Oh Houston, my shining city on a swamp. It looks like we have a visitor for today's lecture, but I guess we'll still go ahead and talk about power. If I had to give my street or perhaps swap definition of power, it is how fast you are working. It is really just the time rate or change of work. So let's look at a few definitions. The average power is W, sorry, W not W, over Delta t, where this is the work done and this is the in this interval, right? So if you do a certain amount of work in a certain amount of time, that ratio is just the power, and it's in joules for seconds which is equal to a Watt, and the abbreviation is big W, right? It's a lot like average velocity, or average velocity was the displacement over a Delta t. We can also think of an instantaneous value, it is really small Delta t. So if we wanted to look at the instantaneous version, then that would just be P, and just like velocity we won't put i and st at the bottom if we see no subscript we mean instantaneous, and that would just be the time rate, the derivative dW, dt. So we can calculate it in a couple of different ways. We can also say these are before after problems or maybe this would be an initial final state problem. Maybe this is from a graph or something, but if you had a Kinematic situation, you want to do apply it too. There's one more useful way to look at it. So let's look at power in Kinematics. By that I just mean if P is d work, d time, we could plug in our integral, F dot dr at our definition of work. So it's d, dt, integral of F dot dr. So if you had some varying work and some varying curve, you could figure that all out from this. But there's a simpler version, I want to get to say for constant force and say for straight line motion, because that would cover a lot of simple Kinematics situations. Then what you would have is that the dot product would be a constant. So if the force is always in the same direction, the motion is in the same direction, they would go together. So we could write this P then as d, dt of the force magnitude, cosine Theta is between them, and then you're just left with the integral of dr, the magnitude dr. But then that's just the distance, that's just what you would call d or r, that's just how far it went. So then you could say, well these are constant, then come out of the d, dt, right? So it would be F cosine theta dr, dt. But that's just the velocity. This is a very long way of saying if work is F dot d, and you take the time derivative and the force is constant, it's the time derivative of the position or the displacement, which is as you know the velocity. So you could say this is F cosine theta V. But you realize, oh, that's the dot product. So now we can make it back, make it general again, and say basically the instantaneous work is the dot product of force and velocity. So now we've gone away from these approximations actually. At anytime, the institute is power, I mean to say is F dot V. So with these three definitions for these different cases before, after instantaneous Kinematics, you can work most problems quickly because that's power, work them quickly.
