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## 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

[itex]v(t)=gtsin(\alpha)[/itex]

Considering the halfpipe an infinitesimal sum of inclined planes we'd get

[itex]\int^{0}_{t}gsin\alpha(t)\partial\alpha[/itex]

However I've failed to derive [itex]\alpha[/itex] in terms of t.

How can I model such a problem in differential equations?