’s book based on the lectures from this course, Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics, visit 

Fundamentals of Physics (PHYS 200)

Professor Shankar introduces the course and answers student questions about the material and the requirements. He gives an overview of Newtonian mechanics and explains its two components: kinematics and dynamics. He then reviews basic concepts in calculus through two key equations: x = x0 + v0t + ½ at2 and v2 = v02+ 2 a (x-x0), tracing the fate of a particle in one dimension along the x-axis.

00:00 – Chapter 1. Introduction and Course Organization
21:25 – Chapter 2. Newtonian Mechanics: Dynamics and Kinematics
28:20 – Chapter 3. Average and Instantaneous Rate of Motion
37:56 – Chapter 4. Motion at Constant Acceleration
52:37 – Chapter 5. Example Problem: Physical Meaning of Equations
01:08:42 – Chapter 6. Derive New Relations Using Calculus Laws of Limits

Complete course materials are available at the Yale Online website: online.yale.edu

This course was recorded in Fall 2006.

this is a first part of a year-long course introducing you to all the major ideas in physics starting from Galileo and Newton right up to the big revolutions of the last century which was on relativity and quantum mechanics the target audience for this course is really very broad in fact have always been surprised at how broad the representation is so I don't know what your major is I don't know what you're going to do later so I picked the topics that all of us in physics find fascinating some may or may not be useful but you just don't know some of you're probably going to be doctors and you don't know why I'm going to do with special relativity or quantum mechanics but you don't know when it'll come in handy if you're a doctor and you've got a patient who's running away from you at the speed of light you will know what to do or if you're a pediatrician with a really small patient who will who will not sit still it's because a loss of quantum mechanics don't allow an object to have a definite cannot have a definite position and momentum so these are all things you just don't know when they will come in handy and I teach them because these are the things that turn me on and got me going into physics and whether or not you go into physics you should certainly learn about the biggest and most interesting revolutions right up to present day physics all right so that's what the subject matter is going to be and I'm going to tell you a little bit about how the course is organized first thing is this year it's going to be taped you can see some people in the back cameras as part of an experimental pilot program funded by the Hewlett Foundation and at some point they will decide what they will do with these lectures most probably they'll post them somewhere so people for elsewhere can have the benefit of what you have sitting in the Yale classroom so I've been told that from now on we just ignore the camera and do business as usual nothing's going to be changed I try to negotiate a laugh track so that the jokes don't work super focusin laughter I was told no I just got to deal with it as it happens so it's going to be it's like one of the reality shows where things are going to be as they are and hopefully after a while we'll we'll learn to act and behave normally and not worry about his presence then coming to the rest of the details of course by the way there are more details on the website that I posted that was given to me by the University if you want to know more about what all this is about the course organization is fairly simple we're going to meet Monday and Wednesday in this room 11:30 to 12:45 I will give you some problems to do on Wednesday and I'll post them on the website you guys should get used to going to the classes website I'm really really dependent on that now finally learn how to use it I will use that to post information maybe once in a while send email to the whole class if you want to get those emails you got to sign up for the course because I push a button and it goes to anybody who signed up there the homework will be given on Wednesday and it's due before class the following Wednesday and I let me introduce you to our head ta Mara Daniel who's recently Mara Barabbas right she will come a little okay so Mara is going to be the person who will see you after class and she will take the problem sets that you have submitted before class and she will give you the graded ones from after class okay that would be sorted out will be up there so you should drop the homework before you come into class rather than furiously working on it during class and the solutions will be posted the same afternoon so there is not much point in giving a homework that's late but once in a while you know you come up with a reason that I just cannot argue with you got married getting a transplant whatever it is that's fine you've got a transplant I want to see the old body part you got married I want to see your spouse if something happened to a grandparent I'm counting and up to four I don't get suspicious go five six seven eight I will have to look at a family tree but you know look any reasonable excuse will be entertain and the relative importance given to these different things is 20% for your homework 30% for the midterm which will be sometime in October and 50% for the final that'll be the weighted average but I have another plan called the amnesty plan in which I also compare just your final grade what your in the final exam and whichever is higher of the two is what I will take to determine your overall course grade this is something I used to announce near the end but then some people felt that it's not fair not to know this from the beginning so I'm telling you from the beginning but don't don't dream and think that somehow the final is going to be so much different from your regular day to day performance but give you some reason to live after the midterm you feel that is hope I can change everything overnight it does happen I put that in for a reason because sometimes some of you have not taken a physics course and you don't know how to do well in physics and slowly you catch on and by the time it's final exam you crack the code you know how to do well as far as I'm concerned that's just fine if at the end of the semester you take a three-hour exam in a closed environment and you answer everything I don't care what you did in your homework or your midterm that's not relevant so that's how the grading will be done we have a maras group of TAS she is the head TA and she's the one you should write to whenever you have a problem then we also have two faculty members one is a postdoctoral fellow mark Caprio mark okay so he will have a discussion section on Tuesdays between 1:00 and 2:00 in Sloan lab and Steve for Lynette oh I don't know if Steve is here or not there's Steve our new assistant professor he will have a section on Tuesday night in Dunham's lab a room 220 Tuesday night is the night when you people realize homework is due on Wednesday so we know that so he will be there to comfort you and give used whatever help you need all right my own office hours I've not determined yet I'll have to find out when it is good for you I used to you know I live and work out of sloan lab up on the hill and it was easy to have office hours before or after class but now you have to make a special trip so just give me a little bit of time to find out maybe by soliciting email responses from you what would be a good time for my office hours but for any procedural things like