I derived the general equation of a Brachistochrone, which is a cycloid.
$y=A(1-\cos\theta)$
$x=A(\theta-\sin\theta)$
I'm now trying to calculate the time needed to go from the origin to a point $(x,y)$.
From previous analysis I found that the time is equal to
$T=\sqrt{\frac{1}{2g}}\int_0^x\sqrt{\frac{1+(y')^2}{y}}dx$
I'm struggling to solve this. This is what I found so far
$\frac{dx}{d\theta}=A(1-\cos\theta)$
$\frac{dy}{d\theta}=A\sin\theta$
Applying the chain rule: $\frac{dy}{dx}=\frac{dy}{d\theta}\frac{d\theta}{dx}=\frac{A\sin\theta}{A(1-\cos\theta)}=\frac{\sin\theta}{1-\cos\theta}$
Therefore, $(\frac{dy}{dx})^2=(\frac{\sin\theta}{1-\cos\theta})^2$
Therefore,
$T=\sqrt{\frac{1}{g}}\int_0^x\sqrt{\frac{1-\cos^2\theta}{(1-\cos\theta)^2}}d\theta$
Is this right? How do I go on from here? I have the coordinates of two points (and therefore I could derive the equation of the brachistochrone curve between them) and I would like to find the time taken to fall from the initial to the final point along the brachistochrone under acceleration g. Thank you!
Best Answer
Changing variables from $x$ to $\theta$ via $x=A(\theta - \sin\theta)$, the integral that represents the time to go from $(x,y)=(0,0)$ to $(x_0,y_0)$ becomes $$ T=\sqrt{\frac 1{2g}}\int_{\theta=0}^{\theta_0}\sqrt{\frac{1+(y')^2}y}\frac{dx}{d\theta}\,d\theta $$ where $\theta_0$ is the value of $\theta$ that corresponds to $x_0$. You've calculated $y'=\sin\theta/(1-\cos\theta)$ and $dx/d\theta=A(1-\cos\theta)$ correctly, now plug everything in. The integral will simplify to $$ T=\sqrt{\frac A{2g}}\int_{\theta=0}^{\theta_0}\sqrt{\frac{(1-\cos\theta)^2+\sin^2\theta}{1-\cos\theta}}\,d\theta=\sqrt{\frac Ag}\int_{\theta=0}^{\theta_0}\sqrt{\frac{1-\cos\theta}{1-\cos\theta}}\,d\theta=\sqrt{\frac Ag}\theta_0. $$