 So, here are two remarkable facts. If you're on a moving car, part of the car is motionless. And if you're on a moving train, part of the train is actually moving backward. This is easiest to show using calculus. So, remember the curve traced by a point on the rim of a wheel as it moves is called the cycloid. And if the wheel has radius one, the parametric equations for the cycloid are x of t equals t minus sine of t, and y of t equals one minus cosine t. Now, we have the x and y coordinates as functions of t, so let's differentiate them. And solving to find our critical points for x will give us, and so x prime of t is going to be zero for t equal to any multiple of 2 pi. But notice that the derivative of y at any multiple of 2 pi will be zero. So, at t equal to any multiple of 2 pi, the point on the wheel is not moving horizontally since the derivative of x is zero. It's also not moving vertically since the derivative of y is equal to zero. And this means the point isn't moving at all. And what this means is that any given point on a rolling wheel will be motionless at some time. A train wheel is essentially a disk on a disk. A point on the rim of the inner disk makes contact with the rail, and it will move like a cycloid. But what about a point on the rim of the outer disk? It will trace a path known as a prolaptochoid. So, we can derive the equation of this path as follows. Let's assume the inner disk has radius a, and the outer disk has radius b. If the disk has rolled through some angle theta, the center of the inner disk will have moved a distance of a theta. That's just like the cycloid. And a point on the outer disk will be located at x of theta, a theta minus b sine theta, and y theta, a minus b cosine theta. Now, if we differentiate both x and y, we see that x prime is a minus b cosine theta. And since a is the radius of the inner disk and b is the radius of the outer disk, a is less than b, and so there will be times when x prime is negative. And this means the point on the wheel is moving to the left, even though the wheel itself is moving to the right. And as they say, nothing is believing. What? Nobody says that? Oh, C! OK, well let's take a look at that. So let's take a point on the outer rim and see what happens as we move forward. And see that if we get to here, let's mark this point. This point on the wheel will actually move back a little bit to the left before it starts moving forward again. And so even though the train as a whole is still moving forward, there's part of it that's moving backward.