you know this problem set was not greater properly and so on there's no point emailing me because I'm going to send it to Mara anyway so directly deal with the powers that be okay finally I want to give you some tips on how to do well in this course and what attitude you should have first likewise this you should come to the lectures it's not self-serving it's not so much for my benefit I think there's something useful about hearing the subject presented once orally secondly the book you can see one of you had a book here it's part it's part of 1100 pages and when I learned physics it was like 300 pages now I look around this room I don't see anybody whose head is three times bigger than mine so I know that you cannot digest everything the books have so I have to take out what I think is a really essential part and cover that in the lecture so you come to class to find out what's in and what's not in if you don't do that it's a danger you will learn something you don't have to and we don't want that okay so that's where you come to class second thing most important thing for doing well in physics is to do the homework the 20% given to the homework is not a real measure of how important it is homework is when you really figure out how much you know and don't know if you watch me do the thing on the blackboard it looks very reasonable it looks like you can do it but only we are going to find out is when you actually deal with the problem that's the only time you're going to find out so I asked you to do the problems as and when they're posted so if you post it on Wednesday is to cover the material for that week then you should attempt it as quickly as possible because I'm going to assume you have done the problems when you come for the next few lectures and in doing the homework it is perfectly okay to work in groups you don't have to do it for yourself that's not how physics is done I'm now writing a paper with two other people that are my experimental colleagues who write papers with 400 other people maybe even a thousand other people when they do the big collider experiments in Geneva and Fermilab collaborations can run into hundreds so it's perfectly okay to be a part of a collaboration but you got to make sure that you're pulling your weight you got to make sure that if you explain to others how to do this problem then somebody else contribute or something else but you know what everybody contributed in the end so the game is not just to somehow rather get the solution the problem set but to fully understand how it's done and the TAS will be there to help you there's every day that's going to be TA in the undergraduate lounge I would urge you to use that there's a beautiful new lounge that the Provost's office allowed us to build for physicists and chemists whoever happens to be in the building if you go there on the third floor of Sloan you may run into other people like you we're trying to work on problems you may run into upper-class students students or more advanced you run into your TA so that's a good climate that our coffee machine isn't that allowing sofas and everything else that our computers that our printers so it's a good lounge and I think if you go there one day a week to do your problem sets or more often that's a good meeting place I recommend that the final piece of advice this is very important so please pay attention to this which is I ask you not to talk to your neighbors during lecture now this looks like a very innocuous thing but you will find out it's the only thing that really gets my back up most of the time I don't really care I'm very liberal but this disturbs me because I'm looking at you I'm trying to see from your reaction how much of your my lecture you're following and then it's very distracting when people are talking so please don't do that if you talk I'm going to assume you're talking about me if you laugh I'm going to assume you're laughing at me let's make that that's not really what I think but that's how disturbing it is when people talk and very nice students who do not realize this often disrupt my straight my line of thinking so I ask you to keep that to minimal once in a while you'll have to talk to your neighbor and say can you please pass me my pacemaker that fell down that's fine then you go back to your business but don't don't do too much of that finally that is this ancient issue about sleeping in the class now my view is it's just fine okay I know you guys need the rest and interestingly the best sleepers are in the first couple of rows I haven't met you guys it's not person I have found some people really have to come to the first a second row because they claim that if they don't hear me they cannot really go to sleep now that was true in Sloan but I think loose has got very good acoustics so you can stretch out in the back but my only of only criterion is if you talk in your sleep that's not alarming it's talking it's not what next if you're going to sleep I ask you to sit between two non sleepers sometimes what happens the whole row will topple over we don't want the domino effect now it's going to be captured on tape and that's going to be really bad for my reputation so spread yourself thread yourself around the other people all right so that's it in terms of class you know logistics and everything I'm going to start going into the physics proper I will try to finish every lecture on time but sometimes I'm in the middle of a sentence or in the middle of a derivation and I have to go over a couple of minutes there's no need to shuffle your feet and move stuff around I know what time it is I also want to get out like you guys but let me finish something other days I may finish a few minutes before time that's because the ideas of physics don't fall into seventy five minute segments and sometimes they spill over a little bit also I'm used to teaching this course three times a week and now it's certainly twice a week and so things that fell into a nice 50-minute units are now being snipped up different ways so it's pretty difficult so even for me some of it will be new and the timing may not be just right okay anybody have any I should tell you first of all that in this class the taping is not going to affect you because the camera is going to be behind your head I mentioned to you in the website that this is not the big opportunity you've been looking for to be a star only the back of here it will be seen in some cases the back of the head could be more expressive than the front in which case this is your opportunity and I wish you luck but otherwise just don't worry about it because you will be only heard you may not even be heard so I've been asked that if a question is not very clear I should repeat it so that people listening to this later will know what the question was but I would ask you let me make one thing very clear that is I'm not in favor of you are talking to each other because it's distracting you're stopping me anytime is just fine I welcome that because I've seen the subject for god knows how many years the only thing that makes it different for me is the questions that you people have so you can stop me anytime and you should not feel somehow you're stopping the progress of the class there is no fixed syllabus we can move things around it is far more exciting for me to answer your questions than to have a monologue so don't don't worry about that so stop me anytime you don't follow something and don't assume that you're not following something because there's something wrong with your level of comprehension quite