 Motion is so common in the world, we probably really don't even give it much thought every day. I mean, things move all the time in the world around us, it's so average as to be easily dismissed. But thankfully, a lot of people who have come before us have thought a lot about things moving, why they move, how they move, why they don't move, and so forth. As a result of that, we have a very complete understanding of motion in the natural world. I mean, this is really something we're going to spend the bulk of the course on, is objects in motion or objects coming to rest. That's no accident, and after all, the universe itself appears to be constantly in motion. Consider riding in this car. We speed by buildings that appear to be fixed with respect to the ground, but from our perspective, they're moving past us from the forward part of the car to the backward part of the car. We're passing other cars, for instance, this one right here, and we're being passed by other cars over on the left side that you can't see. We change lanes, the car moving from one side of the highway to the other side of the highway, even as it continues its forward motion. We can look at the speedometer, which is a measure of the speed of the car, the distance that it's traveling at each moment in time with respect to the roadway, which is considered fixed on the earth. Now, one thing to keep in mind about speedometers in the United States is that because this is a country that has not adopted the SI system of units, we don't use the standard units of meter and second or kilometer and second to make measurements of speed, that is, values and distance with respect to time. Instead, we use the unit of miles per hour. Now, of course, one can convert seconds to hours and one can convert meters to miles. It's all a simple algebraic exercise that will exercise a lot in this class, and in fact, being able to convert from one system of units to another is an important skill that all scientists need to have, because after all, numbers come in many shapes and sizes. This is all totally normal to us, though, this kind of motion, or as we begin to slow down here in a traffic jam, this sort of lack of motion. But again, what is speed? I mean, we can think about those units again for a second, meters per second or miles per hour. If I'm driving, for instance, at a speed of 25 meters per second, what that converts into in miles per hour is a convenient speed in the United States. It's about 55 miles per hour, which used to be a typical maximum speed on many roadways. Now, what this means, of course, is if I'm moving at 25 meters per second, it means that after one second, the vehicle that I'm in will have traveled 25 meters, and that seems like a simple, perhaps even a trivial statement. But in fact, it's quite powerful. For instance, I can make predictions about the future based on this information. If this vehicle's speed remains a constant 25 meters per second, then I know that after two seconds, I'll have gone 50 meters. After three seconds, I'll have gone 75 meters, and so forth. It seems so simple, but it's extremely powerful predicting the outcomes of the natural world. It's one of the tiny powers that physics is a study of energy matter space and time has and gives to us as human beings. Now, in fact, if I were able to maintain that speed from the front door of my house to the parking lot at SMU, where eventually this car will come to rest, I can tell you a lot of things. For instance, I can tell you confidently the amount of time it would require me to make that journey from front door to parking lot at work. The power of such information, the ability to predict outcomes, that is really the power of physics as a way of understanding the natural world. So, for instance, if I were able to maintain that speed of 25 meters per second or 55 miles per hour, constantly throughout the entire commute with no speed ups or slow downs, it would take me about 1,288 seconds to make the journey from front door to parking lot. Now, if you convert that into something a little bit more relatable, that's about 21 minutes. And remarkably, under good traffic conditions, given the highways we have accessible to us, a commute under those ideal conditions takes about 25 minutes door to parking lot at work. That's not bad, considering the fact that the highway that we're on now experiences traffic jams and that its typical maximum speed is actually more like 70 miles per hour and not 55, but things kind of average out. And you wind up with an answer that's about correct. If I maintain a speed of 55 miles per hour from my door to the parking lot at work, it takes about 21 minutes to commute, I would predict. And in reality, on ideal conditions on these highways, it would take about 25 minutes. Of course, with traffic, it's really more like 40 minutes or so. Now, of course, I've made an assumption here. That is that the vehicle that I'm in is traveling on an exact straight line the entire time from the front door to the parking lot. In physics, we would call that kind of thought game an approximation. It's not really what driving from home to work is like. It comes close enough for us to make some reasonable predictions about what's going to happen. And of course, motion is not always on a straight line. It's very complicated. As I've said here, we change lanes. We accelerate. We slow down. We respond to traffic conditions. We exit the highway. We get on the highway again. We get on side streets. We turn left at stop signs. Motion is very complicated. In reality, it takes place in three dimensions. Left to right, going over bumps or into dips in the road, takes us into the vertical dimension up and down. And of course, ideally, we