Using my numbers above, getting up to 100 km requires u = 1 MJ/kg of energy. For the trebuchet model, this number is u = ½ v² and you're presumably going to be flying in a circular arc before you are released, where your acceleration is a = v²/r, or r = 2 u / a.

If we want to keep the maximum acceleration beneath 4g's or so, so that it's uncomfortable in the roller-coaster-thrill-ride sense, not in the 'crap I'm going to die' sense, we'd still need a trebuchet of radius 50,000 m. Even the LHC isn't that long across. :P

I mean, you can in principle get those sorts of speeds, but don't expect a human to easily survive it.