often you guys come up with questions that never crossed my mind very interesting and things we've been repeating year after year after year because they sound so reasonable suddenly sound unreasonable when some of you point out some aspect of it that you didn't follow so it could be very interesting for all of us to have issues discuss in class and quite often some presents are very common and your classmates will be grateful to you that you brought it up otherwise you know you get 10 TAS get 10 emails all with the same question okay so I'm going to start now anybody have any questions about class the format the midterm exams all right yes oh you mean my office hours no the discussion sections are Tuesday afternoon from 1 to 2 and Tuesday night from 8 to 10 and the the website has got all the details on when and where yes yeah there are many many lab times and you have to go to the website for the lab and by the way that reminds me I got here lots of flyers given to me by the director of the laboratories which will tell you which lab is the right lab for you they're offered many times a week yes I think it's a good idea to take the lab because if particularly in this particular class because I don't have any demonstrations or they're all in the other building so if this will remind you that physic is after all an experimental science and you will be able to see where all the laws of physics come from so if you're going to take it you should take it at the same time yes ah very good this is a calculus based class and I expect everyone to know at least the rudiments of differential calculus what's the function what's the derivative what second derivative how do you take derivatives of elementary functions how to do elementary integrals sometime later I will deal with functions are more than one variable which I will briefly introduce to you because that is may not be prerequisite but certainly something you will learn and you may use on and off but there are different ways of doing physics – to demonstrate over and over how little mathematics you need to get the job done there are others who would like to show you how much mathematics you could somehow insinuate into the process okay there are different ways of playing the game and some of us find great pride in finding the most simple way to understand something that's certainly my trademark that's how I do my research also so if you feel that's not enough math used I guarantee you that I certainly know enough eventually to snore the whole class but that's not the point I will use it in moderation and use it to use it to the best effect possible rather than use it because it is there okay so I don't know your mathematical background but the textbook has an appendix which is reasonable measure of how much math you should know you got to know your trigonometry you got to know what's the sine and watch the cosine you cannot say I will look it up your birthday and social security number is what you look up trigonometric functions you know all the time okay I will assume you do all right and of course trigonometric identities you know from high school pages and pages of them so no one expects you to know all those identities but there are a few popular ones we will use all right anything else yes yeah midterm will be sometime around 20th of October I have to find out exactly the right time we have 24 lectures for this class and the first 12 roughly will be part of the midterm but after the 12th lecture I may wait a week so that you had time to do the problems and get the solutions then I will give you the midterm yes yeah this tells you in detail this flier answers exactly that okay there's one more question somebody yes I don't have a strong view I think you should take the lab sometime but I don't know how many semesters semesters that you have to take but I would say the advice of your predecessors is very important if they tell you this is what works that's better than what somebody like me can tell also you should talk to Stephen ions who is the director of the labs he has seen every possible situation he will give you good advice I start now okay so we are going to be studying in the beginning what's called Newtonian mechanics it's pretty remarkable that the whole edifice was set up by just one person Newton and he set us on the road to understanding all the natural phenomena till the year 1800 and something when Max will invented the laws of electromagnetism and wrote down the famous Maxwell equation but except for electromagnetism the basics of mechanics which is the motion of billiard balls and trucks and marbles and whatnot was set up by Newton so that's what we are going to focus on and you will find out that the laws of physics for this entire semester certainly can be written on one of those blackboards or even half of those blackboards and the purpose of this course is to show you over and over and over again that starting with those one or two laws you can deduce everything and I would encourage you to think the same way in fact I would encourage you to think the way physicists do even if you don't plan to be a physicist because that's the easiest way to do the subject and that is to follow the reasoning behind everything I give you and my purpose will be not to say something as a postulate but to show you where everything comes from and it's best for you if you try to follow the logic that way you don't have to store too many things in your head in the early days when there are four or five formulas you can memorize all of them and you can try each one of them till something works but after a couple of weeks you will have one formulas cannot memorize all of them you cannot resort to trial and error so you have to know the logic so the logical way is not just the way the physicists do it it's the easier way to do it if there is another way that will work for non physicists I won't hesitate to teach it to you that way that turns out to be the best way to try to follow the logic of everything okay so Newtonian mechanics is our first topic so Newtonian mechanics has two parts or all of physics consists of tube is a two-part program the plan every time is to predict the future given the present that's what we always do when we do that right we are satisfied the question is what do you mean by predict the future what do you mean by the future what do you mean by the present by present we mean you will pick some part of the universe we want to study and we will ask what information do I need to know about that system at the initial time like right now in order to be able to predict the future so for example if you were trying to study the motion of some object here's one example here see that's an example of Newtonian mechanics I'll give you one more demonstration let's see we can catch this one that's a good example so that was Newtonian mechanics at work because what did I do so I released a piece of candy threw it from my hand and the initial condition had to do with where did I release it and with what velocity that's what he sees with his eyes then that's all you really need to know then he knows it's going to go up it's going to curve follow some kind of parabola then his hands go there to receive it that is verification of a prediction his prediction was the candy is going to land here and he put his hand there he also knew whether candy land but he couldn't get his hand there in time but we can always make predictions but this is a good example of what you need to know what is it he had to know about this object that was