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# Power - Derivation

last edited by 2 years, 4 months ago

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