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.  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.  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?  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.


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My name is Jimmy. His name is Jake. And this is the Weekly Dumb in the year 2022. 2021 would have rhymed. But didn't, Jake, how are you? I'm not doing well with the new year, Jim.  Just let us do the show. I'm good. 2022 big year.  Like if recent success shows you that it's not recent success shows you it's not. Only success was the word you were looking for. We'll be back 6pm tonight. Nightly news. Jim, new year, new you. New me. New me. New me. New me. New me. Healthy. Be healthy. Have me tell you about the sports. Jake, can you tell me about the sports? Jim, everyone is talking about it. Zach Skye, Antonio Brown, little bit of a meltdown. He is a silly goose and that's why we're talking about the goose feathers in the Kansas City Chiefs game. Yes, the brown stuff's wild. I'm going to try and do a breakdown of it, but it's interesting touchy subject. This one's not. We had another football coach burn himself on a space heater. No, we are officially a burn pod and it's a solo show now. That's some new year stuff for you. Oh my goodness. I think I just manifested a frog in your throat. I just did my frog thing and then you just went full ribbit on us. I did rib it. Play it back. Yeah, so we had another coach stab next to a space heater and lit himself on fire. This one's a little different than the pants which Harbaugh did. This is his jacket so you can understand how he didn't feel it. More depth.  Feathers everywhere. Anytime there's feathers. We like a good spill. Yes, I want people start burning themselves more on the sidelines of NFL games. Take it back. Jim, speaking of people you don't like, here's almost a breakdown. Almost a breakdown today. We have the Royal Guard trampling a child. The Royal Guard used to be a very real thing. Now they're just kind of like a tourist attraction, but no one told them they're not a real thing anymore. So they're marching around protecting the royal family also not a real thing anymore. It's just a farce. It's just a farce. But there's a lot of people defending this and you can see a little kid gets in the way. And I mean you shouldn't get in the way, but little kids get in the way sometimes. The guy just screams in the most English voice, Make way! Make way! Which we have to admit that part's cool. That's a fun way to announce that. But running over little kids. And then he just tramples this little kid and like steps over him, like posterize him. And everyone in the crowd is like, what? Yeah. But then there are people supporting the Royal Guard. Zach put the YouTube comments right here and it's actually pretty, like everyone's like, no, that's what the Royal Guard does. It's a serious thing. It's like, no, it's not. It's not a serious thing.  No, it's a serious thing trampling little kids. No thanks. Let's go back to the sports, back to the football, back to things going wrong. What happened, Jake? Jim, arguably things went right. Jim, your Philadelphia Beagles, they're winning football games. Hey, proud of you, Philly. Don't care. Ha! It was a record. They're nine and seven. Facts only. Jaylen Hertz was leaving the game. He's getting his claps. He's waving and whammy. High five. Crowd Falls. Yeah. My grandma, my great-grandma used to tell me a story about the ship that was taking off from the dock and everyone ran to one side of the ship to wave goodbye to the people on land and the ship just straight tipped over and a lot of them died. Is that a real story? Just a life lesson. No, it's a real story that my nana used to tell me and she'd giggle when she told it. The equivalent in this would be if the captain of that boat took a picture with all the corpses. What's your hand? 2022. Hertz is obviously the big winner. Inside Job, people are asking, who asked? People are asking, do you think Hertz people, you know, kind of rig this railing? Because he comes out looking great. Yeah, can we get my mic here? I feel like this is a bad look. I don't want that.  Jim, speaking of inside jobs, you're made of fish. In the not sports news, it rained fish in Texarkana. Anyway, I can't find the story, but I loved the interview. It rained fish and the news interviewed a guy to ask him about the fish raining and he gave them all the insight they needed. It was raining real hard and a fish hit the ground. I said, it's raining fish. Well, I saw a fish on the ground. I said, it's raining fish. Wow. Made the cut. How many interviews did it make the cut? Tough on him. He's a hundred percent. I'm not tough on him. The news running that, as if it adds to the story. They have to share the news. He gave a hundred percent factual statement. Local news is hilarious. Hey, what happened here? It's raining fish. Wow. Great insight. So this happens, Jake, when there's something called a tornado, heard of it, and there's something called a water spout. Same exact thing. But if it's over water, it's a water spout. Why don't they call them land spouts? Were tornadoes ever called land spouts? Seems like they should have been called. All the storms are the same stuff. Tsunamis or hurricanes. Like it's all the same.   Scientists. So anyway, the goes over the water and some little fish gets sucked up into the vacuum. Then it goes on the land and they fall. So it can rain fish. You know what the article said? That it can rain frogs as well. I did see that. I know you saw that. Employee of the week. Should we give out the employee of the year from last year? Zach. I was going to give it to us. Oh, I'll take it. Zach. Zach gets it. Zach, to start to know you, let's like, bygones be bygones. Bygones. How good of a phrase is that? Let's let bygones be bygones. Let's let bygones be bygones. If you counted every time the word by God was said. It's a double word. In the country. It's a double word. How many times do you think it was that specific sentence versus any other sentence that has the word bygones? I'd say 80% of the times a human used the word bygone, it was said in let's let bygones be bygones. I might be higher. Yeah, I think that might be low. I've never used bygone in a bygone error. That's another way you could say it. Good job, Zach. Let's have a clean year, Zach. Maybe no death edits. Oh, he's so dead. Your mom's going to be poosed.  