thrown I claim is the initial location of the object the initial velocity the fact that was blue or red not relevant and if I threw a gorilla at him it doesn't matter you know what the color of the gorilla is what mood it is in they said things we don't deal with in physics or if there is a tall building its standard physics problem and object falls off a tall building object could be a person so we don't ask why is this guy ending it all today that's not we don't know and we cannot deal with that so we don't answer everything we just want to know when is going to hit the pavement and with what speed so we ask very limited questions which is why we brag about how accurately we can predict the future so we only asked limited goals and we are very successful in satisfying them so we are basically dealing with inanimate objects so the project of Newtonian mechanics of predicting the future given the present has got two parts and now one is called kinematics and the other is called dynamics so kinematics is a complete description of the present it's a list of what you have to know about a system right now for example if you're talking about the chalk if I throw the chalk you will have to know where it is and how fast it's moving the dynamics doesn't dynamics then tells you why the object goes up why the object goes down why is it pulled down and so on that's dynamics the reason it comes down is gravity is pulling it in kinematics you don't ask the reason behind anything you simply want to describe things the way they are and then dynamics tells you how they change and why they change so I'm going to illustrate the idea of kinematics by taking simplest possible example that's going to be the way I'm going to do everything in this course I'm going to start with the simplest example and slowly add on bells and whistles and make it more and more complicated so some of you might say well I've seen this before so maybe there is nothing new here that may well be I don't know how much you've seen but quite often the way you learn physics earlier on in high school is probably different from the way professional physicists think about it the sense of values we have the things that we get excited about it different problems maybe more difficult but I want to start in every example in every situation that explain to you the simplest example then slowly add on things so what we are going to study now is a non living object and we are going to pick it to be a mathematical point so the object is a mathematical point has no size if you rotate it you won't know it's not like a potato you take a potato you turn it around it looks different so it's not enough to say the potato is here you got to say which way the notice is pointing and so on so we don't want to deal with that now that comes later when we study what are called rigid bodies right now we want to study an entity which has known spatial extent so it just a dot and the dot can move around all over space so we're going to simplify that too we're going to take an entity that lives along the x axis it moves along a line so you can imagine a bead with a wire going through it and the bead can only slide back and forth so this is about the simplest thing I cannot reduce the number of dimensions one is the lowest dimension I cannot make the object simpler than being just a mathematical point then you got to say what do I have to know about this object at the initial time what constitutes the present or what constitutes maximal information about the present so what we do is we pick an origin call it zero we put some markers there to measure distance and we say this guy is sitting at one two three four or five this is sitting at X equal to five now of course we got have units and the units for length is are going to be meters units for time will be second at time will be measured in seconds then we'll come to other units right now in kinematics this is all you need now there are some tricky problems in the book sometimes they give you the speed in miles per hour kilometers per year pound dollars per square foot whatever it is you gotta learn to transform them that I won't do that I think that's pretty Elementary stuff but sometimes I may not write the units but I weren't the right to do that and you guys have it so you still have to keep track of your units everything's got to be in the right units if you don't have the unit's then if you say the answer is 19 then we don't know what it means okay so here's an object at a given instant it's got a location so what we would like to do is to describe what the object does by drawing a graph of time versus space and the graph could be something like this you gotta learn how to read this graph I'm assuming everyone knows how to read it this doesn't mean the object is bobbing up and down I hope you realize that even though the graph is going up and down the object is moving from left to right so for example when it does this it's cross the origin and it's going to the left of the origin now which is the left with the on it and it turns around to start coming to the origin and going to the right that is X versus T so the language of calculus X is a function of time and this is a particular function this function doesn't have a name that other functions which have a name for example this is x equals T X equal to T Square you can have X equal to sine T and cosine T and log T so some functions have a name some functions don't have a name what a particle tries to do generally is some crazy things doesn't have a name but it's a function except tea so you should know when you look at a graph like this what it's doing so the two most elementary ideas you learn are what is the average velocity of an object it is denoted by the symbol V bar so the average is found by taking two instance in time say T 1 and later T 2 and you find out where it was at T 2 where as where is was a T 1 divided by time so the average velocity may not tell you the whole story for example if you started here and it did all this and you came back here the average velocity will be 0 because you start and end at the same value of x you get something 0 divided by time will still be 0 so you cannot tell from the average everything that happened because another way to get the same 0 is to just not move at all the average is what it is it's an average it does not give you enough detail so it's useful to have the average velocity is useful to have the average acceleration which you can find by taking similar differences of velocities but before you even do that I want to define for you an important concept which is the velocity at a given time so this is the central idea of calculus right I'm hoping that if you learned your calculus you learned about derivatives and so on by looking at X versus T so I will remind you again this is not a course in calculus I don't have to do in any detail I will draw the famous picture of some particle moving and it's here at time T at some value of x a little later which is time T plus delta T the delta T is going to stand always for a small finite interval of time infinitesimal interval of time not yet 0 so during that time the particles gone from here to there that is X plus Delta X and the average velocity in that interval is Delta X divided by delta T graphically this guy is Delta X and this guy is delta T and Delta X or Delta t is a ratio so in calculus what you want to do is to get the notion of the velocity right now we all have a intuitive notion of velocity right now when you're driving in your car there's a needle and