want to continue going forward the whole time. But sometimes you have to go a little bit backward to make progress in a traffic jam. We'll explore these topics of motion, starting with motion in one dimension in this section of the course. Let's move now from a more conceptual attitude toward motion and in specific terms, motion along one direction in space. That is one dimension. It does something a bit more concrete and mathematical. The key ideas that we're going to attempt to absorb in this section of the course are as follows. We're going to try to find a way to describe an object's motion. And that's going to be the case even if the object is large and extended. Now, the trick we're going to use here is that if we have a case where all points on an object, even an object that takes up substantial volume in space, if all of the points on that object are moving together at the same time in the same direction, then we can get away with a bit of an approximation as we'll see. We're going to learn to locate an object in space along a coordinate axis and describe that location. We're going to learn how to describe the displacement that is the change in spatial location of an object from some reference point. And we're going to define things like the average velocity and the average speed of a moving body. And we're also going to see how that's different from what is known as the instantaneous velocity. That is, the speed or velocity of an object at a very specific moment in time, as opposed to over a long period of time. Let us begin by thinking about how to describe complicated objects. Now, the world is full of such things. In fact, they're far more common in the world than the oversimplifications that you'll often find in textbooks. But those oversimplifications have a purpose to them. And that is that as long as we can get away with a few key assumptions, and as long as we are clear with ourselves about what those assumptions have been, we can make approximations. That is, reasonable representations of the natural world. And those representations are, in a sense, capturing the very essentials of a given physical situation. So for instance, consider a light rail train. This is not a simple object. It has many moving parts, for instance, the wheels or the arm on top of the train that makes contact with the electrical transmission lines that power the vehicle. It's got doors that slide open and slide closed. It's got passengers in some cars in some places and not in others. And of course, it has an engine that's completely distinctive from a passenger car that tapers in the front to make it aerodynamic and so forth. But the good news is that despite the fact that a light rail train is an extremely complicated thing, we can actually describe such a complex object by making a very simple observation. And that is that for a properly working train, even though it is big and it's extended in space, it takes up a large volume, when it moves along a rail, all the atoms that make up the train are bound together and they all move together in the same direction all at once, say forward along the track or backward along the track, depending on which way the train is going. So we don't need to describe every single point, every single atom that makes up this train. We only need to pick one representative point and all of the other points on the train are going to do the exact same thing. If this point is moving forward, all the other points are moving forward. If this point is moving backward, all the other points are moving backward. So this will be our approximation. We will imagine taking all of the train and compressing it down to just one tiny representative mathematical point in space and wherever that point moves, the whole train can be thought to be moving. All the atoms, even if they're distributed around that point in the real train, would be moving in the same direction at the same time as this one representative point. And so instead of drawing an entire train and trying to describe its motion atom by atom by atom, which is an impossible task, let's pick a single point near the center of the train and imagine all of it as being described by just that one point. And there you go. This little green dot is our train. So it doesn't seem so silly anymore when you look in your physics textbook or in another science textbook and you see a simple dot or a circle or a sphere being used to represent something that's actually in reality much more complicated. As long as you are honest and state upfront what your approximations are and the motivation for them, you can go ahead and do things like this because I promise you it will greatly simplify the act of sketching up the problem which is something you should be doing when you are asked to do this or when you are trying to sketch a problem in order to think about it a little bit more carefully and clearly. Now that we have learned to make an approximation that allows us to say something about where the train is by representing it by a single physical point, we now want to be able to represent the location of the train, that point at a given moment in time. And so to do that, we need to define a coordinate system. Now, conveniently since trains move along tracks this restricts their motion. They can only move forward along the track or backward along the track. And of course, forward and backward are subjective directions but we can imagine standing next to a train track with the front of our body facing one way along the track and thus facing away from the other direction along the track and to find the direction we're facing in is forward and the other direction is backward. That would allow us to establish a