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Power - Derivation

Page history last edited by Mark Eichenlaub 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

 

 

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