Kinematic quantities: Rates of change

0:00- [Lecturer] Humanity is incredible at prediction, we can figure out the exact velocity an airplane needs to take off, we can even precisely time solar eclipses years in advance. But how do we do that? One of the most fundamental tools is kinematics. (indistinct) does analyze, positions, velocities and accelerations, the building blocks of motion.

0:24But how exactly? Let's find out. So let's start with an example. We have a car at rest, and let's say it starts as engines and it goes in reverse for some reason, it stays there for a while and then it goes forward. Okay, we want to analyze this motion, how do we do that?

0:41Well, our first step would be to convert this animation into some kind of a still image. And one of the best ways to do that is to take snapshots at regular time intervals. So let's say we take snapshots of this car's position, its location every second, and we'll stack it up.

0:56Let's see what that looks like. Here goes. Wow, beautiful, right? Does this make sense? We have taken the snapshots every second. So notice the horizontal now represents the time in seconds. And so what does the vertical represent? Well, it represents the car's position?

1:16So, we can plot the position in meters. So for example, it's saying, it's saying that at time T equals zero, the position of the car is zero, but at time T equals four seconds, the car's position is -6 meters. At seven seconds, the car's position is -10 meters and so on and so forth.

1:33But wait a second, you may be wondering, well, why are the positions all negative? What does that mean? And why is the zero chosen to be over here? Well, the short answer is it doesn't mean anything, you are completely free to choose where your zero is. So for example, we could have chosen our zero position to be somewhere down over here, and the whole graph would shift down.

1:52And now look, all the positions would be positive values. So positive values, negative values completely depends upon your choice of origin, which you know, you can choose whatever you want, so it doesn't matter. So let's just choose this to be our origin, okay.

2:06And now if we join all these points with this, you know, smooth curve, we now get what we call the position time graph. Now remember, this graph does not represent the path of the car, the car is always moving up and down, that's it, right? What this represents is the position of the car at any instant of time.

2:29So now that we know what the car's location or car's position is at every given time moment, we can do some interesting things. First, we can define something called the average velocity. It's a measure of how quickly the car's position is changing, an objects position is changing.

2:45And so mathematically you write it as the change in position, delta X, divided by the time taken for that change. For example, let's say we want to calculate the average velocity of the car from four to seven seconds. Well, how do we do that? Well, for that we'll calculate how much is the change in position.

3:06Well notice, its position has changed, it's gone from -6 to -10, that's -4 meters because it's gone in the negative direction. And how much is the change in time? Well, from four to seven seconds, well that is just three seconds. So the average velocity of the car is -4 divide by three seconds giving us about -1.33 meters per second.

3:28Now, what does this value mean? Does it mean that the car's position changes by -1.33 meters every second? No, because in some time intervals the position might change slower, in some other time intervals, the position might be changing faster. So what this number represents is an average change in the position, that's why it's called the average velocity.

3:51But what does it represent geometrically? Well, geometrically, if you draw a line that passes through these two points, which is called the secant, then look, this ratio represents the slope of the secant. So the average velocity geometrically is the slope of the secant line join the two points.

4:11And I absolutely love this because first of all, notice it's sloping downwards. So as I move forward, it's kind of like I'm going downhill. That tells me immediately that, hey, the average velocity is negative. And secondly, if you look at how steep this secant line is, it gives me a measure of the magnitude of the velocity.

4:32So for example, if I were to ask you what is the average velocity between these two points? Well, I don't know what the actual value is, but look, it's sloping downwards, I'm going downhill, so I know immediately it's still negative. But if you look at it's, you know how steep it is, it's not as steep as this one, so immediately I can say, hey, the average velocity over here, its magnitude must be smaller compared to over here.

4:56Amazing, right? In contrast, look at the average velocity between these two points. Ooh, first of all, notice that now we are going uphill. And so this is a positive velocity, but look at its slope. Oh, it's much steeper compared to this one, and therefore I can guess that its magnitude must be higher compared to this one.

