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I've done some math on this concept albeit with recoil action and hit an issue of too high power requirements to maintain 12rps. The problem is motors are ~80% efficient, while making a liner motor in this manner that is 10% is already great.
I haven't considered providing enough energy to propel the bb only, this might actually be realistic.
Another solution to this is to use neodymium magnets with a large surface area piston and short stroke but this is unlikely to fit into v2/3 form factor.
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