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The Shortest Timed Path

  • Thread starter puneeth
  • Start date
9
0
1. 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 travelling along a straight path.
2. Homework 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 [tex]\sqrt{2gy}[/tex]
[tex]\frac{dx}{dt}[/tex]= vcos[tex]\theta[/tex]
[tex]\frac{dy}{dt}[/tex]= vsin[tex]\theta[/tex]
i am unable to integrate and apply the condition for minimum time- please help.
 

Answers and Replies

Dick
Science Advisor
Homework Helper
26,258
618
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.
 

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