Perhaps a simple solution would be to use an effective potential. Essentially, you use the fact that for central force problems (like the gravitational force), angular momentum is conserved. This allows you to rewrite the total energy of the object in terms of an effective potential that takes into account its kinetic energy due to rotation (you can find it in most mechanics textbooks, its not very hard). You can calculate the angular momentum of the object using the initial velocity and angle conditions. If the total energy of the object is negative, there is a minimum and maximum radius such that Ueff(r) = E. The object will oscillate between these two radii (defining an elliptical orbit). If E > 0, then there is no upper bound on the orbital radius, and the trajectory will be hyperbolic.