# The Shortest Timed Path

1. Mar 13, 2008

### puneeth

1. The problem statement, all variables and given/known data
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 travelling along a straight path.
2. Relevant equations
Energy conservation,

3. 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.

2. Mar 13, 2008

### 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.