[hermitcrab]

Assuming no air resistance and 10m/s^2 gravity. I calculate that you would have to be doing ~1,100m/s (~mach 3.2 at sea level) straight up at 37km to reach 100km. Or ~1,600m/s (~mach 4.7 at sea level) at 45 degrees to perpendicular.

[wizzwizz4]

Seems about right. https://en.wikipedia.org/wiki/Project_HARP (1966) got 2100m/s from a ground-based cannon: this got the projectile to 179km.