Brachistochrone homework

1. Sep 17, 2011

Liquidxlax

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) = $\sqrt{\frac{a}{g}}\int \sqrt{\frac{1 - cos\vartheta}{cos\vartheta_{0} - cos\vartheta}}d\vartheta$

integral is from theta naught to pi

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

$\frac{df}{dx}=\frac{d}{dy}\frac{df/dx'}$

3. The attempt at a solution

1/2 mv2 = mg(y-y1

v = $\sqrt{2g(y-y1}$

dt = ds/v

T = $\int\frac{\sqrt{1+(x')2}}{\sqrt{2g(y-y1}}$ 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 = $\int\frac{\sqrt{1 + (a( 1 - cos(\vartheta))2}}{\sqrt{2ga(cos(\vartheta) - cos(\vartheta0)}}(asin(\vartheta)$

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. Sep 18, 2011

Spinnor

Re: Brachistochrone

Do this help?

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3. Sep 18, 2011

Liquidxlax

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

4. Sep 18, 2011

Spinnor

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?