5:18And what about average velocity between these two points? Hey, now it's not sloping at all. It's neither sloping down nor sloping up, it's average velocity is zero, so slope is zero. Amazing, right? That's why I absolutely love looking at it geometrically. And another important thing is that when you want to calculate the slope of a line, you can calculate it anywhere between any two points.

5:37For example, you could calculate the slope between these two points or between these two points, you're completely free to do that, you'll get the same value. Okay, your turn, why don't you calculate the average velocity between nine and 13 seconds? Why don't you pause the video and give it a try?

5:52Alright, what's the change in position delta X. Well, we go from -10 to -2, that's eight meters. This time we are going in the positive direction. And how much is the time interval? Well, from nine to 13 seconds, well that's just four seconds. So the average velocity is eight meters per four seconds, giving you two meters per second.

6:11And geometrically, that is the slope of this secant line. And you can see that, hey, it is sloping upwards, so it should be positive. We get positive, everything makes sense. Now here's an interesting question, ready? What is the velocity at a particular moment in time, say at 10 seconds?

6:27Now there is no time interval, delta T is zero, so there is no delta X, that's also zero. So at first it seems impossible, zero by zero, mathematical error, right? But here's a way to think about it, first, let's calculate average velocity between two points nearby.

6:43Well, we know how to do that, we'll draw a secant line and calculate the slope of that line. But that is not the actual value, that's a good estimate, but that's not the exact velocity at 10 seconds. So what we can do now is we can make that estimate better by choosing the two points, making them closer.

7:02So you see where we're going with this? The closer the two points, the better the estimate. And in the limiting case, when the two points just merged together, look, our secant line turns into a tangent at that particular point, and so that means the slope of the tangent represents velocity at that particular instant.

7:21So the way we write it, we say velocity equals just like before delta X or delta T, but we now take the limit that delta T tends to zero, and we call this the derivative of position with respect to time. So, when you calculate average velocity, it's the slope of the secant line between the two points.

7:41But instantaneous velocity is the slope of the tangent line at that given instant. And what is special about instantaneous velocity is that it directly tells you how fast the car is moving. If this number is bigger, that means it has a very high speed. The magnitude of instantaneous velocity directly tells you the speed.

7:59Okay, now that we have a tool to calculate the instantaneous velocity at every single point, let's use that to draw a velocity time graph. In the velocity time graph, we plot velocity versus time. And how do we draw this? Well, we just have to draw the tangent at every point and analyze it slow.

8:20So let's do that, let's start with zero. What's the velocity at T equal zero? Well, for that we have to draw a tangent line over here. And the tangent over here look is perfectly horizontal, which means the velocity is zero, the slope is zero, so the graph is gonna start from zero.

8:37Okay, what happens to the tangent as we move forward? Well, let's see. Ooh, ooh. Notice that the tangent line is becoming more and more steep, isn't it? And it's in the negative direction. So the velocity is increasing in the negative direction. At about two seconds in this slope, let's say it's about -2 meters per second.

8:56That means from zero to two seconds the velocity increases in the negative direction, that means the car is speeding up in the negative direction. And that's exactly what we saw in our animation, isn't it? Okay, what happens after two seconds? Well, let's see, from two seconds to about five seconds, look, the slope remains exactly the same.

9:17So can you pause and think about what the graph would look like now from two to five seconds? Well, since the velocity is a constant at -2 it just stays -2. So it's gonna be a horizontal line until five seconds. Okay, what happens next? Well now look at about seven seconds, the slope goes back to zero.

9:40So that means the car comes to a stop and we saw that yeah, it did come here and then it stops. So it slows down and stops. So from five to seven seconds its velocity goes back to zero. Then from seven to about nine seconds it stays zero. And we did see the car was stationary for some time over there.

10:00So from seven to nine seconds it stays zero, the velocity stays zero. And then what happens after that? Well, from nine seconds onwards, look, the velocity just keeps increasing in the positive direction. That means the car keeps speeding up in the positive direction.