The employee of the week.  The employee of the week. I don't know. Who is it? We just did it. We gave out the employee of the year. That was last year. No employee of the week to start the week. Oh, I have three people in mind. Seems high. Maddie Mass for making the Jeopardy! Unfarmed to fame. Rob Schrocco for doing the Mario Kart, but they both don't get it. Job, Zach.  I was trying to figure it out with my wife. We guess it's somewhere between 500 and 700, about 10 a year, but then about 15 a year from like fourth grade to high school. You slept in 500 houses. You did the math. Kate, you were there being bees? I slept in a lot of places. Do hotels count as houses? No, but apartments do. A living, like someone lives there. Like Katie and I slept in 12 in this year.  If I go back, 30 is 300.  Sleepovers and stuff. Yeah, but you sleep over a lot of the same houses. But you just in over the course of a year, it's a couple, especially in college. Comment below. Comment below. That was the Weekly Dome. Today's episode is brought to you by Noom. If you aren't already signed up, Noom is an app that uses a combination of psychology, technology, and human coaching to help you meet the personal health and wellness goals you would like to achieve. If you want to stick to your New Year's resolution in 2022, you can get a free Noom evaluation and a seven day trial when you use our link at Noom.com slash Johnboy. It's a new year. It's a new you. It's a Noom. .com slash Johnboy. Guy dunking today, a drive through. Guy dunking the drive through. Say good morning. Happy New Year. How long can that last? Well, today is the first day to be fair. I know, but how long? The fifth? I took up a day at a face this week. Do you want to hear about it? Yeah. I thought it was that I'd never seen like a full bidet. I never had full access to a bidet. So I was messing with it. And I was like, this is cold. I thought the days are supposed to be warm. And I was touching the sprinkling of the water. And I twisted the knob thinking I was doing the heat full blast bidet into my eye, into my face. Sometimes your fingers freak me out just a little bit. That's not a trick. Do you know who used to do this? Who used to do that? The wood shop teacher. That's how he explained gears to the students. Right.


I mean, we're a court. We really don't know about these things. You know, these are not like the nine greatest experts on the internet. You just listened to associate Supreme Court Justice Elena Kagan joke about how the Supreme Court may not necessarily be the best place to make sweeping decisions about something that they don't entirely understand. And that sentiment right there I think encapsulates the broader view that the justices had towards section 230 in the case of Gonzalez v. Google. And court watchers seem to agree that the court is really apprehensive about making any sweeping changes here. For example, the Hill reports that the justices seemed puzzled during oral arguments, the LA Times deduced that the justices seemed wary of weakening section 230, and the Washington Post gathered that they seem to want to move more cautiously with regard to this particular case. Now, if you watched my coverage of this last week, you'll know that I was sounding the alarm because I was very worried that the Supreme Court justices wouldn't necessarily understand what's being argued here. They don't have a good grasp of what the internet even is and why it's a public utility and why section 230 is absolutely necessary for the internet itself. But it seems as if the Supreme Court justices actually understand the gravity of their decision should they choose to side with the plaintiffs here. Now, let's go to the Washington Post where they're going to give us a broad overview of the justices reaction to the oral arguments that they heard today. Supreme Court justices suggested Tuesday that they might move cautiously in their first examination of the federal law that protects internet companies from lawsuits concerning platforms posting of content from third parties. The justices heard more than two and a half hours of arguments regarding the claim by the family of an exchange student killed in an Islamic State attack that Google's YouTube should be liable for promoting content from the group. But justices across the ideological spectrum said they were confused by the arguments offered by the family's lawyer and worried that the court could undermine an effort by Congress to provide immunity for the platforms decades ago when lawmakers wanted to encourage the development of the internet. Now we're going to listen to a little bit more of what Elena Kagan said during questioning because I think that her questions here in comments kind of demonstrate the court's overall view, at least in my assessment. And that is that any changes that they make could have huge ramifications. And there's a lot of uncertainty. They don't necessarily know what is going to happen if they side with the plaintiffs. So let's listen. Every other industry has to internalize the costs of its conduct. Why is it that the tech industry gets a pass? A little bit unclear. On the other hand, I mean, we're a court. We really don't know about these things. These are not like the nine greatest experts on the internet. And I don't have to accept all Ms. Blatch, the sky is falling stuff, to accept something about, boy, there is a lot of uncertainty about going the way you would have us go. In part just because of the difficulty of drawing lines in this area, and just because of the fact that once we go with you, all of a sudden, we're finding that Google isn't