# Is energy conserved when the potential is discontinuous in time?

I know that energy conservation is related to time-translation invariance, but I don't understand how to check this invariance and what it means.

I have an example case, that confuses me:

Assume you are holding a bowling ball and you are standing still on ice. (no friction as usual)
This means both you and the ball have no kinetic energy.
If you now throw the bowling ball both you and the ball will have kinetic energy, that means energy is not conserved, but only momentum is conserved.

Can you add a potential energy term to the total energy, so that energy will be conserved in such a case?
This would have to be a potential energy that is a step function regarding the time.
Would such a potential energy be possible / would it still conserve energy?

Or is this kind of potential (discontinuous in time) the definition of non-time-translation-invariance?

Let $$V= K_1 + K_2$$ where $$K_1$$ and $$K_2$$ are the kinetic energies of you and the ball respectively. $$V_1$$ is the energy stored up in your muscles to just execute that function of throwing the ball. $$V$$ is the potential energy of the ball.