 I'm being passed by a truck, but I'm in motion relative to the Earth. I'm passing this car. It's in motion relative to the Earth as am I, but I'm going faster than it, so it appears to me as if the vehicles are going backward. Am I at rest or is the other train at rest? Is it moving or am I moving? Are we both moving? Absin external reference points in these kinds of questions. We find ourselves often lost and even a little bit confused about what's really going on. Is the other train moving or am I moving? These are simple questions, but they lead to extremely deep insights into the nature of the cosmos. Let's begin to explore this concept, relative motion. Let's begin a mathematical exploration of relative motion or simply relativity. I'm calling this part one because as I will hint at at the end, the subject of relative motion or relativity is quite a bit deeper than we would get into early on in an introductory course, but I would like to preview some of the importance of this subject at the end of this lecture. The key ideas that we're going to explore in this first lecture on relativity are as follows. We're going to recognize that when two observers use different coordinate systems, each one affixed to their own frame of reference, those two observers may draw different conclusions about the reasons for events happening in the universe. We're going to learn to relate those observations however in those two distinct coordinate systems using something which is known as the Galilean transformation. It has some observations that are a core part of the transformation, but it also has at least one key assumption built into it as well as we will see later. We will then understand that observations and assumptions that underlie that transformation have consequences, and that's what I'm going to close with at the end here. That's why this is only part one of this particular subject for a physics course at this level. To proceed, let's conduct something known as a thought experiment. One of the most famous physicists in the history of the field who was originally born in Germany would have called this a gedanken experiment, but the translation of that roughly speaking is a thought experiment. An experiment we carry out based on the principles of what we know about the world, but we only do the experiment in our mind. Let's consider two actors in this experiment, one person named Sally and one person named Bob. Let's say Sally and Bob are friends, and they happen to have both gone to the train station at the same time, but they're going to different destinations, and so Sally gets on one train and Bob gets on another, and they arrange it so that the windows of their seats are facing each other from one train to the other. They're sitting on these adjacent trains, and the trains originally begin parked in the station. While they can't talk to each other through the glass of the window, so they get out their mobile devices, and they begin chatting with each other using those devices, so they're using Skype or FaceTime or just the plain old phone or something like that, and they're so distracted with their conversation that neither of them notices the very slightest acceleration that suddenly causes one of the trains to start moving forward. Instead, they merely notice by looking out the windows at each other and suddenly realizing that somebody's moving, that one of the trains must be in motion, although they don't know which one because they didn't notice the acceleration. And so they each look out their windows, and they look at each other's trains, and they remark simultaneously on their mobile devices, hey, you're moving. And then a disagreement begins, no, says Sally, you're moving backward, my train is standing still. No, says Bob, it's you who are moving forward, and my train is sitting still. So they both agree on what forward and backward are, forward perhaps is east along the tracks, backward is west along the tracks. And there they don't disagree, you know, Sally might say you're moving westward and Bob might say you're moving eastward, no problem there. But what they disagree on is who is doing the moving. Now, the reason that they have a disagreement at all is neither of them notice the acceleration that started one of the trains moving, the train that's moving is now doing so at a constant speed. So there are no sudden surges that can tell one of them, hey, my engine's powered up. And neither of them can see the ground or any other external reference points. They can't see the platform, they can't see a lamp post on the ground, something that would be fixed to the ground, and if that were moving, they would know that they were on the train that's moving. Okay, it'd be ludicrous to think that suddenly the ground is rotating underneath the train, it must be the train that's moving over the ground. But they don't have any reference points like that. All they can see are the other person's train moving past them, either eastward or westward at a constant speed. So the question here from the physics perspective is, well, who's right? Who is the keeper of the correct perspective in all of this absent any reference points in the outside world that would immediately tell who's actually moving, let's say with respect to the surface of the earth? Well, if you ponder that for a little bit and realize that there may be situations, in fact, you may have even been in one yourself, where you don't have any external reference points and you're stuck thinking about what might be going on with no information to make the decision, you can tell what a difficult situation this must be. And from a physics perspective, the answer is that in truth, they're both correct. Each of them is right in their own way. They certainly agree on the sort of gross events, the most general things that are going on. A train is certainly in motion, but absent any external reference of any kind, they are going to always disagree on which train it is that's moving. That said, thanks to motion and its nature, which we have been exploring now for a little bit in this course, they are actually able to use space and time information to relate one observer's set of observations to the others. And using the mathematics of motion, they can provide explanations, Sally to Bob and Bob to Sally, as to why the other person observes what they observe. And this whole concept is known as relative motion or very simply relativity. We're going to begin to study the subject now using what we already know about motion. But one thing I would like to emphasize at the beginning is in the popular misconceptions of what relativity is, it's often said that relativity is a statement that anybody is correct independent of an absolute truth. But in fact, in physics when we talk about relativity, it's not about what is different for both observers that is the key. Certainly Sally or Bob could disagree on whose train is moving. If he had a third observer, we might be able to sort out that