"Caching the momentum" is essentially what "regenerative breaking" does.

An engineering student designed something like that, using a flywheel: https://www.treehugger.com/bikes/engineering-student-builds-...

But if you want something commercially available, you have to get an e-bike with that feature, and those models seem quite pricey.

I have an ebike w/ regenerative breaking (a Bionix conversion kit, $1500 w/ labor) but I can't say it seems to do much. But then again, with no other non-regen bike to compare to I can't be sure. It doesn't recover anything like the charge it spends going up a hill (with assisted pedaling) when you're coming back down, but that's to be expected, eh? Thermodynamics and whatnot.

The flywheel thing is exactly it, except I think it would be safer and lighter to use compressed air, also stores longer than a flywheel.

With something like that you'd eliminate most of the work penalty for brief stops.

Compressing air isn't a very efficient operation, pumps usually have leak, spring compression, friction, and heat losses.