# 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.

formula not implementedWe 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:

formula not implementedNow we got the pieces to derive the equation for kinetic energy.

formula not implemented`(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. We start with:

formula not implementedNext we substitute

directly instead of *F*

, and get:*a*

Then `a`

is easily eliminated and we end up with: