Space Catapault Model
I won't be displaying mathematics or even a guesstimate here because this subject is somewhat beyond my scope of competency. However:

To shoot something into orbit with a catapult from ground level is not feasible, as the object would immediately burn to ashes, no matter what the material would be. The required velocity to reach orbit is about 40 000km/h, and to make things worse, you'd have to reach that speed at basically zero altitude, with a very dense atmosphere creating friction.

See this question and answers from physics stack exchange for further information.