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.

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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Geek dad, living in Oslo, Norway with passion for UX, Julia programming, science, teaching, reading and writing.

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