Developing kinematic equations from data | AP Physics 1 | Khan Academy

How do we figure out how big a runway we need so that the plane that's landing, for example, can stop safely?

Or imagine there's an asteroid that's hurling towards the Earth. Don't worry, it's not going to hit the Earth; it's going to come very close by. But again, how do we figure out how long it would take to come close to the Earth so that we can be prepared to see it?

Well, this requires us to predict the future, and one of the ways to do that is by building what we call mathematical models. But what exactly are they, and how do we do that?

Well, let's find out. The best way to learn what this is, is by taking a concrete example. Now, these are pretty complex, so let's take a much simpler example, an example in which an object is moving with a constant or uniform acceleration. For example, here's a demo: we can take a toy car on a ramp, let it go, and that will be in a uniform acceleration, right?

We can now measure its positions, times, and velocities. Let's say here's a sample data. So for different values of position, we have recorded the times at which that position was reached, and we've also recorded the velocity. Now, a question we could ask is: how exactly is velocity related to time?

Answering that question is what we call building a mathematical model. But wait a second, isn't that just building an equation? Yes, mathematical models are equations, but they're not any equations; they're equations that connect physical quantities, physical measurable quantities.

So, equations that connect measurable quantities are what we call mathematical models. So how do we get a mathematical model that connects velocity and time? Well, for that, we can plot a graph of velocity and time.

We can choose time as our independent variable because its passage is not controlled by anything. We'll consider velocity as the dependent variable; the dependent variable is usually drawn on the vertical, and the independent variable is drawn on the horizontal.

So let's plot that graph. So here's a graph. This is what we get, and you can see that this graph is pretty much a straight line. So whenever a graph is a straight line, we say it's a linear graph, and the beauty of linear graphs is that this means these two quantities become proportional to each other.

So in this case, we can say that velocity is proportional to time. Now, what is the meaning of that? Well, this means if the time changes by some factor, velocity will also change by the same factor.

So let's see that actually in the graph. Okay, so consider time changing from 0 to 1. All right, so here it is: time changing from 0 to 1 second. Let's see how much the velocity changes. The velocity starts with 20; you can see over here that's the initial velocity, 20.4, roughly 20, okay?

And from here, it changes to... Let's see how much it changes to. It changes to 38. So the change in the velocity over here is 18 cm/s. So when the time changes by 1 second, velocity changes by 18 cm/s.

Okay, now let's consider what happens if the time changed by 2 seconds. So from here to here. So let's do that over here. So now the time has changed by 2 seconds. Let's see how much the velocity changes by. Again, it starts with 20, and it goes till here, and that is about 56.

You can see halfway is 50, and this is somewhere in between, so it’s about 56. And again, how much is this change from 20 to 56? Well, subtract the two, and you get 36. So the velocity this time has changed by 36, and this is 2 seconds.

Now you can clearly see that when the time changed by a factor of two, look, the velocity also changed by a factor of two. If the time had changed by a factor of three, the velocity would have changed by a factor of three, and so on and so forth.

That's what it means to be proportional: when one quantity changes by some factor, the other quantity also changes by the same factor.

Okay, now how do we build a model or an equation that connects velocity and time? Well, we know the equation to a straight line: an equation to a straight line is y = mx + b, where B is the Y intercept and M represents the slope.

Now let's say if we know the values of the slope, and this slope represents the rise over the run, and we have these values. It's 18 / 1, or you can do that over here: 36 / 2, which is 18. So we know m is 18, and B is the Y intercept. That's over here; you can see that's about 20.

That's 21.4, but we can just round it off and just call that as 20. Um, values are not really important over here; our goal is to build the model, right? So we get y = 18x + 20.

But now let's see what these numbers represent. Y represents our velocity; so this will be our velocity. What does 18 represent? Well, look at the units: it's 18 cm/s per second. That is a slope, right? That is cm/s squared. That is your acceleration, okay?

That is the acceleration. So 18 here represents the acceleration. X represents the time, and what is this 20? This 20 represents the initial velocity, velocity when time was zero.

So we can call that as v0, and there we have it. This is the mathematical model that connects velocity and time. Now if you knew what the acceleration was and the initial velocity was, then I can use this to predict the future.

If that object continued with the same, you know, constant acceleration, then if you input the value of time, I can then figure out what would be the value of the velocity at that particular time. Predicting the future using mathematical models is pretty cool, right?

Let's try to build another model. This time let's ask ourselves what is the connection between position and time? And again, we can plot this time the graph of position and time. This time, position will be dependent, time will be independent.

The graph we get looks like this. What's interesting is that this is not a linear graph; in fact, this is a quadratic graph. We call this a parabola.

So now one way to build the model for position and time is to try and write an equation for a parabola, but for that, we now have to remember the equation for the parabola and we need to think about which kind of parabola it is and stuff like that.