problem. But fundamentally, in order for them to relate their observations to one another, it's not what's different that matters, but what is constant or the same for both of them, that is what matters. And fundamentally, as we're going to see, relative motion is pivoting not so much on what's distinct about what two observers observe, but rather what uniform things, what constancies in nature that they believe they can use to relate their observations. Now to begin this discussion, we have to establish some kind of mathematical framework, some kind of toolkit that's going to allow us to describe what's going on here, keeping this analogy of the trains and Sally and Bob in our minds. Now I picked trains for a reason. They're long, straight, and they mostly travel on straight tracks, and so they make an excellent one-dimensional motion example. Now one key idea that we're going to need here to describe that kind of situation in our thought experiment is something known as a frame of reference. And a frame of reference is merely a coordinate system, imagined or otherwise, you could literally make a coordinate system and mark it. And that coordinate system is possessed by each observer so that each observer can make measurements and then those measurements can be compared from one frame to the other. For instance, by communicating them over a mobile device. They carry that coordinate system with them. So we can imagine that wherever Bob is sitting, that that defines the origin of Bob's coordinate system. And east along the train is the positive X direction and west along the train is the negative X direction. And Sally similarly has her own coordinate system, Sally's coordinate system. She is the origin of that coordinate system. When she moves, the origin goes with her, just as if Bob were moving, the origin of his coordinate system would go with him. And again, we have to think about what's the same here. And since these two trains are parallel to each other and both observers agree what east is and what west is, for Sally as well, even though the origin of her coordinate system may move with her, eastward along that coordinate system is positive and westward is negative. So they both agree on what are positive directions in space and what are negative directions in space. They may simply disagree on who's at the origin of each other's coordinate systems because each is at the origin of their own. And that's illustrated here on the right. I've tried to generate two one-dimensional coordinate systems. You can almost think of these as trains that are parked next to each other at the station. But this is a moment in time when, for instance, if we imagine that Sally is on the train on the top and Bob is on the train on the bottom and each is sitting at the origin that is zero of their coordinate systems, they're slightly displaced from one another. Their origins are not in the same place. This may represent a moment in time after one of the trains has started moving and one person is displaced along the origin of the other person's coordinate system. So one person says, oh, well, I'm still at the origin of my coordinate system. I haven't moved. And then they look over at the other person and they see that person is not sitting at the origin of their coordinate system and they say, you've moved. And of course, the other person can look at the first and say, no, no, no, I'm at the origin of my coordinate system and in my coordinate system, it is you who have moved and you can begin to see the origin of the disagreement here. And since either Bob or Sally can make the claim that each of them is at rest and the other person is moving, that's all perfectly valid, we as thinkers about this problem, we have to make a choice. We have to define one of them as being at rest. We have to make that decision. We may not even really know who's moving, but it turns out it won't matter. The mathematics works out the same. All that changes in the end is if you made the wrong decision, you wind up just realizing that they're, for instance, real speed of the train that's moving your points in the direction opposite the one you thought it did. So we will always define one coordinate system in a case where we have two as being at rest and the other as being in motion relative to the first. And really that's what relativity is. You make a choice. I choose this coordinate system to be at rest. And then based on that, if any other coordinate system is moving relative to that one, the other one is in motion. That's how you make these definitions sane at the beginning. Otherwise, there's no way to make any decisions whatsoever. Now, the one that's defined to be at rest is typically denoted as the frame capital S. And the one that's assumed to be in motion is typically denoted as the frame capital S with a little tick mark next to it known as a prime. So I'm gonna call this S prime for the remainder of certainly this lecture and likely every other time I talk about this when you interact with me about the subject. Now, we could imagine in order to kind of concoct a situation that you might be more comfortable with that there's a third person and we are that third person and we're standing on the platform near the two trains and because we're standing on a platform that is a fixed to the surface of a planet and we define the planet to be at rest and the platform therefore a fixed to it is at rest and we standing on the platform not moving with respect to the earth are also at rest. We could make the decision that we in our privileged position are the absolute frame of reference where all things are defined to be at rest. And one of the trains which is not moving is also in our frame of reference and therefore is at rest, but the other one which is moving relative to our frame. And then that would allow us to make the decision about which one is S and which one is S prime, okay? And the train that's moving of course is said to be the frame S prime. It carries a coordinate system with it that with respect to the other trains or to ours is moving. Now let's continue to use this concept of a frame of reference and let's think about some relative measurements. Let's do a simple relative measurement at first to build up a little bit of comfort with this idea. Let's begin by considering a situation where yes, Sally and Bob, S prime and S, they do in fact have their own coordinate systems and that they begin displaced from one another. Sally maybe picked a window seat at one end of a car and Bob picked a window seat at the other end of the car next door. So they're not sitting window to window facing each other. One is at one end of a car, one ends at one's at the other. They can talk to each other over their mobile devices. They can look out their windows and down the line of the train and see each other but they're not at exactly the same horizontal location along the track. In that case, their origins are displaced from one another but neither train is moving yet. So