 Energy can either be created or destroyed, so when we use energy, what we're really doing is we're taking one form of energy and we're transforming as much of it as possible into another form that we prefer, and the rest of it typically goes into other sources of energy like sound, light or waste heat. The form we prefer, we often create that via a force, like the push of a muscle or an engine. But force is not a unit of energy. So what is the relationship between energy and force? We know that if we have some object and we apply a force to it, then it's going to accelerate. And so we know that it's going to change its speed, and we know that if we change the speed of an object, then we're going to change its kinetic energy. And so that should be able to tell us enough to work out the relationship between energy, change and force. Now if we're applying forces to things and they're going to start accelerating around and their velocities are going to change, everything's going to change, they might even change direction. And so we're going to simplify things for our first look at just looking at a very small amount of time where it doesn't have a chance to change the velocity very much. And so we've got our particle before, and then we're going to have our particle after. We've got that constant force on it, and they're each going to have a velocity. So initially it's going to have a velocity v, and then afterwards it'll have a very slightly different velocity, v plus delta v, a little change. And we're going to look at the work done, and that's going to be the change in the kinetic energy. And we know the relationship between the kinetic energy of a particle and its velocity. It's just a half times the mass of the particle times its velocity squared. And so after it's had some acceleration, it's going to be going a little bit faster. Let's expand out that bracket, and of course those two things cancel. And we can also ignore this, because as we make delta v smaller and smaller compared to v, then this term is going to be increasingly small compared to the other term. And so we get to neglect it because we make delta v as small as we like until we're happy with that approximation. So all we have to do is relate this delta v, this change in velocity, to our force. And that's easy to do because our force is causing acceleration, and a change in velocity is just the acceleration times the time. Newton's second law tells us the acceleration is the force divided by the mass, and so the acceleration times the time is just the force divided by the mass times the time. And then we note that the masses cancel. And so we're left with the force, and we know that the velocity times the small amount of time is going to be basically how far it travels. Now of course that's just the work done over that very small amount of movement. And so in order to get the total work done for it moving at an appreciable distance, we have to add all these up. And if we add all these up, we're obviously going to get the force times the sum of all the distances. So we're just going to get the force times the total distance traveled. So if we look at the force times the time for which we're pushing on the particle, we'll learn how much its momentum changes. And if we take the force times the distance the particle moves, then we'll figure out how its energy changes. So my two assumptions there are that the force is constant over that distance. Otherwise I have to add up all the pieces of work done as I change the force. And the other assumption I've made there is that the force is in the direction of the motion. Now we haven't thought about what happens if my force is in a different direction, but if it was exactly backwards, then all that would have happened is that that change in velocity would have been negative. So that change in velocity would have been negative, in which case I would have got a minus sign there, and I would have got a minus sign for my work. And it's okay to have negative work. What that means is that my force would have been slowing my particle down and taking kinetic energy away from it. And so that's perfectly reasonable. Sometimes people get a little bit confused with the idea that work is force times distance precisely because of this reason. When I detest it and think, I'll just hold this. I have to apply all this force. Oh, but I feel it's taking energy to hold something still. And so distance doesn't seem to have anything to do with it. And the reason that's a kind of mistake is fairly clear if you put it down on a bench. And you see the bench is now providing the force, but the bench doesn't get tired. The bench isn't spending any energy at all. It could do that for 100 years if it wanted to. And the reason we find so much difficulty holding it with our muscles is that our muscles are really inefficient and they're wasting all sorts of heat to just hold it there just from the way that they work. So while it's possible to waste energy in myriad ways, if you're actually applying a force to a particle, then its energy will be changed by the force times the distance traveled. So a good example of that is to try and work out the stopping distance of a car. If a car's going at velocity v and you want to see how long it takes to stop, one thing you could do is you could look at its acceleration. One thing you could do is you could look at its kinetic energy. So you could say that the work done is going to have to equal its initial kinetic energy. To stop the car, all the kinetic energy of the car has to be gotten rid of by the work done by the force of the wheels acting on the road. So the work done is going to be that force times the stopping distance, and it's going to be equal to the kinetic energy, which is a half m times the velocity of the car squared. So by rearranging, we can see that the stopping distance is going to go up as the square of the velocity and inversely proportional to however good traction we have with our tyres. And that tells us why we have to leave more and more space to stop as we go faster and faster.