10:15And again, that's what we saw. So what would we look like over here? Well, let's say the velocity at this point at 14 seconds is about five meters per second. So then that means at 14 seconds, the velocity over here would be five meters per second. So that means from nine to 14 seconds the velocity increases from zero to five meters per second.

10:35And this is our velocity time graph. Now guess what we can analyze using the velocity time graph? We can analyze the acceleration. I mean just like how we defined average velocity as, you know, a measure of how quickly the position is changing, we can now define average acceleration as a measure of how quickly the velocity is changing, change in velocity divide by the time taken for the change.

11:01For example, if you want to calculate the average acceleration between three seconds and let's say 12 seconds, how do we calculate that? Well, we'll first figure out how much the velocity has changed. It has gone from -2 to three, so the change in velocity is five meters per second.

11:20And how much time did it take for that change? Well, from three to 12 seconds, so that's nine seconds. So the average acceleration is five meters per second per nine second, giving you about 0.55 meters per second per second is meters per second square. That is the average acceleration and it's unit becomes meters per second squared.

11:42And just like before, what does it represent geometrically? It represents the slope of a secant line connecting the two points. And just like before, how we define instantaneous velocity as taking the limits as delta T tends zero, giving us a derivative of position with respect to time, we can now define instantaneous acceleration as taking the limits as delta T tends zero, giving us the derivative of velocity.

12:08So derivative of position with respect to time gives you velocity, and the derivative velocity with respect to time gives us acceleration. And just like before, if you now find the slope of a tangent in the velocity time graph, that slope represents the instantaneous acceleration at that moment.

12:26Okay, now that we have a tool to calculate instantaneous acceleration at every moment, let's use that to draw an acceleration time graph. It'll be a great idea to again pause and see if you can plot this yourself. Alright, let's do this. What we notice from zero to two seconds we have a straight line, and you have a straight line, the slope is a constant, which means the acceleration at every point from zero two seconds must be the same value, it must be a constant.

12:56And we can see that because it's sloping downwards, that must be a negative value. And how much is the slope? Well, we can just do -2 divided by two, you get -1. So the acceleration must be -1 from zero to two seconds. So it must be a constant. Okay, what about from two to five seconds?

13:17Again, from two to five seconds we have a straight line and that straight line is horizontal, which means the slope is zero everywhere. So two to five seconds, our acceleration is zero. That means that from two to five seconds the car was moving at a constant velocity.

13:32Okay, what about five to seven seconds? Notice again, we have a straight line, a constant slope. And again, if you look at it, it's gonna be two, positive two divide by two so you get plus one. So from five to seven seconds, acceleration is plus one. What about seven to nine seconds?

13:52Again, we have a horizontal line, slope is zero, so the acceleration must be zero. And from nine to 14 seconds, again we have a constant acceleration because we have a straight line and you will see it is five divide by five, which is plus one. So you get a constant acceleration of plus one from nine to 14 seconds and so on.

14:13And there you have it, that is the acceleration time graph. So long story short, velocity is the rate of change of position, its unit is meters per second, then you calculate over some time interval you get the average value. But if you take the limiting case, when delta T tends to zero, you get the instantaneous value, we also call that the derivative.

14:34Now, geometrically, in a position time graph, the average value, average velocity represents the slope of a secant line and the instantaneous velocity represents the slope of a tangent line. Using this we can draw a velocity time graph, and we define acceleration as the rate of change of velocity.

14:52And just like before, if you measure it over a time interval, delta T, you get the average value. On the other hand, if you measure it in the limiting case, when delta T tends to zero, you get the instantaneous value. So derivative of velocity gives you acceleration.

15:06And just like before, geometrically, in the velocity time graph, the average acceleration represents the slope of the secant line, and the instantaneous acceleration represents the slope of the tangent line. And using this we can draw the acceleration time graph.