 So now let's examine the work kinetic energy theorem. We're going to start with a couple of definitions for network and kinetic energy. Network is going to be the total transfer of energy due to all of the forces. So not considering just each single force by itself, but the total of all of those. And we're going to give that the symbol of work net. For kinetic energy, that's the energy of motion. And we're currently using the symbol ke for kinetic energy. Well, these two things that we've just defined are related to each other. Starting with the conceptual connection between these two things, we recognize that network happens when you have a net force. If there is no net force, you're not going to end up having any network. Well, if I've got a net force, that's going to cause an acceleration. That's Newton's first law. Well, an acceleration is a change in velocity because acceleration is the rate the velocity is changing. So if I have an acceleration, I will have a change in velocity. And a change in velocity means I'm going to have a change in the kinetic energy. So starting from the very beginning to the end, the network causes a change in kinetic energy. And not only are these things conceptually connected, but the work kinetic energy theorem says that they're exactly equal to each other, the network is equal to the change in kinetic energy. We can see that this is true if we start by looking at the work done by a constant force. If I have a constant force, then I'm going to have constant acceleration. So I can use one of those equations for constant acceleration that I had back in previous chapters, where the final velocity squared minus the initial velocity squared equals 2 times the acceleration times the displacement. And for right now, we're sticking with just one dimension. Well, if I take this equation and multiply both sides of the equation by the same factor of 1 half m and then multiply that factor through, what I find is that I've got 1 half m vf squared minus 1 half m vi squared for my left-hand side of the equation. And on the right-hand side of the equation, the 1 half and the 2 cancel each other out. And I'm left with ma times my delta x. I recognize that over here on this left-hand side, what I really have is the final kinetic energy and the initial kinetic energy. And over here on the right-hand side, I can recognize this ma is the force net. Well, that means the stuff I have over here on the left-hand side is the change in the kinetic energy. And the stuff I have over here on the right-hand side is the network. So the network is equal to the change in kinetic energy. And we found this by starting off with our constant acceleration equation. Well, there's a similar proof if I have variable forces. But it's a little bit more complicated. I need to integrate over the path to find the network. And I can't start with the constant acceleration equations. Because of those complications, the proof of this is a little bit beyond what we're presenting in this course. But we still have the relationship that the network done is equal to the change in the kinetic energy, even if the forces are variable forces. So that ends the introduction to the work kinetic energy theorem. We'll see other opportunities to apply that very soon.