Developing kinematic equations from data

0:00- [Instructor] 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 gonna hit the Earth. It's gonna 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, right?

0:18Well, 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.

0:31Now 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'll be in a uniform acceleration, right?

0:46And we can now measure its positions, and times, and velocities and let's say here's the 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. A question we could ask is how exactly is velocity related to time?

1:05Answering 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.

1:22So equations that connect measurable quantities are what we call mathematical model. 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 and we'll consider velocity as the dependent variable.

1:41Dependent variable is usually drawn on the vertical. 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 that these two quantities become proportional to each other.

2:05So 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 zero to one.

2:22All right, so here it is, time changing from zero to one 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.

2:38So the change in the velocity over here is 18 centimeters per second. So when the time changes by one second,, velocity changes by 18 centimeters per second. Okay, now let's consider what happens if the time changed by two seconds, so from here to here. So let's do that over here.

2:54So now the time has changed by two 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 the halfway is 50 and this is somewhere in between, so it's about 56. And again, how much has this changed?

3:10From 20 to 56, we'll subtract the two, you get 36. So the velocity this time has changed by 36 and this has to be centimeters per second, and this is two seconds. And 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.

3:30If the time had changed by a factor of three, the velocity would've 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?

3:45Well, we know the equation to a straight line. An equation to a straight line is y equals mx plus b where b is the y-intercept and m represents the slope. Now let's say we know the values of the slope and this slope represents the rise over the run, and we have these values.

4:03It's 18 divided by one or you can do that over here, 36 divided by two 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 20.4, but we can just round it off and just call that as 20. Values are not really important over here.

4:18Our goal is to build a model, right? So we get y equals 18x plus 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 centimeters per second. Per second, that's a slope, right?

4:39That is centimeters per second 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.

5:03This is the mathematical model that connects velocity and time. And now if we 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 constant acceleration and 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.

5:24Pretty 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, and the graph we get looks like this.

5:42What'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 a 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 have to think about which kind of parabola it is and stuff like that.

6:02So 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, that 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.

6:28So 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.

6:44We can see a rectangle and a triangle. Well, now let's think about the sides. Well, this represents our velocity 'cause this is a velocity axis. This represents the time because it's the time axis. This side represents the initial velocity. Does that make sense?

7:00So 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-naught times t plus the area of this triangle which is 1/2 times base which is t times height. Height, this height would be v minus v-naught.

7:23All right, we have a model but we can simplify it further. v minus v-naught we can see from here is just at. So let me just take a copy of this and substitute this as at, giving us 1/2at squared, and finally, what is delta x? Delta x is displacement which is change in position, which is final position minus the initial position.

7:45So 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 'cause it's exactly what we saw.

8:06It 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?

8:23Then this thing would reduce to x equals x-naught. This would be zero plus 1/2at squared, and what's interesting about this is that this is also of the form y equals mx plus b provided the horizontal axis is not t, but t squared. In other words, if you plot a graph of position where it says t, that's a parabola because this is quadratic.

8:50But if in this special case, we drew the graph of position and t squared, that would be a straight line. And 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 a 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.

9:11So over here notice we would get position to be proportional to not time, but time squared, and this method where we clearly 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.

9:32So 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.

9:49You 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.

10:03It'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 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 we get a straight line.

10:29If we get a straight line, that's great, linearization worked. 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.

10:43Here it is, so we just squared these values and now let's take a plot of velocity square 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.

11:02Linearization worked, so this is indeed working. So now, so we can now 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, y equals mx plus b.

11:19We know what t 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. So we can do rise over run.

11:36If I consider 40 centimeters, then the rise over here would be subtracting these two values which I got from here. These two values which is 1,444 centimeters squared per second square. So the slope would be this divided by this and if we do that, we get roughly 36 centimeters per second squared.

11:55First of all, this is the acceleration units, so this must be having units of acceleration. So there must be acceleration over here, and then last time we found the acceleration was 18 centimeter per second square. So that means this is twice the acceleration, so this represents twice the acceleration.

12:12So 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 v-naught squared. If we rearrange this, we get this and then finally, remember that actually x is the displacement.

12:40That happens because our initial position was zero. If it was not zero, then we should replace this with x minus x-naught. 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.