What is the Shortest Timed Path for a Body to Reach Two Points in Space?

[puneeth]

Homework Statement

A frictionless path is set up between two points in space at different heights, such that a body released from the higher point along this path takes minimum possible time to reach the lower point. Show that the path is a cycloid. Also show that a body released along this path takes about 25% lesser time as compared to traveling along a straight path.

Homework Equations

Energy conservation,

The Attempt at a Solution

Let us assume that the total height difference is H and the horizontal distance is X.
when the body falls through a height y its speed is $$\sqrt{2gy}$$
$$\frac{dx}{dt}$$= vcos$$\theta$$
$$\frac{dy}{dt}$$= vsin$$\theta$$
i am unable to integrate and apply the condition for minimum time- please help.

 

[Dick]

This is a calculus of variations problem. If you write the curve as y=f(x) you want to express the time to fall as an integral expression involving f. Then change it into a differential equation using Euler-Lagrange. Then solve the differential equation.