protected. And maybe Congress should want that system. But isn't that something for Congress to do, not the court? I think that you did a really good job of explaining the difficulty in trying to draw these arbitrary lines. And as a liberal justice, in theory at least, you expect her to kind of want to be more rigorous in her questioning against big tech companies. But in this instance, it's not just about protecting big tech companies, it's about protecting all of us. So in order for free speech, and individuals like myself to win, we need Google to win in this instance. Because this is about the internet as a whole, being a public utility. And these social media platforms that exist because of these tech giants, if they are fearful of lawsuits, and they think that allowing us to post content on their websites is going to lead to them going bankrupt or getting sued, then they're just not going to allow us to post. So in order for the internet to exist in the way that it is today, we need section 230, which is essentially arguably the law that created the internet as it is today, that needs to stand. Now, one thing that the plaintiffs were arguing, at least the solicitor general, because they are citing with the plaintiffs here, is that, well, sure, you're saying that this could open them up to a lot of lawsuits, but those lawsuits won't be one. But they're not buying it. And you may be surprised to learn who is actually citing with us here, with internet freedom. And I'm actually honestly surprised by Brett Kavanaugh of all people. The Washington Post continues, Kagan and Justice Brett M. Kavanaugh suggested a ruling on behalf of the Gonzales family could unleash a wave of lawsuits. Kavanaugh did not seem persuaded when Deputy Solicitor General Malcolm L. Stewart, representing the Justice Department and citing in part with the plaintiffs, said few lawsuits would have much likelihood of prevailing. Quote, isn't it better to keep it the way it is? Kavanaugh replied, for us to put the burden on Congress to change that, and they can consider the implications and make these predictive judgments. Justice Clarence Thomas, who has been a critic of big tech companies and the protections they received, said Tuesday that he was unsure how YouTube could be said to be aiding and abetting terrorism when its neutral algorithms worked the same way whether a viewer was seeking information on the Islamic State or how to make rice pilaf. Chief Justice John G. Roberts Jr. wondered whether recommending a similar video to someone who has expressed interest in a subject is not the 21st century equivalent of a book seller pointing a customer asking about sports books to the section of the store where they are kept. Justice Sonia Sotomayor and Kagan told Eric Schnappner, a lawyer for the Gonzales family, that his argument about algorithmic recommendations was very broad. So at the risk of sounding overly optimistic here, because I'm still cautiously optimistic, it does seem, at least to me, as if the justices understand what's its daycare. And they realize that the plaintiff's argument is very, very broad. Now to give you a sense as to what the justices are responding to here, let's hear a little bit of the arguments from the plaintiffs. I think that the task for this court is to apply the statute the way it was written. And if I might return to a point that Justice Alito made, much of what goes on now didn't exist in 1996. The statute was written to address one or two very specific problems about defamation cases. And it drew aligns around certain kinds of things and protected those. It did not and could not have written, been written in such a way to protect everything else that might come along that was highly desirable. Congress didn't adopt a regulatory scheme. They protected a few things. It will inevitably happen, and has happened, that companies have devised practices which are maybe highly laudable, but they don't fit within the four walls of the statute. So what he's essentially arguing is that Section 230 was originally written to protect websites from defamation, but to extend that to algorithms, something that didn't exist at the time that the law was written, that violates the law, right? That violates the intent that the lawmakers had. So in other words, what he's arguing is that an algorithm can in effect aid and abet terrorists by recommending an Islamic State video to a viewer. But the justices aren't buying that, including Justice Clarence Thomas. Now what's remarkable about this next clip that we're going to listen to here is that Clarence Thomas was originally the only justice to my knowledge that was vocally against Section 230, and that's because he's a Trump sick event, and Donald Trump was saying we should overturn Section 230. But Clarence Thomas here is not buying the argument from the plaintiffs here. What if the YouTube, instead of automatically providing this list, which is hard, it's hard for me because I don't see this, I see these as suggestions and not really recommendations because they don't really comment on them. But what if you had to click on something like, for more like this, click here? Would that also be as far as your concern, aiding and abetting? In terms of aiding and abetting, if someone comes to me and says, what's Abaddadi's phone call, phone number, I'd like to call it, and I give him the phone number, I'm aiding and abetting, even if I don't say and I hope you'll join ISIS. Whether we're able to recommendation or not on our view is not the issue here. We tried to say that in our brief. I mean if you call information and ask for Abaddadi's number and they give it to you, I don't see how that's aiding and abetting and I don't understand how a neutral suggestion about something that you've expressed an interest in is aiding and abetting. I'm actually shocked because I never knew that Clarence Thomas had the capacity to be reasonable even minimally. So I agree with what he's saying here. I think that for you to claim that YouTube, but in having this content there and recommending