arbitrary but nonetheless useful sense of forwardness and backwardness. So this would represent a single dimension in space forward and backward. So we can say because the train is restricted to moving along the track either forward or backward that it executes a kind of one dimensional motion. That is its motion can be entirely described as forward or backward along the length of a track system. So we might represent that movement in space, that ability to move forward or backward using a single line, a coordinate axis which we will label for the purposes of simple references to geometry as an X coordinate axis. A vertical axis would be a Y axis and an axis that goes into and out of the screen would be a Z axis. But since we're not concerned with motion up and down or left and right in this case we are really going to restrict ourselves just to a X axis, a single axis in space. We might imagine that where the train begins its motion we can label as point zero. The units of direction, the units of distance along this axis are kilometers. And so the train can be said for instance to begin its life at zero kilometers. Now, if the train moves in what we have defined as the forward direction that is in the direction you are facing along the tracks then we can say that it's moving along the positive X direction. And so for instance along this direction numbers increase in magnitude from zero to one to two kilometers to three kilometers, et cetera. Of course all numbers in between these tick marks are possible but for now just let us concern ourselves with the numbers that are labeled here on this coordinate axis. If the train were instead to reverse its direction and go backward from point zero then we would note that it is going in the direction in which negative numbers decrease in size. Negative two is smaller than negative one, negative three is smaller than negative two, et cetera. So this is the negative X direction and this allows us to denote whether or not the displacement that is the change in location and space of the train is in the positive direction, forward or in the negative direction backward. So now we can mathematically represent our notions of forward and backward. So here for instance is the train at time zero seconds. So we will label this T with a subscript one. We might imagine that 60 seconds later at a second time T with a subscript two that the train has now moved two kilometers in the forward direction. So here now is the train dot and we can define the displacement of the train over that 60 second time period using some mathematical notation. Displacement in general will be denoted with the Greek letter delta preceding the coordinate that is changing. So in this case, the train is moving along an X coordinate axis. It's X coordinate in kilometers in space is changing with time and we would denote that displacement as a capital Greek delta preceding X or delta X. So in shorthand verbal nomenclature in physics, this displacement would be referred to as delta X. This is defined as the location of the train at the later time T two minus the location of the train at the earlier time T one. And we would denote that as X two and X one. So we simply take the difference of these two numbers and if we read those off of our coordinate axes, the later displacement is two kilometers. The earlier location is zero kilometers. And so the full displacement over the 60 seconds is a two kilometer positive displacement in the forward direction. Let's imagine taking our train back to its original location X equals zero kilometers at time T one equals zero seconds. And this time when we run the clock forward 60 seconds, we observe that the train displaces not in the forward direction, but in the backward direction. So now it's a spatial location after 60 seconds is not positive two kilometers in front of where it started, but negative two kilometers behind where it started. This is perfectly fine. Trains go in reverse all the time. So this time we observe that the displacement defined identically as before delta X is the later distance minus the earlier distance X two minus X one. This time we have the later distance is negative two kilometers. The earlier distance is zero kilometers and negative two minus zero leaves negative two kilometers. This is now a negative displacement. Delta X is a quantity that is less than zero. So it is important to note at this point that when we speak of displacement, we are speaking of something that is a directional quantity. That is the sign positive or negative in front of the resulting number carries information about what has just happened. If the sign is positive, the train has moved in the forward direction along the X axis. That is it has moved in the direction of increasingly larger spatial locations in kilometers. If instead the sign of the displacement is negative, it tells us that the train has been moving backward. It is displaced backward from the origin zero kilometers. The negative sign tells us that we are moving into increasingly smaller negative numbers. So negative one, negative two, negative three, negative four kilometers, et cetera. The sign carries tremendous information. You must never lose the sign positive or negative in front of the displacement when you are asked about the displacement. Now, let us consider questions about how the spatial displacement and the time information taken together can give us new information about what the train is doing. If we want to understand how place changes with time, then we want to define the quantity known as velocity. So imagine again the train at time zero, that is the first time T1, zero seconds. And then 60 seconds later, the train is now at two kilometers along the