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He calls it a trebuchet but I'd call it a slingshot.

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?

My understanding is a trebuchet operates using some heavy weight. This is more of a slingshot with an arm attached.
It depends on the weight and ballistic coefficient of the projectile and the medium through which the projectile travels (as far as "muzzle" velocity goes). Using compressed air and a small enough projectile you get well over 1400 feet per second/ 425 meters per second. Using helium you can get it a little higher. That's about on par than what the guy in the youtube video achieved.

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 don't think the rubber is the limiting factor. You can always use more 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.

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