A clever “perpetual motion” device [video]
This circuit is awfully complex....    it's possible to do the same with a far simpler circuit if you use the same coil for sensing and accelerating the ball.A small microcontroller could do both - perhaps even with low enough power that the whole circuit could stay turned on for years on a charge (when not flinging the ball).Looking at the total energy you need to impart on the ball, you should be able to do that with a far smaller coil and many fewer capacitors as long as you have a suitably shaped steel core to keep the flux path low.   I suspect you might be able to do it with no capacitors at all, since modern lithium cells are perfectly happy to deliver 100 amps for a few milliseconds.