X direction in the forward direction. This is the same situation we had a couple of slides ago. We can define a quantity known as the average velocity. And you'll see why it's called average velocity in a moment, it will become crystal clear. The average velocity of the train is merely defined as the displacement, which remember is a signed quantity that carries information about direction, positive or negative. It's the displacement divided by the time taken to make the displacement. So if we want to run through the math of this, V for velocity with a subscript AVG for average, so V average or average velocity is defined as the ratio of the displacement in space and the displacement in time, delta X over delta T. And just like displacements in space, displacements in time are the later time minus the earlier time. So this ratio just winds up being X2 minus X1, all of that divided by T2 minus T1. And if we plug in numbers, we find of course here that the displacement is two kilometers, the time displacement was 60 seconds. And if we take the ratio of those and put them in a calculator and calculate it out, we find that this comes out to be positive 0.033 kilometers per second, about 33 meters per second. So this is very close to typical highway speeds. This is not an unreasonable speed for a train to be executing when it's in a straightaway section of track and able to increase its velocity in order to move passengers from point A to point B more quickly. Now, the reason that this is known as the average velocity is because between zero seconds and 60 seconds, for all we know, the speed of the train might actually vary. Maybe the train speeds up on a little straightaway, but then as it approaches a slight bend in the track, although it's still moving forward along the track, it slows its speed so as not to fly off the track and cause a derailment, which of course is very bad. But what we all care about is just how long it took to get from point A to point B and what the distance was between point A and point B. If all we care about are those two pieces of information, then we can immediately compute the average velocity that the train was moving at, which takes account of the fact that maybe it spent more time moving more slowly and less time moving more quickly, and that ratio gives us some information about the typical speed that the train was moving on as it's made its journey from point A to point B. So really all we're doing is just using the beginning point and the endpoint information to bracket the motion of the train and get some quantity that gives us an average amount of information about the kinematics, that is the way that the movement occurs, the actual details of the motion occurs in that time period. Now, we can again do this exercise with displacement and time, but this time consider a time zero, the train located at zero kilometers, and at time two, 60 seconds later, the train located at negative two kilometers. What this time has been the average velocity? Well, again, we run through the calculation. The average velocity is defined as the displacement in space divided by the displacement and time. That gives us negative two kilometers of displacement over 60 seconds in displacement and time, and this time we wind up not with a positive number, but with a negative number, negative point 0.33 kilometers per second, or again, about 33 meters per second. This time, however, it's in the backward direction. So again, this sign in front of the displacement carries information that's crucial to understanding the motion. It makes a big difference if the train you care about is moving in the wrong direction while you're waiting for it. So it's important when somebody tells you how fast a train is going that it also is moving in the direction you care about that is toward you to pick you up and take you to your final destination. If it's moving away from you at that speed, it doesn't help you very much. Now, speed is a related concept to velocity. Speed is the magnitude of the velocity. It is a quantity that is always positive, regardless of whether or not the velocity is positive. So in order to get the speed, or the average speed in this case, all you have to do is ignore the positive or negative sign in front of the velocity and just quote the number after the sign. So in this example, and in the previous one, the speeds have been exactly the same. The speed was always 0.033 kilometers per second. Whereas in the first example, the velocity was positive 0.033 kilometers per second. And in this example, the velocity was negative 0.033 kilometers per second on average. Now let's do one last exercise where we do a graphical representation of changes in space with changes in time. So we're still talking about motion only along one dimension, that is along one coordinate axis. But we can represent all of the information of the motion, the displacements along the spatial dimension, as well as the displacements in time using a two-dimensional graph with the vertical axis representing where in space, for instance, the train is located, and the horizontal axis representing where in time those spatial measurements are being made. So it is true that we're now looking at a two-dimensional graph, but it's still a representation only of motion in one dimension, one spatial dimension. So this is something to keep in mind when you're reading these graphs. Always read the axes, always read the units on each of them. Note I've switched to meters on distance, and I'm still keeping seconds on time. Don't just look at the graph and go, ah, it's two dimensions, therefore this must be two-dimensional motion. Read the graph, and when you're making a graph, think very carefully