the needle says 60 that's a velocity at this instant it's very interesting because velocity seems to require two different times to define it the initial time in the final time and yet you want to talk about the velocity right now that is the whole triumph of calculus know that by looking at the position now the position slightly later and take the ratio and bringing later as close as possible right now we define a quantity that we can say is the velocity at this instant so V of T is the limit delta T goes to 0 of Delta x over delta T and we use the symbol DX DT for velocity so technically if you ask what is the velocity is stand for it stands for let me drop general situation if a particle goes from here to here Delta x over delta T I don't know how well you can see it in this figure here is the slope of a straight line connecting these two points and as the points come closer and closer straight line will become tangent to the curve the velocity at any part of the curve is a tangent to the curve at that point it's tangent of that theta this angle is Theta then Delta X and or delta T by trigonometry stand data okay once you can take one derivative you can take any number of derivatives and the derivative of the velocity is called the acceleration and we write it as the second derivative of position so I'm hoping you guys are comfortable with the notion of taking one or two or a number of derivatives interestingly the only the first two derivatives have a name first one is velocity second was acceleration the third derivative unfortunately was never given a name and I don't know why I think the main reason is that there are no equations that involve the third derivative explicitly F equals MA the a is this fellow here and nothing else is given an independent name of course you can take a function and did it take derivatives any number of times so you are supposed to know for example if X of T is T to the N you're supposed to know DX DT is n to the T to the N minus 1 then you're supposed to know derivatives of simple functions like sines and cosines so if you don't know that then of course you have to work harder than other people if you know that that may be enough for quite some time okay so what I've said so far is a particle moving in time from point to point can be represented by a graph X versus T or any point on the graph you can take the derivative which will be tangent to the curve at each point and this numerical value will be what you can call the instantaneous velocity at that point and you can take the derivative of the derivative and call it the acceleration so we are going to specialize to a very limited class of problems in the rest of this class the limited class of problems is one in which the acceleration is just a constant now that is not the most general thing but I'm sure you guys have some idea why we are interested in that so anybody know why that's such a big so much time is spent on that yes for me right the most famous example is that when things fall near the surface of the earth they all have the same acceleration and the acceleration that's most common is called G and that's 9.8 meters per second squared so that's a very typical problem when you fall in the surface of the earth you are describing the problem of constant acceleration that's why there's a lot of emphasis on sharpening your teeth by doing this class of problems the question we are going to ask is the following if I tell you that the particle is a constant acceleration a can you tell me what the position X is normally I will give you a function and tell you to take any number of derivatives that's very easy this is the backwards problem here you're only given the particle as acceleration a and you're asked to find out what is X in other words your job is to guess a function whose second derivative is a and this is called integration which the opposite of differentiation an integration is just guessing integration is not a an algorithmic process like differentiation if I give you a function you know how to take the derivative change the independent variable find the change in the function take the ratio and that's the derivative the opposite is being asked here I tell you something about the second derivative of a function and ask you what is the function the way we do that is we guess and the guessing has been going on for 300 years so we sort of know how to guess so let me think aloud and ask how I will guess in this problem I would say okay this guy wants me to find a function which ready is to the number a when I take two derivatives and I wrote somewhere here this result which says that when I take a derivative I lose a power of T in the end I don't want any powers of T very clear I got to start with a function that looks like T Square this way when I take two derivatives there will be no T left unfortunately we know this is not the right answer because if I take the first derivative I get 2t I take the second derivative I get two but I want to get a and not two then it's very clear the way you find pat's it up is you multiply it by this constant and now we are all set this function will have the right second derivative so this certainly describes a particle whose acceleration is a but a is not dependent on time but the question is is this the most general answer or is it just one answer and I think you all know that this is not the most general answer it is 1 answer but I can add to this some number like 96 they'll still have the property that if you take two derivatives you're going to get the same acceleration so 96 now we said typical constants I'm going to give the name see the consulate everyone knows from calculus that if you if you're trying to find a function about which you know only the derivative you can always add a constant to one person answer without changing anything but I think here you know you can do more right you can add something else to the answer without invalidating it and that is anything with one power of T in it because if you take one derivative it'll survive but if you take two derivatives it will get wiped out now it's not obvious but it is true that you cannot add to this anymore basic idea in solving these equations in integrating is you find one answer so that when it derivatives the function does what it is supposed to do but then having found one answer you can add to it anything that gets killed by the act of taking derivatives if you take only one derivative you can add a constant if you're taking two derivatives you can add a constant and something linear in T if you knew only the third derivative of the function you can add something quadratic in T without changing the outcome so this is the most general position for a particle of constant acceleration a now you must remember that this describes a particle going side to side I can also describe a particle going up and down if I do that I will I would like to call that coordinate Y then I will write the same thing you got to realize that in calculus the symbols that you call x and y are completely arbitrary if you know the second derivative of Y to be a then the answer looks like this if you knew the second derivative of X the answer looks like that now we have asked what are these numbers B and C so let me go back now to this expression X of T equals 8 e squared over 2 plus C plus BT it is true mathematically you can add two numbers but you can't ask yourself what am i doing as a physicist when I add these two numbers what am I supposed to do with a and B I mean with this B and C what value should I pick