Work is defined as the energy, you add to an object by applying a force
F over some distance
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.
r with equation
(3). Then we we can substitute
a in equation
(9) with definition for acceleration in equation
(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.