about what you need to represent and how you're going to represent it. So let us now draw two points on this graph. Point one is down here at the origin in both time and space, zero and zero, zero seconds, zero meters. Point two, which is at four seconds, represents a distance along the x-axis of three meters. So I've now drawn two points on this space-time graph, and I can ask a question. What is the average velocity represented by this drawing? Think about that for a moment. What would you say in response to that question? Given a first space-time point here, and a second one later in time, four seconds later giving a distance of three meters, what would you say is the average velocity represented by this drawing? Pause the video here, think about this for a moment, make some notes on a piece of paper, and when you're ready for the answer, resume the video. Here's the answer. In order to get the average velocity, all you have to do is take the displacement in space, divided by the displacement in time. Well, the displacement of space can be read off of the vertical axis. The final displacement point is three meters from the origin, and the original one is zero meters from the origin. In time, the later time is four seconds, and the earlier time is zero seconds. So this is just three meters of displacement divided by four seconds of time. And if you take that ratio, three meters over four seconds, you should have found that the average velocity is 0.75 meters per second. And the fact that it's a positive number is important. In this case, the displacement was all in the positive direction, but nonetheless, that sign, that positive sign is implied here. If you're feeling really paranoid, you can always write the positive sign at no extra cost. So if you got this, three meters divided by four seconds by reading off this graph, three meters minus zero meters, four seconds minus zero seconds, and take the ratio of those two differences, and you got 0.75 meters per second, you're doing well. If you didn't, you'll have another opportunity in a moment to think about this again in the context of the definition of average velocity, and to try again. The insight that you should take away from this is that graphically, when you represent this kind of motion using a space-time graph like this, the average velocity is just the slope of the line connecting the first point to the second point. If you want to check the math on that, pause the video here, write down the equation for a line y equals mx plus b, and keep in mind that y in this case is distance, and x is time. All right, so really, we have x equals mt plus b. b is the y-intercept of that line, which in this case is zero. It intercepts the y-axis at zero. And m, of course, is the slope of the line. So given that information from a high school geometry class, that y equals mx plus b gives you the information about a line, and from m, you can get the slope. So solve for m, verify, in fact, that graphically, the average velocity is just the slope of the line connecting point one and point two. Now, what if I leave this graph here? I have point one and point two, point one, point two, nothing has changed. What if I now show you how the speed of the object actually changed from time to time during this whole four-second interval? So what if I now show you this? Aha, it wasn't that the object, whatever it was, kept a constant speed the entire time. In the first second, it went from zero to two meters, but in the last three seconds, it went from two meters to three meters. And if you think about that for a moment, the speeds are obviously changing from the first second to the last three seconds. So given this information, the blue line followed by the red line now representing the way that distance changes with speed during the whole four-second trip, what now is the average velocity of this object? Again, pause the video, take some notes, write some things down, think about this and see what answer you can come up with. If you said that the average velocity is still 0.75 meters per second, you are absolutely correct. What? Why? How can that be? In the first second, you go two meters, but in the last three seconds, you only go an additional meter. So how can it be that the answer is still 0.75 meters per second? I mean, obviously this is a much more complicated motion than we thought was going on originally. But in reality, nothing has changed because after four seconds, we have still displaced only three meters from the origin. And it's still true that three meters divided by four seconds is 0.75 meters per second, the average velocity of the object over the whole period of time. Now, of course, if I asked you what's the average velocity between zero and one second, you're gonna get a different answer. And if I asked you what's the average velocity between one second and four seconds, you're gonna also get a different answer. But from zero to four seconds, the answer is still the same, 0.75 meters per second. And so the insight you should take away from this is that this graph really begins to illustrate the difference between the concept of instantaneous velocity, which we'll dig into more in a minute, that is the velocity at a given moment in time, an average velocity, which maybe over a long period of time when fluctuations in the velocity might kind of cancel each other out to give you the same number that you were going to get anyway over the four seconds. It is true that in the first second, the speed is much greater. The slope of the blue line is quite steep compared to the black line. And it's also true that in the next three seconds, which is represented by the red line, the slope is much less steep than the black