the answer is that simply knowing the particle has an acceleration is not enough to tell you where the particle will be for example let's take the case where the particle is falling under gravity then you guys know you just told me acceleration is minus 9.8 but G's minus 9.8 we call it – because it's accelerating down and up was taken to be the positive direction in that case Y of T will be minus 1/2 G t square plus c plus bt so the point is every object falling under gravity is given by the same formula but there are many many objects that can have many histories all falling under gravity and what's different from one object the other object is when was it dropped from what height and with what initial speed that is that's what these numbers are going to tell us and we can verify that as follows if you want to know what the number C is you say let's put time T equal to zero in fact let me go back through this equation here if you put time T equal to zero X at zero it doesn't have this term doesn't have this term and it is C so I realize that the constant C is the initial location of the object and it's very common to denote that by X not so the meaning of the constant is where was the object at the initial time it could have been anywhere simply knowing the acceleration is not enough to tell you where it was at initial time you get to pick where it was at the initial time then to find the meaning of B we take one derivative of this DX DT that's velocity as a function of time and if you took the derivative of this guy you will find it's a t plus B that's the velocity of the object then you can then understand that V of zero is what B's which we write as B zero okay so the final answer is that X of T looks like X naught plus V not T plus one-half a t-square okay so what I'm saying here is we are specializing to limited class of motion where the particle has a definite acceleration a then in every situation where the body has an acceleration a the location has to have this form where this number is where it was initially this was the initial velocity of the object so when I threw that thing up and you caught it what you're doing mentally was immediately figuring out where it started and at what speed that was your initial data then in your mind without realizing it you found the trajectory at all future times now that is one other celebrated formula that goes with this I'm going to find that then will I'll give you an example now I'm fully aware that this is not the flashiest example in physics but I'm not worried about that right now you will have you'll see enough things that will confound you but right now I want to demonstrate a simple paradigm of what it means to know the present and what it means to say this is what the future behavior will be and we want to do that in the simplest context then we can make the example more and more complicated but the phenomenon will be the same so what we have found out so far I'm purposely going from X to Y because I want you to know that the unknown variable can be called an X or can be called a Y it doesn't matter as long as the second derivative is a that's the answer now that's the second formula one derives from this you guys probably know that too from your days of the day care but I want to derive that formula and put it up then we will see how we use them second formula tries to relate the final velocity at some time T to the initial velocity and the distance traveled with no reference to time so the trick is to eliminate time from this equation so let's see how we can eliminate time you know that if you took a derivative of this you will find V of T is v-0 plus 80 what that means is if you know the velocity at a given time and you know the initial velocity you know what time it is the time in fact is V minus v-0 over a if I don't show you any argument for V it means V at time T and the subscript zero means V of zero so what this says is you can measure time by having your own clock the clock tells you what time is this but you can also say what time it is by seeing how fast the particle is moving because you know it started with some speed it gaining speed at some rate a so if the speed was so and so now then the time had to be this time can be indirectly inferred from these quantities then you take that formula here and you put it here wherever you see a time T you put this expression so what will you get we get an expression in which there is no T T has been banished in favourites so I'm not going to waste your time by asking what happens if you put it in I will just tell you what happens what happens is you will find the v square is equal to V 0 square plus 2 a times X minus x0 how many people have seen this thing before ok that's a lot look I know you've seen this at the moment I have to go through some of the more standard material before we go to the more non-standard material if this part is very easy for you there's not much I can do right now so let me draw a box just drawing a box you guys means important so these are the two important things I claim now remember I want you to understand one thing how much of this should you memorize suppose you've never seen this in high school how much are you supposed to memorize I would say keep that to a minimum because what the first formula tells you should be so intuitive that you don't have to cram this we are talking about particles of constant acceleration that means when I take two derivatives I want to get a then you should know enough calculus you know it has to be something like a t-square and the half comes from taking two derivatives the other two you know are stuff you can add and you know what what you're adding those things because the particle has a headstart it's got an initial position even at t equal to zero it has an initial velocity so even without any acceleration it will be moving from why not – why not plus VT the acceleration gives you an extra stuff quadratic in time once you got that one derivative will give you the velocity then in a crunch you can eliminate T and put it into this formula but most people end up memorizing these two because you use it so many times eventually it sticks in you but you should try to memorize everything so we are now going to do one standard problem where we will convince ourselves we can do we can apply this formulas and predict the future given the present so the problem I want to do there are many things you could do but I'd espect one this is the one with their own numbers so I can do it without a calculator here's a problem that is this building and it's going to be 15 meters high and I'm going to throw something and it's going to go up and come down something I throw up has an initial speed of 10 meters per second so we have to ask now now that my claim is you can ask me any question you want about this particle and I can answer you you can ask me where it would be nine seconds from now eight seconds from now how fast will it be moving I can answer anything at all but what I needed to do it to do this problem was to find these two unknowns so you got to get used the notion of what will be given in general and what is tailor-made to the occasion so we know in this example the initial height should be 15 meters and the initial velocity should be 10 and for acceleration I'm going to use minus G and to keep life simple I'm going to call it minus 10 as you know the correct answer is 9.8 but we don't want to use the calculator now so we call it minus 10 consequently for this object the position Y at any time T is known to be 15 plus 10 t minus 5 T squared that is the full story of this object of course you got belittle when you use it for example let's put T equal to 10,000 years what are you going to get if you put t equal to 10,000 years of 10,000 seconds you're going to find Y is some huge negative number now said right there what's wrong with that reasoning so you cannot use a formula once it hits the ground because once it hits the ground the fundamental premise that a was a constant of minus 9.8 or minus 10 is wrong so that's another thing to remember once you get a formula you got to always remember the terms under which the formula was derived if you blindly use it beyond the bella D you will get results which don't make any sense conversely if you get an answer and it doesn't seem to make sense then you got to go back and ask am i violating some of the assumptions and here you will find the assumption that the particle had that acceleration is true as long as it's freely falling under gravity but not when you hit the ground now if you dug a hole here till there then of course it may work till that happens okay but you thought you got into every time when you this is so obvious in this problem but when you see more complicated formula you may not know all the assumptions that went into the duration and quite often you'll be using it when you should all right see this you agree is a complete solution to this miniature tiny Mickey Mouse problem you give me the time and I'll tell you where it is if you want to know how fast it's moving at a given time if you want another velocity I just take the derivative of this answer which is 10 minus 10 T so let me pick a couple of trivial questions one can ask one can ask the following question how high does it go how high will it rise to what height will it rise so we know it's going to go up and turn around and come down we're trying to see how high that is so that is a tricky problem to begin with because if you take this formula here it tells you why if you know T but now we are not saying that we don't know the time and we don't know how high it's rising so you can ask how am I supposed to deal with this problem then you put something else that you know in your mind which is that the highest point is the point when it's neither going up and are coming down if it's going up that's not the highest point which is coming down that's not the highest point so at the highest point it cannot go up and it cannot go down that's the point where velocity is zero if you do that let's call the particular time T star then 10 T star minus 10 is equal to 0 or T star is 1 second so we know that it will go up for one second then it will turn it on and come back now we are done because now you can ask how high does it go then you go back to your Y of 1 and Y of 1 is 15 plus 10 minus 5 which is what 20 meters by the way you will find that I make quite a lot of mistakes on the blackboard you're going to find out you know one of these years when you start teaching that when you get really close to blackboard I you just cannot think it's definitely some inverse correlation between your level of thinking and the proximity to the blackboard so if you find me making mistake you got to stop me why do you stop me for two reasons first of all I am very pleased when this happens because I'm pretty confident that I can do this under duress but I may not do it right every time but if my students can catch me making a mistake it means they're following it and they're not hesitating to tell me secondly as we go to the more advanced part of the course we'll take a result from this part of the blackboard let's take it into the second part and keep manipulating if I screw it up in the beginning and you guys keep quiet we'll have to do the whole thing again I would I would ask you when you follow this thing to do it actively try to be one step ahead of me for example if I'm struck by lightning can you do anything can you guess what I'm going to say a next you have any idea where this is going you should have a clue if I die and you stop that's not a good sign okay you got to keep going a little further because you should follow the logic so for example you know I'm going to calculate next when it hits the ground you should have some idea how I'll do it because this is not a spectator sport if you just watch me you're going to learn nothing like watching the US Open and thinking you're some kind of player you will have to do you will have to shed the tears and you got to bang your head on the wall go through your own private struggle I cannot do that for you I cannot even make it look hard because I raised this problem from childhood so there is no way I can make this look difficult that's your job all right so we know this point at one second is 20 meters so let's just ask one other question we'll stop one other question maybe when does it hit the ground and with what speed at typical physics question so when does it hit the ground well I think you must know now how to formulate that question when does it hit the ground is when is y equal to zero by the way I didn't tell you this but I think you knew that I pick my origin to be here and measured Y positively to be upwards and I call that 15 meters you can call that your origin if you call that your origin your y 0 will be 0 but ground will be called -15 so in the end the physics is the same but the number is describing it can be different but you have to interpret the data differently but the standard origin for everybody is the foot of the building you can pick your origin here some crazy spot it doesn't matter but some origins are more equal than others because there are some natural landmark there here foot of the building is what I call the origin so in that notation I want to ask when is y equal to 0 so ask when Y is equal to 0 then I say 0 is 15 plus 10 t minus 5 T Square or I canceling the 5 everywhere and changing the sign here I get T Square minus 2 t minus 3 equal to 0 that's when it hits ground so let's find out what the time is so T is then 2 plus or minus 4 plus 12 over 2 which is 2 plus or minus 4 over 2 which is minus 1 R 3 okay so you get two answers when it hits the ground so it's clear that we should pick three but you can ask why is it giving me a second solution anybody have an idea wives yes that's correct so her answer was if it had if it was the full parabola then we know it would have been at the ground before I set my clock to zero first of all negative time should not bother anybody T equals zero is when I set the clock a measured time forward but yesterday would be T below minus one day right so we don't have any trouble with negative x so the point is this equation let's not know about the building doesn't know the whole song and dance that you went to building any threw up a rock it knows nothing what does the mathematics know it knows that this particle happened to have a height of 15 at time zero and a velocity of 10 at time zero and it is falling under gravity with an acceleration of minus ten that's all it knows if that's all it knows then in that scenario if there is no building at anything else it continues a trajectory both forward in time and backward in time and it says that whatever seconds one second before you set your clock to zero it would have been in the ground what it means is if you release the rock at that location one second before with a certain speed that we can calculate it would have ended up here with precisely the position and velocity it had at the beginning of our experiment so sometimes the extra