Sally's not moving, Bob's not moving. Each of them agrees that the other's train is at rest with respect to theirs. Let's consider that situation. And to illustrate this, for instance, imagining that maybe there's a suitcase, you know, one unit of distance ahead of Sally on the train. She might look at that suitcase and say, ah, that suitcase, that suitcase in my coordinate system for I am at zero, is located at one unit of distance in my frame ahead of me. And she might, if she were to write down her observations and maybe using some program, share them with Bob via her mobile device, might draw a picture like this. So here is Sally, here is the suitcase and she notes this as position X prime one in her coordinate system and it's one unit of distance, whatever the units are ahead of her on her train. Now, where would Bob, where would an observer at the origin of frame S claim that to be located? Well, I drew a little dotted line to help you out here, to help you connect points on Sally's coordinate system, frame S prime with Bob's coordinate system, frame S. And if you follow the line down to Bob's coordinate system, you see that while Bob is sitting way over here, he's displaced back from Sally in relative space. But that's the origin of his coordinate system, where he's sitting. And he sees that same suitcase located not at X equals one in his coordinate system, but rather at X equals two units of measurement. So the answer here is that in frame S, X is equal to two. In frame S prime, X prime, the measurement along that coordinate system is equal to one. Now, it's not that one of them is wrong, they're each right in their own coordinate systems. The question is, can we relate their observations? Can we find a mathematical equation that given Bob's observation, we could get Sally's or given Sally's observation, we could get Bob's? Well, sure, neither train is moving. There's a fixed displacement, a constant displacement between where Sally is and where Bob is. Let's denote that displacement as just a lowercase letter C for the time being. And since Sally, S prime, is shifted ahead of S, Bob, by one unit along the respective axis that Bob has, we can relate their observations. So for instance, if Sally reports a measurement of the suitcase being at one unit ahead of her in her coordinate system, we would note that to get Bob's observation, all we have to do is add one unit to Sally's, and that will give us what Bob saw, which is two, so X prime plus one equals X, and that gives us two, in fact, the answer that we can see by reading it off of this picture over here. So we have a mathematical rule that we could generically apply to relate observations in one frame to those in another. If we know the offset at any moment in time, then we can relate observations in the S prime frame to those in the S frame. If we know where the S prime frames, say, origin is with respect to the origin of the S frame. So we take like coordinate points in the two systems, and we determine their offset, one frame relative to the other. We call that C, and we simply add that. So I'm taking the offset to be measured from frame S to frame S prime. So any number we get in S prime, we have to add C to it to get the number in S. And so the equation is quite simple in one dimension. It's that any position measurement in X is given by the position measurement in X prime plus the offset. And you can rewrite this equation to solve if you know the position in S, if you know X, you can get X prime by subtracting C from X. So you can go from measurements in the S system, Bob's system, to measurements in the S prime system, Sally's system quite easily. Just rearrange this equation. From Sally's perspective, Bob is offset by negative one unit in her frame. So you can see a minus sign just crops up here in order to relate these observations. It's easy to relate these observations when something is the same for both of them. Now we've looked at relating positions. What happens now if something is moving in frame S prime? What would frame S see? So what about speeds of objects that we now place in this frame? I mean, we can imagine that Sally, for instance, throws a ball forward in her frame or something like that or somebody is running down the aisle of the train in Sally's train. These are all things that we can imagine. So let's imagine that. Let's imagine that now there's an object that can move in frame S prime. What does the motion look like in frame S? And again, let's consider a situation here where the two frames are both at rest with respect to each other. Frame S prime is not yet moving, but they're displaced from one another. So let's consider an object that in frame S prime is moving to the right with a speed of one meter per second. So I've kind of illustrated that here. It starts off originally at negative one in the S prime coordinate system. And then one second later, it's at zero. So it's where Sally is located, for instance. And then another second later, the object is again moving at one meter per second, gone one meter forward. So now the units on this scale could be meters, for instance. So it's gone one meter forward in frame S prime. And the question we wanna answer is, okay, it's moving at one meter per second in frame S prime and Sally's frame. What's its speed as observed in frame S in Bob's frame of reference? Well, from the sequence of time snapshots that are available here, the object at zero seconds at negative one meter, the object at one second at zero meters, the object at two seconds at one meter, all in frame S prime. We can see that for every second and in both frames, the object actually advances by a displacement of one unit in the positive direction. Let's look at Bob's perspective on this object. Bob says, oh well, the object began not at negative one, as you say, Sally, but rather at zero, where I am. And then one second later, it was one unit ahead of me, one meter ahead of me. And two seconds later, it was two meters ahead of me. It went another meter in that second. So you'll note that the deltas from one unit of time to the next in both frames are indeed the same when these frames are at rest with respect to each other. In both frames, both observers agree that delta X and delta X prime are the same. They're one unit because the speed is one meter per second. So this is great because when the frames are at rest with respect to each other, the velocities of objects are the same as measured by observers in each frame. Both agree that the object is moving forward. Both agree that the object is moving at one meter per second. They don't agree on where it is in space, but we know how to relate those observations to one another using what we did on the previous slide. Let's look at this mathematically, and the derivative calculus offers us a glimpse as to why it worked out this way. If the offset between the frames is, again, a constant and unchanging value, frames are not moving yet with respect to each other, and we call that value