# 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 and then we substitute directly instead of , and get instead. Then a is easily eliminated and we end up with .

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