line or the blue line. So it's moving more slowly. So it moves quickly at first, then much more slowly later. And that just averages out to the same answer we got in the first exercise, 0.75 meters per second, the beauty of physics. Now let's take a look at a much more complex motion using this space-time graph. So again, you're gonna get one more crack at this concept of average velocity. Ready? Here we go. Now, this very fancy curve represents the way the distance changes with time during the four-second trip. I mean, look at this, it's magnificent. It seems to be relatively straight, but then it bends very gently until it finally sort of straightens out again and we arrive at the last space-time point, too. So what this time is the average velocity that the object has during this trip. Again, pause the video, take some notes, write some things down. And when you're ready for the answer, go ahead and unpause the video. Well, here's the answer. If you said nice try, but the average velocity is still 0.75 meters per second, you are right. What? Why? Look at this curve, it's ridiculous. This isn't two straight lines stitched together. Look how much this thing bends like a gentle elbow and then heads into 0.2. It doesn't matter. Again, nothing substantially has changed. We start out at zero, zero. We end at three meters at four seconds. The average velocity is just the displacement divided by the time required to make the displacement. Nothing has changed. It's still three meters over four seconds. It really doesn't matter that the speed was doing all these fancy things and in between that space was changing in some very non-linear way with time. The average velocity remains the same. So the insight we can take away from this is that really this graph better illustrates the difference between instantaneous and average velocity. In this representation of this motion, we have a very complex curve that relates displacement and time. And at each moment, the velocity is clearly varying. I mean, not so much here, maybe not so much here, but definitely in this region, there is a gentle change in the way the displacement occurs with time. But it doesn't matter because the ultimate endpoint in space is still three meters at our endpoint in time of four seconds. And so the average velocity remains exactly the same regardless of all this fancy behavior in between. Now, looking at this curve for a moment, however, we really can begin to dig into the concept of instantaneous velocity and do so mathematically. I mean, for instance, it's very clear that at this tiny moment in time, if I could slice through this curve, that the slope of this part of the curve is much steeper than over here, where if I slice in a very tiny moment in time here, the slope of the curve is more close to level. It's a much flatter slope than the one over on the left side of the curve. It's very hard, obviously, to exactly describe this in terms of a bunch of little straight lines adding up to make this curve, but nonetheless, mathematics allows us essentially to do that. It allows us to ask questions like, if I were to look in a very thin little slice of time and define the average velocity in just that little slice, that might get me closer to understanding what the exact velocity at that exact moment in time is. And in fact, that's the basis of calculus. It's taking problems where you have, say, a continuous change in space with respect to time, and it's breaking the problem up into tiny little slices, let's say, in time in this case, and looking at what the displacement is doing in those tiny little slices as we make the slices thinner and thinner and thinner. And that allows us to get a mathematical definition of velocity at an exact instant in time, or instantaneous velocity, rather than the velocity averaged over some long period of time, in this case, four seconds. And the mathematical notation for this is that the instantaneous velocity, or simply just the velocity v, is defined as indeed the average velocity, delta x over delta t, but in the limit that we make very tiny slices in delta t, smaller and smaller and smaller sending the size of the slice down to zero dimension in time, giving us access to the smallest unit of space. And this can be represented, this whole cumbersome thing, this limit as delta t goes to zero of delta x over delta t. This thing can be represented by the much more compact notation of calculus, dx over dt, where d is a symbol that's sort of like the Greek delta, except that the Greek delta represents large chunks and the little d represents infinitesimal little chunks, the tiniest thin slices of distance you can imagine. That's what dx stands for. It is representative of the linguistic concept of a thin slice. And so we're looking at thin slices in space over thin slices in time, and that allows us to define mathematically velocity at a given instant in time. We'll come back to this concept a bit more, but this is the foundation of calculus in mechanics and the study of motion. So if we wanna understand this a bit graphically, we could imagine, again, first computing the average velocity over the whole spatial time of four seconds. So I could enclose the curve in a box whose corners are at the locations on the function where four seconds is located and zero seconds are located. And then just draw a box like this. This is my time slice. It's not very impressive because it encloses all time in this problem. But I could imagine keeping the box centered on two seconds and slicing more thinly. So instead of looking at a delta t of four seconds, I could look at a delta t of two seconds from one second to three seconds centered on two. And again, the box now not only shrinks