# 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.

formula not implemented

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.

formula not implemented

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 implemented

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:

formula not implemented

Now 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 implemented

Next we substitute F directly instead of a, and get:

formula not implemented

Then a is easily eliminated and we end up with:

formula not implemented