C, then we know that in the prime frame, the velocity of the object will simply be given by its rate of change with time of its X prime coordinate or DX prime DT. If you want to think about gross average velocities rather than instantaneous velocities, you could instead say delta X prime over delta T, but you'll get the same answer, and I prefer to use the calculus here to nudge you a little bit to begin to think more in this infinitesimal, tiny, instantaneous, thin slice way. Now, what about in the S frame? This is the speed in the S prime frame. What about in the S frame? Well, in the S frame, it's the same equation, but with the primes gone, the speed of the object or the velocity of the object in the S frame is DX DT or delta X over delta T, if you want to talk about gross average velocities. But we have an equation that relates measurements in X to measurements X prime in the other frame so we can substitute for X in this equation and take the time derivative of that equation, X prime plus C, whatever it is that Sally measured plus the offset of Sally's frame with respect to Bob. Remembering from our previous look at this that when you have the derivative of a sum of two things, that the derivative operation distributes to the two things, you can distribute the derivative with respect to time to X prime and you can additionally distribute it in the sum to C. So you have a time derivative of X prime plus a time derivative of C. We arrive at our final answer and that is that this reduces to just DX prime DT. That is the velocity in Bob's frame of reference which was certainly DX DT in his frame also is DX prime DT which is exactly equal to V prime which we started with over here in Sally's frame. In other words, V is equal to V prime by this long chain of equivalents. Why is that? The offset is constant. The time derivative of any constant is zero because constants do not change in time and the time derivative fundamentally answers the question how does this thing that I'm acting on change in time? And if it's a constant and only a constant, the answer is not at all, zero. So the first term in this may not vanish but the second term certainly does, DC DT is zero. And we get our equivalence. What we observed in a picture over here, we see also confirmed by mathematics which really is the reliable way to do this because mathematics after all is the descriptive language of the natural world. Let's now consider what happens when we're in a situation that's much more like the train thought experiment that began the part of the lecture. Now in that case, we had one of the frames of reference moving with respect to the other. So by our convention, let's denote the moving frame as S prime and so that's going to be the top frame in my cartoon down here. And let's imagine that it is moving relative to S which is down here at a speed of one meter per second. So let's imagine that and let's draw that motion as time moves forward. So starting here at zero seconds, the origins of the two frames, the origins of these two one dimensional coordinate systems coincide. But at one second, frame S prime has moved one meter and so now at time equals one second, the origin of its coordinate system is no longer aligned with zero in frame S, it's now aligned with one meter in frame S. And if we let time lapse one more time, one more second, now the origin, which is still zero in frame S prime, is located at X equals two in frame S. And so we see that the origin of S prime is moving to the right in what both frames agree is the positive direction along their respective X axes. And the displacement now between the origin of the S prime coordinate system and the origin of the S coordinate system is increasing. So unlike before where the displacement, which we denoted as the letter C was a constant, now the displacement depends on time. And in fact, if you sort of play this back a little bit here, you can see that if frame S prime is moving relative to frame S at a speed of one meter per second in the positive X direction, that it's also true that the displacement between these two coordinate systems is likewise increasing at one meter per second. So now, because the displacement is no longer a constant, it varies with time. And any time now that we do some math and we take the time derivative of C, the displacement between the two frames, we have a new relationship. That relationship is that the rate of change of C with time, that is the first derivative of C with respect to time is actually exactly equal to the speed with which the frame S prime is moving. And of course, since displacement is a signed quantity, really this is the velocity of the frame S prime with respect to S. So we are no longer in a situation where frame S prime is at rest with respect to frame S and they just have displaced origins in their coordinate systems. Now it's that the origin of frame S prime is actively moving along the X axis of frame S and increasing the displacement with time. Of course, if we imagine a situation where the speed with which frame S prime is moving is exactly zero, what happens to the time derivative? Well, the time derivative also becomes zero. That is to say the displacement between the two frames of reference is a constant and we return to the situation we looked at in part one of this question of speeds in frames of reference. Let's now revisit the thought experiment regarding Sally and Bob in the trains. A person on the platform outside of the trains in a kind of privileged position where they know for sure that they're standing on a platform that's affixed to the earth and they're able to call that a sort of frame of absolute reference of rest. They're gonna see that Sally's train is actually the one that's moving and it's Bob's train that's actually at rest. But because Sally and Bob both lack any reference points in some other frame, they can't know this. All they see is that one of them is sitting and one train and the other one appears to be moving. However, with our privileged viewpoint, we're able to label Bob's frame S and Sally's frame S prime. S for the frame that we call at rest, S prime for the frame that we say is moving with respect to the one we're calling at rest. Now, if the train is moving at this velocity, VS prime, then we can begin to understand Sally's point of view and of course we can also at the same time understand Bob's point of view. Bob believes that it's Sally that's moving because he can claim he's at rest and that Sally is moving forward at a speed VS prime. But Sally believes it's Bob that's moving because she can claim that she's at rest. Now, we know better, but Sally cannot know that. But something for her is in fact different. In her frame, it's Bob's train that appears to be going in the other direction. So Bob might say, ah, Sally, your train is moving in the positive X direction and Sally would say, no, no, no, I'm at rest. Your train is moving in the negative X direction. Now, they both agree that the speed is VS prime, but the direction is different, which makes the velocity different. And so we can see from this perspective that it must be true that what Sally calls VS is the negative of what Bob calls VS prime. And this allows us to relate their observations one to the other. So what if Sally now throws a ball toward the front of her train while she's moving? Now, she doesn't know she's moving, she just takes the ball and she throws it forward. Let's ignore gravity for the time being. Let's pretend the ball just moves along the X axis and only along the X axis. It doesn't rise, it doesn't fall. What does she see? And what does Bob see? Let us imagine that Sally has thrown the ball at a speed in her frame of reference, which will label VS prime, of one meter per second in the forward direction. Now, at the same time, her train is also moving forward. And she doesn't know this, but from Bob's perspective, that speed is one meter per second in the forward direction. So let's watch what happens in this picture on the left-hand side over here. Let's watch what happens to the ball from Bob's perspective as time marches forward. The ball is represented by this blue dot that begins at time zero seconds at the origin of both coordinate systems. I have chosen time zero seconds to define the moment when the origin of Sally's coordinate system in S prime and the origin of Bob's coordinate system in S are the same. If we let time go forward at one second, we can then begin to observe what each of them sees. So again, at time zero seconds, Bob sees the ball at x equals zero and Sally sees the ball at x prime equals zero. But at one second, Sally's train has moved forward one meter relative to Bob's coordinate system. And so at one second, Sally sees the ball at one meter forward in her frame of reference. Remember, she's still at zero. But her zero point in Bob's frame of reference has now moved forward to one meter in his frame of reference. So where does Bob see the ball? Well, Bob sees the ball at x equals two. So it was at zero, at t equals zero seconds, but now it is at x equals two, at t equals one second. The origin of Sally's coordinate system is only at one meter in Bob's, but the ball is at two because the ball is moving at a speed that appears to be in addition to the speed of Sally's train. Let's look at t equals two seconds. At t equals two seconds, Sally is now at two meters in Bob's coordinate system. In Sally's frame of reference, the ball has only gone two meters from where she threw it. One meter per second, two seconds later, it's gone two meters. But from Bob's perspective, it's now at four meters in his coordinate system. So at time zero, it was at zero. At time one, it was at two meters. And at time two, it's at four meters. So from Bob's perspective, the ball isn't moving at one meter per second as it would appear to Sally, but rather at two meters a second. Every second it moves two meters in Bob's frame of reference, but how? How can the speed be different? What's going on here that makes this happen? And the answer is, is that the speed of the ball has, from Bob's perspective, the speed of Sally's train added to it. So she says, I threw the ball at one meter per second. And he said, well, that's true, but your train is also moving at one meter per second. So I see it moving at two meters per second, and that's how we relate our observations. And we can see this again with calculus. So again, let's remember that in Sally's frame of reference, which is S prime, the speed of any object that she throws will be denoted V prime. So really it's a velocity. It can go in the positive direction. It can go in the negative direction. So it has directional information. And that's the first derivative with respect to time of the coordinate of that object in her frame, which is X prime. Now, transferring over to Bob's frame of reference, frame S, in his frame, the speed of the object will be denoted by V. Again, velocity really. And that is the first derivative with respect to time of the coordinate of the object, the ball in this case, in his coordinate system, which is X, so DX, DT. But again, we know how to relate coordinates in Sally's frame to Bob's frame in frame S prime to frame S. All we do is we take the coordinate as measured in Sally's frame of reference, the S prime frame, and we add to it the offset between her origin and his origin, the origin of frame S prime and the origin of frame S. That's just X prime plus C. We distribute the derivative. So now we have DX prime DT plus DC DT. But remember now that Sally's frame, S prime, is in motion with respect to Bob's frame. So again, consider the picture up here. In one second, her origin had moved one unit in Bob's frame. In two seconds, her origin had moved two units in Bob's frame of reference. She is moving with a frame speed of VS prime. And that frame speed is just the time derivative of the offset C. That's it. So DC DT is not zero this time because she's moving. DC DT is what we would call VS prime, the speed of her frame with respect to Bob's frame of reference. So this is just V prime as defined over here. DC DT is what we call VS prime, the speed of her frame with respect to Bob's. And so in the end, we find out that the speed that Bob sees is the speed she sees plus the speed of her frame. Now, if these are velocities, these can be signed quantities. They could subtract from each other and we'll explore that in a moment. It's also important to note that we now have a relationship as well between the coordinate for the ball that Bob measures and the coordinate for the ball that Sally measures. All we have to do is take Sally's coordinate and because this is right now, constant speeds that we're considering, we add to that the speed of her frame multiplied by the time that has passed since zero time for instance. So the X coordinate in Bob's frame is the X prime coordinate in Sally's frame for the ball plus the speed of her frame times time. This looks familiar, right? This just looks like one of those one-dimensional equations of motion. But here it tells us the relationship between measurements in one frame and measurements in another. It's the same equation, but it's employed for a new purpose. Now, the final consideration we're going to do for all this is what if Sally throws the ball backward in her frame of reference? So instead of throwing it in the forward direction and in the direction of increasing positive numbers, what if she throws it in the direction of increasingly negative numbers? What will be the velocity as observed by Bob? Well, we can work this out. Let's do the mathematics first. From the mathematical perspective, the velocity observed by Bob will be the velocity observed by Sally plus the velocity of her frame with respect to Bob's frame. Well, in her frame, the velocity's negative one meters per second. She threw it backward. It's going more and more negative every moment in time. The speed of her frame, however, is still in the forward