horizontally in time, but because we're enclosing a very specific segment of this function where there's less overall curvature, there's still some curvature, but less compared to including all of this, we now have enclosed a thinner slice in time of the function. We're beginning to just zone in now on what we mean by velocity at a given instant in time. And I could take another slice that is in fact just 0.5 seconds in width but still centered at two seconds. This is allowing us to conceptually understand what it would mean to ask the question, what is the velocity of this object at t equals two seconds? In order to answer that question, we have to take a very thin slice of the displacement in a window of time around two seconds and calculate the average velocity in thinner, in thinner, in thinner slices around two seconds. That will mathematically allow us to answer the question, what is the velocity of this object at two seconds? At means two with no uncertainty on it, but this definition allows us to conceptually slice very thinly in time around two, so thinly that it in fact doesn't even matter that this slice has any width. And you can imagine now going down to a tenth of a second, a hundredth of a second, a millionth of a second, a billionth of a second, a hundred trillionth of a second until we find a slice so thin that we get a very straight little segment enclosed in this little green box and that little segment would allow us to compute an average velocity that so short in time it represents an instant of time in the universe. That's conceptually what's going on in all of this. And so what we learn is that as we narrow the time window and just keep recomputing the average velocity in our heads or imagine trying to measure it on this graph by using thinner, in thinner time slices, we are sampling a thinner and thinner slice of this displacement curve until in the limit that we sample an infinitesimally thin time slice we're ultimately measuring the average velocity over an infinitesimally small moment in the motion. That's what instantaneous velocity means. It is impossible to measure at an exact instant in time, even with the best measuring instruments that humans have ever developed, you'll never do better than a certain tiny uncertainty in time. So it's better to adopt the concept that you think about motion in space and time in thin slices and those thin slices allow you to define mathematically an instantaneous quantity like velocity or as we'll see in a moment this language applies equally to the concept of acceleration changes in velocity with time, not just space but also velocity itself can change in time and that is the concept of acceleration that we will get to next. Let's look at some real one dimensional motion that is movement of an object along only one dimension. I have here a small cart with some very low friction wheels and I've marked it with a green ball so that it's easier to see in the video and all I'm going to do is give it a push and let it go and once I stop making contact with this it will essentially have no forces acting on it any longer. While my hand is in contact with the cart I am accelerating it, I am changing its state of motion from no motion to moving but once I release my hand from the cart in principle there are no more forces acting on it. Now of course in reality that's not that simple and we'll look more at that later but let's take this approximation and run with it for now and see what we can learn. The cart will be in free motion along the x-axis which I have defined as the horizontal direction in this video. Let's watch the motion and see what we can observe. Let's rewind and look at that motion one more time but a bit more slowly. Using special software on the iPad it's possible to analyze the motion in this video and get information about space and time and how they relate for the movement of the object but also how space and time together can be converted into velocity information. This graph shows exactly this kind of analysis. The cart was tracked by the software in the video and as you can see there's a roughly straight linear relationship between the position of the cart shown on the vertical axis in the top diagram and time on the horizontal axis in the top diagram. Each dot represents a sampling data point that's captured by the software and you can see that they line up very well on a straight line that makes an angle with respect to the horizontal. This looks like constant velocity and indeed that is confirmed by the computer software in the lower diagram. The lower diagram is generated by the analysis software using the same kinds of techniques, average velocity and instantaneous velocity that we've developed previously in this lecture. The velocity of the cart is calculated and shown on the vertical axis and time again is shown on the horizontal axis and as we can see apart from a few glitchy places in the time sequence of the data points where maybe there was some small force acting on the cart perhaps my hand at the beginning or a bump in the table, something like that. This velocity versus time is extremely flat. It looks as if the cart has indeed executed one dimensional motion very faithfully with a constant velocity or at least a nearly constant velocity so close to constant that we can't really tell the difference. So in this lecture I hope that you have learned how to recognize how to measure the spatial position of an object and how that position changes with time, how to convert that information into information about the motion, specifically the velocity, the average velocity, the instantaneous velocity, and also come to some understanding of the difference between speed and velocity.