Newton's cradle-gun

Inspired by [baconbrain]'s comment on the Passive Rail Gun idea about the bearings moving "downhill", you drop a ball bearing, guided by a pipe, maybe at a slight angle off vertical. At the bottom of the straight pipe there's a semicircular track so the ball bearing curves around until it's moving up instead of down. Then there's another straight section of pipe. This one has a stack of ball bearings in it (the bottom one can be supported by a wire ring). The moving ball bearing hits the bottom of the stack and stops, transferring its momentum and kinetic energy to the ball at the top of the stack (which also has more potential energy). At the top of the stack, there's a semicircular track to direct the ball back down again, to roughly where it started. At this point you can have another downwards track parallel to the first, (where it loses potential energy and gains kinetic energy) then another semicircle at the bottom, then another upwards stack of magnets (where the top ball gets the kinetic energy from the moving ball, and also has more potential energy), etc, forming a coil-like structure (but with straight sections)

Then at the end the ball comes out with as much speed as if it had been dropped from the height of all the stacks off balls put together. 5 coils with 1 meter stacks should be the equivalent of dropping a ball 5 metres, which would make it move at 10 metres/second. 500 coils should get you 100 m/s

There's another design I thought of but haven't fully worked out which extracts potential energy from the whole stack of balls, instead of just the top one: instead of having multiple coils, there's just one loop, and the second ball follows the same path as the first ball. The first ball might need to get out of the way so it doesn't interfere with later ones. Also the final ball just smashes into the wire stack support ring instead of leaving the device. If this second one still isn't enough power, you can add multiple coils again, so long as the last coil loops back into the first. I calculate that with 100 ball bearings in a 1 metre stack (average height in stack being 50cm) each loop should be equivalent to dropping a ball 50 metres, which gets you about 30m/s speed. With 100 coils of this it's like dropping something 5 kilometres, which may be enough to break the device or reach 300 m/s (nearly the speed of sound). Though it couldn't go faster than the ball's terminal velocity. Hopefully it would explode or something before terminal velocity.