direction. It's one meter per second going forward. Well, we have one meter per second in the negative direction, one meter per second in the positive direction. These cancel each other out. And so we claim that Bob would observe that the ball in his frame of reference appears to be going nowhere at all. So Bob would seem to see the ball just stand still in the air. Again, we're neglecting gravity in this little cartoon example. In reality, we'll come to what would happen in a moment. Of course, it would begin to fall under the acceleration due to gravity in the vertical direction, but it is true that in the horizontal direction, which is independent of the vertical direction, in this case, the ball would appear to just sit still in his coordinate system along x. Let's confirm this using our picture to the left here. So we have the ball at time zero seconds from both Sally's perspective and from Bob's perspective. And then we advance time one second. From Sally's perspective, the ball is moving backward to increasingly negative numbers. Although I haven't drawn in here, this is negative one in her coordinate system. She's still here at zero. Bob is down here at zero. Look where he sees the ball. The ball is right along x where it was a moment ago. From his perspective, it's not moving at all. Yes, Sally has advanced forward one unit in his coordinate system, but that ball is not going anywhere. And if we go to two seconds, the ball appears to hover right where Sally had been at time zero. It's not going anywhere, but from her perspective, of course, it's now negative two units in her coordinate system behind her, where she's sitting at zero in her coordinate system. So indeed, even graphically, even pictorially here, we see that the ball stands still from Bob's perspective. Whereas from Sally's perspective, remember she's sitting here at zero in her coordinate system. That ball is moving backward with a velocity of negative one meters per second. So the two effects cancel out in Bob's frame of reference, but they don't cancel at all in Sally's frame of reference because she thinks she's at rest and she thinks that she's throwing the ball with a velocity of negative one meters per second, which is true in her frame, but not in Bob's. Bob explains this by saying that the speed of her train is canceled by the speed at which she threw the ball in the negative direction, and then it appears to hover there for him. So I hinted that we need to take a look at motion in two dimensions, and this is just going to be sort of a brief tour. We're not gonna go into this in the mathematical depth that we went into this in one dimension. Extending this to two dimensions is just an exercise in carrying an extra dimension around. It's not that bad. But what if we now think about relative motion in two dimensions? And there's a demonstration I'm gonna show you a short movie of right now. I showed it to you before in the lecture on two-dimensional motion when we began this subject as a contrived case where a cart is rolling along a table at a constant velocity and it launches a ball into the air, and then some distance later, it catches the same ball that it launched earlier. We've considered many cases of two-dimensional motion so far, projectile motion for instance, but we've also looked at circular motion. Circular motion is the one case where because of the tethering of an object with some influence or force, the motion in the X and Y coordinates is not independent. But in all the other cases we've looked at so far, the motion in the X coordinate system and the Y coordinate system are totally independent of one another. And we're gonna revisit that kind of motion here again. And let's remind ourselves of the independence of those kinds of motion, one along X, one along Y. By again, looking at the short slow motion video of moving at a relatively constant velocity along a horizontal surface, launching a ball into the air and then it catches it a few moments later. So let's look at that one more time and see this effect. Now let's think about the perspectives that we can have on this motion. From our perspective, at rest with respect to the surface of the earth, we're sitting behind a camera, the camera's sitting on the floor, the floor is attached to the earth, the table is attached to the earth, that stuff we can all say is not moving. So we can say that we're in the stationary frame, the frame S. And from our perspective, it's the cart that's moving to the right where at rest the cart is moving. And when it ejects the ball out of its top, the ball already has a horizontal component of velocity from our perspective. It was part of the cart, it was moving with the cart. And so even though the cart launches the ball vertically into the air, that ball nonetheless already has an existing component of motion along the horizontal direction. So of course, when the ball is now freed from the cart, it's simply going to continue along the x-axis in the forward direction, even while it has this new component of motion that was granted to it by being ejected out the top of the cart. So that additional component of motion is nice, but it comes with some complications. Of course, the minute that the ball escapes the cart, the only force that's acting on it is gravity. Gravity will slow its velocity in the upward direction and then reverse it. And these two vectors, one horizontal and one vertical, they add together to create projectile motion, just like we've seen before. So this is an excellent example of projectile motion. So this is just like throwing a ball into the air at an angle. If you launch it with an initial velocity in the x-direction and an initial velocity in the y-direction, the velocity in the x-direction, which ignoring air resistance is completely unaffected by any other influences, will simply continue as it started. But in the y-direction, gravity will slow the vertical component of the velocity eventually to zero. It'll reach its apex, its highest point, and then it reverses that velocity and it begins to accelerate downward, say, toward the floor. And that's exactly what happens here in this video. The cart is ejecting the ball. The ball has an initial velocity in the y-direction, but gravity slows it down, it comes to a stop, and then it reverses that motion and eventually it comes right back down to where it started because nothing happened to influence its horizontal component of velocity. So that's our explanation from our perspective. But what about the perspective of somebody sitting on the cart? From our privileged position, affixed to the surface of the earth, we make the bold claim that we're at rest and the cart is moving. And so, of course, the cart and the ball have the same horizontal velocity and the cart simply catches the ball later because they're both moving at the same speed along the positive x-direction. In the y-direction, it's complicated, but in the x-direction, it's simple. But somebody's sitting on the cart, how would