Prerequisite:
Power - Introduction
Now let's look at power using calculus. We know that power is the time derivative of kinetic energy
P = d/dt (KE)
so let's plug in our expression for kinetic energy,
P = d/dt (1/2 mv^2)
To take the derivative, we need to recognize a few things:
- the factor 1/2 is just a constant
- the variable m, for the mass, we will assume to be constant in this course. Because it's a constant, we don't have to worry about its derivative, it just gets carried through the calculation just like the factor 1/2 does
- v, the velocity, is a function of time. We have to use the chain rule to take the derivative.
The result is
P = 1/2 m (2 v dv/dt)
We know another name for dv/dt - the acceleration, a. Plugging that in and combining the factors 1/2 and 2, we have
P = m v a
But here we can do another simplification. m*a is the force, F, so
P = Fv
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