So we can go ahead and do that, but here's another way to think about this. Okay, so let me go back to the velocity-time graph that we had plotted. You may recall that the area under the velocity-time graph represents the displacement.

So this means if I figure out what this area is, then I can build an equation for the position and the time. So, it'll be a great idea to pause the video and see if you can try to find the area of this graph and see if you can figure out what the model would be that connects position and time.

Okay, let's do this. There are multiple ways to calculate the area of this graph, but what I like to do is divide this graph into two parts. We can see a rectangle and a triangle. Well now let's think about the sides.

Well, this represents our velocity because this is the velocity axis; this represents the time because it's the time axis. This side represents the initial velocity, so that makes sense.

So from this, we could write Delta X, which is the displacement; that is the area under the graph that equals the area of this rectangle, which is V * t, plus the area of this triangle, which is half * base * height.

This height would be V minus V0. All right, we have a model, but we can simplify it further. V minus V0, we can see from here is just a T, so let me just take a copy of this and substitute this as T, giving us half a T².

And finally, what is Delta X? Delta X is displacement, which is change in position, which is final position minus the initial position. So let's do that, and then finally rearranging this gives me the required model.

It's a model that connects position and time. And now, if you look at this model, we can see that here we no longer have a linear equation; we have a quadratic equation. That's nice because it's exactly what we saw; it was not a linear graph; it was a quadratic graph, and we were getting a parabola.

But what's interesting now is in this particular case, what if the initial velocity was zero? So in our case it's not, but what if we had a different kind of motion with constant acceleration in which the initial velocity was zero?

Okay, then this thing would reduce to x equal to x0. This would be 0 + half a^2. And what's interesting about this is that this is also of the form y = mx + b, provided the horizontal axis is not T but T².

In other words, if you plot a graph of position versus T, that's a parabola because this is quadratic, but if in this special case we drew the graph of position and T², that would be a straight line.

So for this particular case, not ours, but in this particular case, we could have derived this model by drawing a graph of position and time squared, drawing the straight line, and then figuring out what the slope would be and the intercept would be.

And that's how we could have derived this mathematical model. So, over here, notice we would get position to be proportional to not time but time squared.

And this method, where we cleverly pick the axis in such a way as to get a linear graph, is what we call linearization.

Anyways, since we're having so much fun, let's do one more model where we connect position and velocity. So again we'll plot a graph; we will take position as our independent variable this time, and we'll take velocity as our dependent variable.

So it'll be along the vertical; position will be along the horizontal. Let's see what the graph looks like, and we get a graph that looks like this. If we plot them all, you can see it's not a straight line, and again it looks like a parabola.

So here's how we can think about it because this is on its side. See, velocity is not proportional to position; it's not growing as quickly; it's growing much more slowly.

So it's not proportional to position; it's a square root graph. So in this particular case, velocity is proportional to the square root of position, or we can get rid of our square roots.

I don't like square roots; we could say velocity squared is proportional to position. So this is an attempt at linearization, which means let's try to plot the graph of velocity squared versus position and let's see if it gets a... if we get a straight line.

If we get a straight line, that's great; linearization works. If not, we'll probably need to troubleshoot and figure out other ways to linearize it. But anyways, let's see if this works.

So we need to now look at velocity squared. So we can add one more column where we square these. So let's do that; here it is. So we just squared these values, and now let's take a plot of velocity squared versus position and see what we get.

Here we go... Oh, we indeed get a straight line. You can clearly see it's a beautiful straight line. So velocity squared versus position gives us a straight line; linearization worked!

So this is indeed working. So now we can write the equation of the straight line and simplify and see what the model is. Great idea to pause the video and see if you can try it yourself first.

All right, so going back, the equation to a straight line is y = mx + b. We know what Y is: that's our velocity squared. We know what x is: that's our position. We know what B is: B is this Y intercept, which is the initial velocity squared.

The big question is what is M? Well, let's look at the value and try to figure that out. We can do rise over run. If I consider 40 cm, then the rise over here would be subtracting these two values, which I got from here: these two values, which is 1444 cm/s².

So the slope would be this divided by this, and if we do that we get roughly 36 cm/s². First of all, this is the acceleration unit, so this must be having units of acceleration. So there must be acceleration over here, and last time we found the acceleration was 18 cm/s².

So that means this is twice the acceleration. So we can now write all of those things down: this is velocity squared; this is twice the acceleration, so 2a. Then we get the position plus B, and B is our initial velocity squared, because this is the velocity squared axis, which is V².

If we rearrange this, we get this and then finally remember that this actually X is the displacement that happens because our initial position was zero. If it was not zero, then we should replace this with x - X0.

And there we have it, the final mathematical model that connects velocity and position. These three models are what we call the kinematic equations, and they're extremely useful for predicting stuff when we are dealing with motions that have constant acceleration.