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Time in brachistochrone problem

  1. Nov 18, 2005 #1
    I hope that you've heard about Brachistochrone problem: http://mathworld.wolfram.com/BrachistochroneProblem.html
    Given two points, I can find (calculate) the courve, on which the ball needs minimum time to travel from point 1 to point 2.
    I get the equation for the courve, which is cycloid, in parametric form, let say:
    [tex]
    x(\theta)&=&C\left(\theta-\sin{\theta}\right),
    [/tex]

    [tex]
    y(\theta)&=&-C\left(1-\cos{\theta}\right).
    [/tex]

    Now I also need to calculate the time needed...
    How could I calculate it out of formula below using the equation for the courve/cycloid in parametric form?
    [tex]t_{12}=\int_{T_1}^{T_2} \frac{\sqrt{1+{y'}^2}}{\sqrt{2g\,y}}dx, \quad y'=\frac{dy}{dx}[/tex]
    Thanks for your answers!
    Hriby
     
    Last edited: Nov 18, 2005
  2. jcsd
  3. Nov 18, 2005 #2
    how 'bout:

    dy/dx = (dy/d_theta)/(dx/d_theta)
     
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