Thank you for your comments. I'll check the links out.
The problem (in my mind) is that I don't know the trajectory of the ball such that it will strike the surface tangentially. The string is increasing in length with time. As it increases in length the contact point moves and the flight time increases. So I don't have a point on the cylinder from which to calculate a trajectory.
Maybe I just need to sleep on it. Or solve an easier case first, ie not tangential, and I'll see the solution.
It's just math, after all.