they explain all of this? Well, they would make the following argument. Well, no, it's nice that you think you're in a privileged position and you're at rest, but actually I'm also in a privileged position because I'm sitting on this cart and I'm not moving and the cart doesn't seem to be moving with respect to me, so I argue I'm at rest. And then, yeah, at some point, this ball gets shot out of the top of the cart, but I'm not moving and the ball wasn't moving either with respect to the cart in the horizontal direction. And so, of course, the ball goes straight up, gravity reverses its direction and it comes straight back down and the cart, of course, catches the ball. It was under it the whole time. The only reason you think that there's projectile motion going on with some horizontal component is that you're moving backward relative to me. You claim I'm moving forward relative to you, but I say no, no, no. The reason you think that there's a projectile motion is because the ball is staying in the same place along x in my coordinate system, but you're moving backward. So you can see how this relativity stuff can get a little confusing sometimes, but the key thing is to keep in mind what's the same for all observers. We can contrive actually a really complicated situation and in fact, I've done that for your pleasure here. We can actually use cameras in two frames of reference to recreate these two perspectives to show you that each perspective is absolutely valid and we can certainly agree that an object, it gets launched, an object lands someplace. The exact details will vary depending on which frame we're viewing the action from, but we can relate to each other and we'll come back to how it is we're able to relate to each other, really what's going on and why it is we're able to relate to each other, our observations in a moment. But let's take a break and let's take a look at this rather Rube Goldberg-esque contrived experiment that I did along with one of the young physicists in our department, Eric, who assisted me throughout this particular exercise. So let's take a look at this. So what we've done is we have set up a camera in the audience, a camera at the top of a stand and the stand is sitting on a cart that can be pulled. Now Eric is gonna pull me standing on the cart at a constant velocity. In my hand, as you're gonna see in a moment, I'm holding an iron weight with a bright yellow rubber glove on it, don't ask. I'm gonna drop this weight, watch what happens. The weight appears to move along with me even as it's dropped. When it's dropped in the X coordinate system of the room, from this perspective it's starting somewhere over on the left and when it hits the floor, it's moved to the right. Now let's take a look at the exact same experiment, but from the perspective of someone on the cart, we have a camera mounted above the weight wrapped in this yellow glove, again, don't ask. Eric has started pulling me along. I now will let go of the weight and watch what it does in my reference frame. It falls straight down the meter stick that's mounted to this stand. That's my Y coordinate axis and indeed it fell straight down. It never moved forward or backward relative to me on the cart. It's as if it was at rest with me the whole time. And so we can see these two perspectives, one from the frame S and one from the frame S prime, we may disagree on what happened. Either the thing was falling and moving to the right or falling just straight down, but we can relate our observations. So let's just quickly think about the mathematics that would be required to relate observations really in any number of dimensions. And fundamentally this relationship between coordinates in one coordinate system and coordinates in another coordinate system, so one frame and another frame. This thing is known as the Galilean transformation and it's named after Galileo Galilei who did really detailed studies of motion and came to understand things like if I'm carrying an object with me and I'm in motion relative to the ground and I drop that object, the object continues to move along with me. If we have any two frames of reference, we can relate velocities in the two frames of reference using a simple vector equation. It's an extension of the one-dimensional equation that we developed earlier. The velocity in the frame S, which we call the rest frame, is simply the velocity of an object in the frame S prime plus the velocity of that frame S prime with respect to the frame S. This is just the vector version of this addition of velocities equation I showed you a moment ago. Now if we consider a special case where the motion is entirely along one-dimensional axis, so let's say that the motion of frame S prime is entirely along the x-axis in frame S. Then we can say that that velocity vector looks like this and based on this, we can begin immediately relating using the equation we developed earlier for relating positions of moving objects in the two frames just like we did before. In the horizontal direction, in the x direction, the coordinate position x of a moving object in the second frame S prime is given by its coordinate in that frame plus the speed of frame S prime with respect to S times the time that passes. But all the y measurements will be the same because the frame S prime is not moving along y, it's not moving along z, it's only moving along x. And so all the observations that we make along y or z, if you do three dimensions, they're gonna be the same between our two frames. But the x's won't and they'll be related to each other by the speed of the frame times time. And this is where we have to think very carefully about what it is that's been the same. This whole time that allowed us to make observations in one frame of reference map onto observations in a different frame of reference that's moving with respect to the first one. What have been so far the actual observations that we've used to draw these conclusions but what has been the implicit assumption as well that we've used to draw these conclusions? Well, the observations are that observers in different reference frames are absolutely going to disagree on absolute positions in space. There really is no such thing as absolute position in space. I have my coordinate system, I'm at the origin of it. You have your coordinate system, you're at the origin of it. That's okay. You might say that something is at one meter along the x-axis. I might say that something is five meters along my x-axis. That's okay. We can relate using the displacement between our axes, those observations. And that displacement may change with time. We can nonetheless adapt to that situation. We know how to do that by using the relative speeds of the frames to relate coordinates. So we already know that spatial measurements are relative observations. They are relative to each other. They can be different, but we can relate them. But the assumption we have been making in order to relate them has been that something