# 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`.

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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.

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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.

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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:

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

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`(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:

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

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Then `a` is easily eliminated and we end up with:

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