Time ball needs for a certain path/courve

[hriby]

Hi,
it is probably a simple question, but it bothers me for quite some time.
How can I calculate time that ball needs for certain path/courve, where path equation is given?
I could use
$$t=\int_{T_1}^{T_2}\frac{ds}{v}$$
where $$v$$ is velocity and $$ds^2=dx^2+dy^2$$.
Because of
$$\frac{1}{2}m\,v^2=m\,g\,y$$
we get
$$v=\sqrt{2g\,y}$$
and new "formula" for time is
$$t_{12}=\int_{T_1}^{T_2} \frac{\sqrt{1+{y'}^2}}{\sqrt{2g\,y}}dx, \quad y'=\frac{dy}{dx}$$
If we have a curve in parametric form (e.g. cycloide), how can I calculate this equation numericaly?
Any guidance/hint/web link is appreciated.
Many thanks!

 

[benorin]

http://mathworld.wolfram.com/BrachistochroneProblem.html

 

[benorin]

or else

http://mathworld.wolfram.com/TautochroneProblem.html

 

[hriby]

Thanks benorin, but I need to go a little bit further (and that is missing to me :shy:):

let say, that I get exact parametric equation for cycloid (like in brachistochrone problem, I numericaly calculate $$\frac{1}{2}k^2$$). How shall I proceed with integral? Thx.

 

[benorin]

I'll try to come back to this later, after my homework is done. Note to self: mention awesome solution to the brachistochrone problem by means of fractional integration which may be found in Special Functions by Andrews, Askey, and Roy (the big red book upstairs in your room).