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Brachistochrone homework

  1. Sep 17, 2011 #1
    1. The problem statement, all variables and given/known data

    consider a sigle loop of the cycloid with a fixed value of a. A car is released at a point P0 from rest anywhere on the track between the origin and the lowest point P, that is P0 has a parameter 0<theta0 < pi. show that the time for the cart to roll from P0 to P is given by the integral

    time( P0]/sub] -> P) = [itex]\sqrt{\frac{a}{g}}\int \sqrt{\frac{1 - cos\vartheta}{cos\vartheta_{0} - cos\vartheta}}d\vartheta[/itex]

    integral is from theta naught to pi

    and prove the integral equals [itex]\pi\sqrt{\frac{a}{g}}[/itex] the integral may be tricky and you can use theta = pi -2(alpha)
    2. Relevant equations

    [itex]\frac{df}{dx}=\frac{d}{dy}\frac{df/dx'}[/itex]

    3. The attempt at a solution

    1/2 mv2 = mg(y-y1

    v = [itex]\sqrt{2g(y-y1}[/itex]

    dt = ds/v

    T = [itex]\int\frac{\sqrt{1+(x')2}}{\sqrt{2g(y-y1}}[/itex] dy

    test the Euler formula to get

    y = a(1 - cos(theta)) y' = asin(theta)

    x = a (theta - sin(theta)) x' = a - acos(theta)

    This is now the part I am having a problem with

    Now i substitute my y and x' in

    T = [itex]\int\frac{\sqrt{1 + (a( 1 - cos(\vartheta))2}}{\sqrt{2ga(cos(\vartheta) - cos(\vartheta0)}}(asin(\vartheta)[/itex]

    and somehow that equals the top equation. Then i need to integrate it, which i get no where near close to what it wants no matter what the substitution and trig identities

    not sure why all the itex isn't working
     
  2. jcsd
  3. Sep 18, 2011 #2
    Re: Brachistochrone

    Do this help?
     

    Attached Files:

  4. Sep 18, 2011 #3
    Re: Brachistochrone

    it sure did sometimes i'm such a retard, x and y do not depend on each other they depend on theta...

    because i was doing sqrt ( 1 y'^2) dx

    thank you very much
     
  5. Sep 18, 2011 #4
    Re: Brachistochrone

    x and y do depend on each other but the answer had to be in terms of theta so that was a more useful parameter?
     
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