is constant. And we've implicitly assumed that it is time that marches at the same rate for both frames of reference. Go back and look at the exercises that we did before where there's zero seconds, then one second of passing time, then two seconds of passing time. What was implicit in those cartoons that I drew? That time is marching at the same rate for observers in frame S and observers in frame S prime. Now this assumption seemed perfectly reasonable for a long time. And in fact, it was good enough for a long time from the days of Galileo and those first really serious studies of motion using both experiments and mathematics through the time of Isaac Newton, which is basically the next century. And then into the 1700s and 1800s, the development of electricity and magnetism, the laws of thermodynamics, all of that really happened in the 17 and 1800s and culminated in the 1800s. And it was only at the end of the 1800s that it was possible to actually see the effects of time not marching at the same rate for two frames of reference. Now, we'll come to this perhaps later in the course. Certainly if you wanna do special topics on this, I'm happy to do this. But I wanna emphasize that for our purposes for this course, it's pretty safe to assume if I'm in motion relative to you, that my clock advances at the same rate as your clock. And so even if we disagree on spatial measurements, we absolutely would agree on time measurements. And it does turn out that that assumption is fatally flawed. In fact, in reality, time does not move at the same rate for an observer in motion relative to another observer. This may sound a little crazy, but you can do experiments that reveal this. Some of them are quite simple. One experiment you're doing all the time without realizing it is likely that you're using the global positioning system in order to figure out where you are on the surface of the earth. To go to some friend's house, you've never been to before. They tell you their address, you punch it into a GPS map program like Google Maps or Bing Maps or something like that. And you get a dot on a map and it says, here's the best route to get there. How does all that happen? Well, fundamentally it happens because there's a system of satellites in orbit around the earth and they relay information to your phone or some GPS receiver unit in your car. And that information tells you where you are now and how to get to where you want to be in this other location on the surface of the earth. But those satellites are moving extremely fast in orbit around the earth and they have extremely sensitive atomic clocks on them. And it turns out that they're moving fast enough relative to the surface of the earth that the different rate of the passage of time is observable. Their clocks will actually come out of sync with any twin clocks that they were originally synchronized to on the surface of the earth. Now motion plays a big part of this but there are other effects that have to do with relativity that are also in here as well. It's a little complicated to get into now but suffice to say that every time you use the global positioning system you're using a system that is constantly corrected for the drifting of time in two different frames of reference. You might ask yourself, well, if time is different in one frame moving relative to another and spatial measurements aren't the same. What is it that's constant? What is it that allows us to relate measurements in one frame to another and even correct for such effects in the first place? And it's something I hinted at earlier in the course and it is in fact the speed of light. The speed of light is as far as we know a fundamental constant of nature that is unaffected by the motion of the frame in which the light is admitted or observed. So for instance, even though satellites whipping around the earth experience different rates of time and of course different spatial positions as a result of their motion relative to the earth, light signals, radio, that's transmitted to and from those satellites is always observed to move at exactly the speed of light. Even though those satellites are in motion when they emit the radio signals, when they arrive on earth, the speed of those signals is still the speed of light. As Neil deGrasse Tyson has said in the most recent reboot of the series Cosmos, which I would encourage you to watch, there's a law of nature. We don't completely understand where it comes from but we know that it's a law. And that is when it comes to light and the speed of light, thou shalt not add the speed of your reference frame to the speed of light. The speed of light is completely constant and invariant against all motion. It's an interesting fact of nature and we're able to use it to solve this relativity problem, this relative motion problem, but nonetheless, it is one of those interesting features of nature we really have yet to fully grasp. So let's review the key ideas. We've seen that when two observers use different coordinate systems, they may draw different conclusions about the reasons for events happening. Nonetheless, we have learned to relate observations into coordinate systems and really what this boils down to is a series of relationships which are known as the Galilean transformation. They are relationships between spatial coordinates but also velocities between the two frames as measured for different objects in those frames. We've come to understand that observations and assumptions go into these transformations and in the Galilean transformation, the assumption is that time marches at the same rate for all observers regardless of their state of motion and that's gonna work just fine for us in this course. But it is true that as you advance forward in physics and certainly by the time you get to third semester physics, you will have to accept the fact that it is time and space that are both relative to observers in different frames of reference and it is instead the speed of light that is the constant that allows us to unite the observations of those two frames of reference. So that is the reality of the nature that we live in. The fact that time is not quite the same for two observers as long as we keep our frames of reference moving at slow speeds relative to one another far below the speed of light itself, for instance, will be okay. We'll be able to use the Galilean transformations to understand the universe but when you have extremely fast clocks and extremely fast objects holding those clocks they are sensitive to those time drift effects and you can't use the Galilean transformation anymore. You have to use something a bit deeper, something known as the special theory of relativity or if you really wanna go full bore the general theory of relativity. Those are topics for a different lecture but I hope you've enjoyed this little peek into the nuances of space and time revealed simply by a thought experiment about two people who disagree about who's in motion on two trains.