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
Considering the halfpipe an infinitesimal sum of inclined planes we'd get
However I've failed to derive [itex]\alpha[/itex] in terms of t.
How can I model such a problem in differential equations?