The problem with this scheme is atmospheric friction. When you launch with a catapult, the spacecraft is going fastest where the atmosphere is thickest. (This is just the opposite of what happens with a rocket launch.) To maintain a low earth orbit at an altitude of 185 km, a spacecraft would need to be travelling at 7.79 km/s; to have this velocity at 185 km, it would need to start off with a velocity of at least 9.69 km/s at sea level -- that's about 35,000 km/h, or about 35 times the speed of sound. It would therefore experience intense frictional heating -- much worse than the space shuttle experiences on re-entry. The outer surface of the spacecraft would become incandescent, like a meteor.
To get round this problem, you might consider launching the spacecraft inside a tube from which all the air has been removed, with the tube going up the side of a high mountain. Designing the top end of the tube so that the spacecraft could get out, but air couldn't get in, would be a challenge.