Power

Check out these weightlifters. The one on the right is lifting his weight faster, but they're both doing the same amount of work. The reason I can say that is because work is the amount of energy that's transferred. Or to put it a simpler way, this is the way I like to think about it, work is equal to the amount of energy you give something or take away from something. Both weightlifters are giving their weights the same amount of gravitational potential energy. They both lift them two meters, and the masses are 100 kilograms each. Plug those into the formula for gravitational potential energy, and you find that the work done by each weightlifter is 1,960 joules. But the weightlifter on the right is lifting his weight faster. And there should be a way to distinguish between what he's doing and what the other slower weightlifter is doing. We can distinguish their actions in physics by talking about power. Power measures the rate at which someone like these weightlifters or something like an automobile engine does work. To be specific, power is defined as the work done divided by the time that it took to do that work. We already said that both weightlifters are doing 1,960 joules of work. The weightlifter on the right takes 1 second to lift his weights, and the weightlifter on the left takes 3 seconds to lift his weights. If we plug those times into the definition of power, we'll find that the power output of the weightlifter on the right during his lift is 1,960 joules per second. And the power output of the weightlifter on the left during his lift is 653 joules per second. A joule per second is named a watt, after the Scottish engineer James Watt. And the watt is abbreviated with a capital W. All right, let's look at another example. Let's say a 1,000 kilogram car starts from rest and takes 2 seconds to reach a speed of 5 meters per second. We can find the power output by the engine by taking the work done on the car divided by the time it took to do that work. To find the work done on the car, we just need to figure out how much energy was given to the car. In this case, the car was given kinetic energy and it took two seconds to give it that kinetic energy. If we plug in the values for the mass and the speed, we find the engine had a power output of 6,250 watts. We should be clear that what we've really been finding here is the average power output because we've been looking at the total work done over a given time interval. If we were to look at the time intervals that got smaller and smaller, we'd be getting closer and closer to the power output at a given moment. And if we were to make our time interval infinitesimally small, we'd be finding the power output at that particular point in time. We call this the instantaneous power. Dealing with infinitesimals typically requires the use of calculus, but there are ways of finding the instantaneous power without having to use calculus. For instance, let's say you were looking at a car whose instantaneous power output was 6,250 watts at every given moment. Since the instantaneous power never changes, the average power just equals the instantaneous power, which equals 6,250 watts. In other words, the average power over any time interval is going to equal the instantaneous power at any moment. And that means work per time gives you both the average power and the instantaneous power in this case. Let's say you weren't so lucky, and the instantaneous power was changing as the car progressed. Then, how would you find the instantaneous power? Well, we know that power is just the work per time. So something we can try is to plug in the formula for work, which looks like FD cosine theta, and then divide by the time. Something that you might notice is that now we have distance per time in this formula. So let's isolate the distance per time. Distance per time is just the speed. So I can replace d over t with v in this formula. And if you plug in the instantaneous speed of the car at a given moment in time, you'll be finding the instantaneous power output by the force on the car at that particular moment in time. So to find the instantaneous power output by a force, plug in the force on the object at a particular moment in time, multiply by the speed of the object at that same moment in time, then multiply by cosine theta. But be careful here. Theta isn't any old angle. It's the angle between the force on the object and the velocity of the object. But in many cases, the force is in the same direction as the velocity, which means the angle between the force and the velocity is zero. And since cosine of 0 is 1, you don't really need the cosine in the formula at all. And you find that the instantaneous power is just the force times the speed. All right. So what does power mean? Power is the rate at which work is done. What does average power mean? Average power is the work done divided by the time interval that it took to do that work. What does the instantaneous power mean? Instantaneous power is the power output of a force at a particular moment in time.