Time it takes a mass point to go down a curve under gravity

Question

An age old question. How to calculate the time it takes a mass point to go down a frictionless curve under gravity?

P.S. The curve is convex and smooth and can be of any kind of shape.

Answer 1

This is the well known brachistochrone (Greek for short time) problem.

The time it takes to transverse a small element ${\rm d}s$ is

$$ {\rm d}t = \frac{{\rm d}s}{v} = \frac{\sqrt{1+y'^2}\,{\rm d}x}{\sqrt{2gy}} $$

with the time found from the integral

$$ \boxed{ T = \int \limits_{x_1}^{x_2} \dfrac{\sqrt{1+y'^2}}{\sqrt{2gy}} \,{\rm d}x } $$

NOTE that $y' = {\rm d}y/{\rm d}x$