(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am after an expression for the time T it takes to slide down from height zero to -h of the brachistochrone. (Starts from x=y=0, slides along a distance -x, descends a height -h).

2. Relevant equations

I have deduced that

[tex] T = \frac{1}{(2g)^{1/2}} \int _{0}^{-h} \frac{(x^{\prime}^{2}+1)^{1/2}}{(-y)^{1/2}} dy

[/tex]

Along the way, I have shown (using Euler-Lagrange Eqns) that

[tex]

\frac{x^{\prime}}{(-y)^{1/2}(1+x^{\prime 2})^{1/2}} = const = \frac{1}{\eta^{1/2}}

[/tex]

and

[tex]

x^{\prime} = - \frac{y^{1/2}}{(\eta-y)^{1/2}} =>

x = - \int \frac{y^{1/2}}{(\eta-y)^{1/2}} dy

[/tex]

and, after a horrid bit of integration:

[tex]

x(\theta) = -\frac{\eta}{2}(\theta - sin\theta),

y(\theta) = -\frac{\eta}{2}(1 - cos\theta)

[/tex]

3. The attempt at a solution

The question suggests I combine the result

[tex]

\frac{x^{\prime}}{(-y)^{1/2}(1+x^{\prime 2})^{1/2}} = const = \frac{1}{\eta^{1/2}}

[/tex]

with [tex]\eta = h[/tex]

with the equation I derived for T

[tex] T = \frac{1}{(2g)^{1/2}} \int _{0}^{-h} \frac{(x^{\prime}^{2}+1)^{1/2}}{(-y)^{1/2}} dy

[/tex]

(I presume that I have got this equation for T correct?)

On doing so, I simplified the integral to

[tex] T = \frac{1}{(2gh)^{1/2}} \int _{0}^{-h} \frac{(x^{\prime}^{2}+1)}{x^{\prime}} dy

[/tex]

which, using [tex]x^{\prime} = - \frac{y^{1/2}}{(\eta-y)^{1/2}}[/tex] I boiled down to

[tex] T = \frac{h}{(2gh)^{1/2}} \int _{0}^{-h} \frac{1}{y^{1/2}(h-y)^{1/2}} dy

[/tex]

I am at a loss to know how to solve this integral (assuming I've got it right so far, and need to solve it!).

Could anyone suggest the next move to make? This is frustrating!

Cheers,

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: The brachistochrone

**Physics Forums | Science Articles, Homework Help, Discussion**