# Relationship Between Work and Kinetic Energy

Work is defined as the energy, you add to an object by applying a force `F`

over some distance `r`

.

This could mean e.g. lifting an object up a distance `r`

against gravity. The energy added is then potential energy. However in this case I want to show how work related to kinetic energy.

If you apply a force to an object, to make it move faster, you increase its kinetic energy.

I will show how the two are related using one of the motion equations I’ve covered earlier.

We start with the definition `(1)`

and then we simplify `(2)`

it by saying initial velocity `v₀`

and initial distance traveled `r₀`

are both zero.

Finally we rearrange `(3)`

the equation to get a way to express `r`

, so that we can substitute it into `W = Fr`

.

We also want to get rid of acceleration from the equation, because the expression for kinetic energy does not contain it. Let’s rearrange Newtons second law:

Now we got the pieces to derive the equation for kinetic energy.

`(7)`

substitute `r`

with equation `(3)`

. Then we we can substitute `a`

in equation `(9)`

with definition for acceleration in equation `(8)`

.

Finally `(11)`

we can see that work equal kinetic energy.

**Simpler Solution (Edit)**

When reading through this post by chance again I noticed my approach could have been a lot simpler.

If we start with *W = Fv²/2a* and *F = ma* then we substitute *F* directly instead of *a*, and get *W = mav²/2a* instead. Then a is easily eliminated and we end up with *W = mv²/2*.