Catapult?

[KenWittlief]

I think you have made some assumptions that are not necessarily true.

1. whatever your catapult 'arm' is made of, once a ball is fired you must stop it before you can fire again, so the inertia of the arm is always lost (unless you are going to have something more like a bat that wacks balls that are dropped in front of it.

2. an arm that rotates only part of a circle, 90° for example, could hit a spring at the end of its motion that bounces it back into its starting position. If fact, when it gets back to its starting position there could be a second spring that absorbs (recovers) the momentum of the arm?

3. a motor would be my last choice for transferring energy into a catapult. First choice would be a pneumatic cylinder.