solution is very interesting and you should always listen to the mathematics when you get extra solutions in fact when very famous physicist called Dirac was looking for the energy of a particle in relativistic quantum mechanics he found the energy of a particle is connected to its momentum this P is what we call momentum and its mass by this relation so particle of mass m and momentum P has this energy so you solve for the energy you get two answers now your temptation is to keep the first answer because you know energy is not going to be negative particles moving it's got some energy and that's it but the mathematicians told Dirac you cannot ignore the negative energy solution because it tells you there's a second solution and you cannot throw them out and turns out the second solution with negative energy was when the theory is telling you hey there are particles and there are anti particles and the negative energy even properly interpreted will describe anti particles so the equations are very smart the way the physics works is you find some loss of motion in mathematical form you put in the initial conditions or whatever you solve the equations and the answer that comes you have no choice you have to accept the answer if there are new answers besides the one you were looking for you got to think about what they mean and that's one of the best things about physics because here is a person who is not looking for anti particles he was trying to describe electrons but the theory said there are two routes in the quadratic equation and the second route is mathematically as interesting as the first one it has to be part of the theory and in trying to adjust it so it can be incorporated you discovered anti particles so always amazing to us how we go into the problem our eye or mind can see one class of solutions but the math will tell you sometimes there are new solutions and you got respected and understand and interpret the unwanted solutions and this is a simple example where you can follow what the meaning of the second solution is it means that the problem you post there is more than the answers that you could imagine here it meant particle that was released from the ground earlier that it meant something much more interesting namely anti particles accompanying particles they are going to accompany particle surely as every squad ratting equation has two solutions all right so now in this problem we can do something slightly different and that's use this expression here and I will do that then I will stop for today if you are asking questions like how high does it go but you don't ask when go to the highest point then you don't have to go through the whole process of finding the time at which it turned around I don't know where that is that disappear in the blackboard then putting the time equal to one second into this formula if the question of time is not explicitly brought up then you should know that you have to use this formula so how do we get it here well we say at the top of the loop when the course of it comes down the velocity is zero therefore you say zero square equals initial velocity squared plus two times minus G that's my acceleration times y minus y zero if you solve for that you find y minus y 0 equals V zero squared over 2g and if you put in the V zero I gave you which is what 10 100 over 20 which is 5 meters so y equal to y 0 plus 5 meters and that is the height to which it rises I think we got it somewhere else we found the maximum height to be 20 meters another thing you can do is you can find the speed here if you want to find the speed there you put the equation V square equals V zero squared plus 2 times minus G what is y minus y 0 the final Y is 0 the initial Y is minus is 15 you solve for that equation then you will find the final velocity so if time is not involved you can do it that way I want to derive the last result in another way then I will stop and that's pretty interesting because it tells you use and abuse of calculus I'm going to find for you this result using calculus in a different way so from the calculus we know DV DT is equal to a so multiply both sides by V now you have to know from elementary calculus that V times DV DT is really D by DT of B squared over 2 now I hope you guys know that much calculus that when you take a derivative of a function of a function namely V squared over 2 is a function of V and V itself as a function of T then the rule for taking the derivative is first take the V derivative of this object then take the D by DT of V which is this one the right hand side I'm going to write this a DX DT this much is standard but now here is what I'm I'm going to do something which somehow we are told never ever to do which is to just cancel the DTS you all know that when you do the Y DX you're not supposed to cancel that D that's actually correct okay you don't want to cancel the D in the derivative but this happens to be completely legitimate so I'm going to assume it's true and I'll maybe take a second explain why it's legitimate what this really means is in a given time delta T the change in this quantity is 8 times the change in this quantity therefore you can multiply both sides by the delta T but the only thing you should understand is delta T as long as they're small and finite will lead to some small and finite errors in the formula because the formula is really the limit in which Delta X and delta T both code is 0 so what you have to do is multiply both sides by delta T but remember it's got to be in the end made vanishingly small as long as we understand that we can do this cancellation and this says on the left hand side the change in the quantity V squared over 2 is the change a times the change in the quantity X so add up all the changes or what mean by integral same thing add up all the changes the change in V squared over two will be the final V squared over two minus the initial V squared over two and the other side will be eight times the change in X X minus X naught and that's the formula I wrote for you u squared is V zero squared plus two a X minus X naught so the point is whenever you have derivatives with something over DT do not hesitate to cancel the DTS and think of them as a Delta V squared over two is equal to a times Delta X this will be actually true as long as both quantities or vanishingly small they will become more and more true as Delta X and Delta B squared become vanishingly small in the limit in which they're approaching zero the two will be in fact equal if Delta X is a finite amount like one second this will not be true because in the starting equation Delta X and delta T and Delta V Square or all assumed to be infinite decimal so don't hesitate to do manipulations of this type and I will do them quite often so you got to understand when it's okay and when it's not okay what this means is in a time delta T if this quantity changes by some amount in the same time delta T that quantity changes by some amount then keeping the delta T equal to some number we may equate the changes in the two quantities provided it is understood the delta v square over two is a change in V squared over two in the same time in which the particle moved a distance Delta X then by adding the differences we eliminate time and we get this final result alright so if you go to your website today you will find I have assigned some problems and you should try to do them they apply to this chapter then next week we'll do more complicated problems that involve motion in higher dimension how to go to two dimensions or three dimensions