Motion in Nonlinear Differential Equations

kgns

Homework Statement

How do you derive the time-dependent velocity equation for motion along a curve, such as a skateboarder on a half pipe?

For the sake of abstraction, I ask myself the following:

A uniform sphere of mass m and radius r is set free from the top edge of a semicircle half pipe with radius R. If R > r, what is the time-dependent velocity equation v(t) for the sphere in terms of t, m, r, R and g?

(i) ignoring any effects of friction?
(ii) if the sphere is rotating?

2. The attempt at a solution

If it were an inclined plane, we'd have no problem with
$v(t)=gtsin(\alpha)$

Considering the halfpipe an infinitesimal sum of inclined planes we'd get
$\int^{0}_{t}gsin\alpha(t)\partial\alpha$

However I've failed to derive $\alpha$ in terms of t.

How can I model such a problem in differential equations?