it is tantamount to like aiding and abetting terrorists is insane. It's just, it's an insane argument that the plaintiffs are making and I sympathize with them. I understand that they don't want this to happen to another family, right? I understand their fears here, but changing the internet, destroying the internet potentially as we know it is not the way that you seek change. So what we're seeing here is a sort of skewing of partisan lines on the Supreme Court because traditionally you expect the conservative justices to side with big business here and liberals to kind of be a little bit more skeptical towards big business, but both sides seem to understand here that if they side with the plaintiffs, then the internet as we know it could be a thing of the past. And that's just not necessarily something that should be done by the Supreme Court. The plaintiffs are essentially asking them to behave as legislators when this, if there is going to be a remedy at all, it needs to be in the hands of Congress. And I would agree with that. I don't think that there should be a remedy. I think that there are ways to rein in big tech, such as anti-trust legislation, laws requiring more transparency, even nationalization, if you ask me, but having the Supreme Court litigate this here and make this broad change that affects all of us on the internet, I'm just not okay with that. Now, one surprise was Kintanji Brown Jackson. She of all the justices seems to be siding with the plaintiffs here at the most. And you kind of expect liberals in theory to side more with the individuals who want to hold these big businesses accountable. And that's a good instinct to have, but not in this instance. So let me show you what I mean by this. So her line of questioning towards Google, the defense attorneys here, was very, very aggressive. And I don't know if she's just playing devil's advocate, but what she said kind of worried me a little bit. If you look at the statute, it says protection for good Samaritan blocking and screening. If you take into account Stratman, Oakmont, if those things I thought were like a given, what the people who were crafting this statute were worried about was filth on the internet and the extent to which because of that court case and perhaps others. And so the statute is like we want you to take these things down. And so here's what we're going to do. We're going to say that just because they're on your website, it doesn't mean you're going to be held automatically liable for it. And that's C1. And to the extent you're in C2, you're trying to take it down, but you don't get them all, we're not going to hold you liable for it. That seems to me to be a very narrow scope of immunity that doesn't cover whether or not you are making recommendations or promoting or doing anything else. What I understand the government and the petitioner to be saying is that disseminating even 24-7 disseminating of ISIS videos is protected. The only thing that's not protected is whether you can tease out something about the organization and call it a recommendation when there is no express speech recommending it. It's just the placement of where in the order in which content appears. And that same complaint could be made about search engines. So I think under your view, search engines would not be covered because they are taking user information, targeting recommendations in the sense of their saying, we think you would be interested in the first content as opposed to the content on 1,692 sections. I mean, they have millions and millions of hits for any search result. And if you think those are recommendations and the other side gives you no basis for distinguishing between search engines, then the statute is just very different than what I think the one that Congress was talking about. Because again, if you're going to look at findings and history and policy, this is about diversity of viewpoints, jump-starting an industry, having information flourishing on the internet and free speech. Now, it doesn't necessarily mean that she inherently accepts the argument from the plaintiffs here. And she thinks that Section 230 should be destroyed. Perhaps she agrees with me that there needs to be more regulations on these big tech companies. I think that's a perfectly reasonable stance to take. But to get rid of Section 230, that is something that we need to protect all of us. Again, it's a little bit counterintuitive because in order for the little guy to win, we need big tech to win in this instance. Because if these companies are no longer going to allow us to publish content because they're fearful of lawsuits, then we don't have a voice on the internet. Social media as a utility, the internet as a public utility, is no longer a thing. So we need big business in this one limited instance to win. But that doesn't mean that we shouldn't hold these big tech giants accountable, right? And so Kintanji Brown Jackson, she's seemingly siding with the plaintiffs here, but we can't know for sure until the decision is released. So in conclusion, it seems as if at least based on preliminary oral arguments that we're hearing from today, it's not as bad as I thought the Supreme Court justices, even though they may be ignorant to the internet and technology, they understand the gravity of the situation and how their decision here can have monumental impacts on the entire internet and democracy itself. Now, this is just one case. Remember that tomorrow they're going to be hearing Twitter v. Tom and now, but at least the sense that I got today from the Supreme Court is that they're not so willing to just straight up destroy the internet. And maybe that's because the conservative justices and even the liberal justices just want to inherently side with big tech. Either way, I don't care. What I care about is the result. And if them siding with big business, them siding with Google here, save section 230 and freedom of speech on the internet, then I'm all for it. So I'm a little bit encouraged here, but cautiously optimistic to emphasize