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The Shortest Timed Path

  1. Mar 13, 2008 #1
    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 [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.
     
  2. jcsd
  3. Mar 13, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    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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