 Now that we've introduced the concept of power, I can look at the concept of how power relates to force and velocity. Now, we start by remembering that power is the rate of work being done. And one way to write that equation is to look at the average power being related to the work done in a certain amount of time. Now I want to take a look at work done by a constant force. So this is sort of a special case. And in that case, I could replace the work by the equation we used for the force and the displacement. In this case, I'm keeping it just in one dimension for the x direction. Now I can rearrange this equation just slightly to pull my force out front and leave my delta x over delta t over on the side. Now what this really tells me is that that constant force doing some work may be causing a velocity of that object. And if I look at this equation then, my delta x over delta t, that's the average velocity. And that's why I use the average power over here, because I knew I was going to have an average velocity. Now if you happen to be moving at constant velocity, then this ends up being just the velocity and the power is a single value for the power. Now in general, if I'm not using just a constant force in one dimension, then I could write my power out as the dot product between the force and the velocity. I could also write that dot product out in terms of how much force do I have, how fast am I moving, and then the cosine of that angle in between them. Just like when we did work, we used the cosine of the angle between the force and the displacement. If my force and velocity are in the same direction, and that happens quite a bit, that means my angle is 0. And so the cosine of theta is 1. And I could simplify my equation down here to be the power is just the force times the velocity. So let's look at some examples of that. So let's say I've got a heavy box, and I'm moving it across the floor at a constant speed of 2 meters per second. So what's happening here is somehow I'm pushing the box. And yeah, there's probably some friction pushing against me. But if I consider just my pushing the box and I'm exerting a force of 20 newtons to do that, then I've got some power output here. If I look at my equation, the power output of that person equals the force they're exerting and the velocity that they're managing to get the box moving. So that would be our 20 newtons times our 2 meters per second, or 4 newtons meters per second. And newton meter per second is also one of the ways we can write the unit watt. So this is equal to 4 watts. Now here's another example that kind of works it in a little bit of a different order. Let's say I've got a 30 watt motor, and it's used to lift a crate up into the air such that it rises at 0.5 meters per second. How much does the crate weigh? Well, if I draw a quick little diagram, what I see is that that motor is probably connected to some sort of a cable, which is applying a force to my crate. At the same time, gravity is pulling down on that crate. And because I'm moving at a constant velocity, those two forces must balance each other out. So when I look at the power of the motor, the force that I'm actually working with is the applied force. So putting our numbers into this, what I see is my 30 watts is my power, my 0.5 meters per second is my velocity, and I can solve for the applied force. So this would give me an applied force of 60 newtons, because that's going to be 30 divided by 0.5. And because I saw that those two forces were balancing each other out in this case, that means that the force of gravity, or the weight, is 60 newtons. So that's an introduction to how force and velocity are related back to this concept of power.