A trebuchet can be supersonic
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In any case, a very cool project IMHO. I've long wondered how fast modern technology could throw a projectile without resorting to explosives. I now know it can at least be supersonic.
Can it go even faster, though? Is there an even better mechanical design than a slingshot?
https://en.wikipedia.org/wiki/Trebuchet
https://exarc.net/issue-2018-3/at/traction-trebuchet
This seems like a form of traction trebuchet with traction provided by the elastic.
Railguns can get projectiles up to mach 8. The magnets at the HLC can get particles pretty close to the speed of light.
I think if you continued with the trebuchet/rubberband approach you'd really only be limited by the amount of tension you'd be able to put your band under. There are probably better materials than rubber.
I think the main stopper for higher speeds would be materials strength, especially with that design relying on a cable, but even so I doubt the limits are anywhere close from what was achieved in this video.