# I'm confused about the work energy theorem

reyrey389
The reason why the work energy theorem " ΣW = Δ KE " only includes kinetic and not potential energy, is because gravity undoes the potential energy ? What I mean by "undoes" here is, If you were to, for e.g., lift an apple up a vertical distance, but also accelerate it while lifting it. Then you would do +mgh joules of work to lift it and +1/2 mv^2 to accelerate it, but gravity did -mgh work, so the total was just kinetic. So that theorem will only contain kinetic energy ? Or is it only for this certain case where it does not ? If the former is true, and mass stays constant, can the theorem simply be stated as net work is only done if objects change their speed ?

This somewhat seems to make sense because if you lift the apple at a constant speed instead of accelerating it you did +mgh joules of work on it , but gravity did -mgh work, so the net work was 0. And since the apple didn't pick up or lose any speed, there was no net work done on it.

Then again, it somewhat doesn't make sense because even though the speed of the apple didn't change, we expended energy to lift it, and at its highest point it now has stored energy. There was no energy initially (assume initial position = 0) and now there is energy mgh, so shouldn't there be some work done on the apple ?

## Answers and Replies

nasu
The potential energy is just an alternate way to take into account the work done by gravity or other conservative forces.
If you include the potential energy then the work in that formula won't be the net (total) work but just the work of the forces not associated with potential energies.
So in your formula, the sum over W is the net work, the work of all forces, including gravity. Gravity is not excluded.
However, by definition, the work done by gravity is equal to the change in potential energy, with a minus sign. So the potential energy is not excluded but counted in the work side of the formula. If you want to count it as explicit;y PE, you move in on the other side and the negative sign becomes positive.

But you don't need to look for complicated reasons. The work-energy theorem is a direct consequence of Newton's second law. Introduction of potential energies is like a further "refinement".