Introduction

We say power is one of the three things people generally want, along with money and happiness. However, the way we think about power is more analgous to thinking about force or respect, rather than actual power. In fact, horses and oxen have much more "power" than humans. There's a reason why car engines are often measured in horsepower and not manpower!

Power is actually the rate at which something does work, or the rate at which it can change the energy of a system. Both definitions are pretty much analogous and can be referred to interchangeably. The SI unit for power is the watt, after the "inventor" of the steam engine, James Watt.

If we don't want to invoke calculus, we have to deal with the average power exerted on a system. This is much like the other average values we've dealt with before and involves the change in the energy in the system over the change in time. It's also equivalent to the amount of work done over the time interval. Most of the time, this is enough.

$$ P = \frac{\Delta E} {\Delta t} = \frac{W}{\Delta t} $$
The more accurate definition of power involves a derivative. Shocking, I know. Instantaneous power is the time derivative of energy or work with respect to time.

$$ P = \frac{dE}{dt} = \frac {dW}{dt} $$
Most of the time, we will be dealing with constant power, so we don't need to worry too much about the effects of a varying power. IAfter all, most engines and motors have a consistent power output rather than a varying one. Moreover, in a lot of problems we only really care about the average power, as figuring out the instantaneous power would require quite a bit of work. There is actually a formula for instantaneous power not requiring calculus that is really useful, however.

$$ P = \vec{F} \cdot \vec{v} $$
The exact reasoning for why this is true requires a bit of calculus, but we can understand it another way. Recall that power could be defined as the rate at which work was done on something. But, work is equivalent to the dot product of force and displacement. The rate of change of displacement is equal to the velocity, which is how we get that the power is equal to the dot product of the force and velocity of an object. This can easily be shown to be true with a bit of basic calculus. Recall how power could be defined as $P = \frac{dW}{dt}$. Work is equal to the dot product of the force and displacement. This allows us to just distribute the derivatve inside and differentiate the displacement with respect to time. This gives us the aforementioned formula since the time derivative of position is velocity. Simple, really.

You will find that this concept is rather simple, which is why sometimes problems will arbitraily add difficulty by asking about power in a roundabout way. For instance, they may ask what the rate of change of the system's energy is. However, if you have a good handle on what power is, this should be no issue.

We have already covered all the concepts behind what power is, so we can move on to a practice problem. This is probably the best way to explain what power actually is, because the actual defintion can be a bit vague and mystical until you actually apply it to a scenario.

This is a short lesson because there is not too much to cover on power. You'll likely see this concept pop up again in the future whenever we have discussions of systems with changing amounts of energy, but overall it's a really simple concept that just requires understanding and applying simple formulas. The next section also only contains one concept, but that's where the similarities end. It's a bit of an abstract concept, but with some effort you'll definitely understand it. If you're ready to dive back into gravitation, move on to the next lesson!