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.

Image for post

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.

Image for post

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.

Image for post

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:

Image for post

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

Image for post

(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 W = Fv²/2a and F = ma then we substitute F directly instead of a, and get W = mav²/2a instead. Then a is easily eliminated and we end up with W = mv²/2.

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

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store