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Find a shape of inclined plane(not necessarily plane) that takes shortest time to reach from top to bottom. Assume the ball is sliding without any friction.

Mathematics
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|dw:1347894497129:dw|
geowotsit?
assume both h and w are not zero.

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Other answers:

|dw:1347894819011:dw|
could you all delete your answers. I missed to put that the ball is sliding gliding with no friction. yep cycloid.
it's a pretty famous problem ...
http://en.wikipedia.org/wiki/Brachistochrone_curve
yep ... this is brachistochrome problem.
cycloid is a shape made my rolling circle. http://en.wikipedia.org/wiki/Cycloid
I was thinking geodesic at first, shortest distance....
i did this with calculus of variation. haven't done this with purely calculus.
What about when h=w and h and approaches infinity
h and w*
take a circle with zero radius, you will get a straight line.
* if w = 0 ...
http://www.cmsim.eu/papers_pdf/january_2012_papers/25_CMSIM_2012_Pokorny_1_281-298.pdf (brachistochrone as special case of geodesic)
|dw:1347895370580:dw|
I got that cycloid part........
What if h and w approaches infinity.....
It will never get to the bottom and you will never know..:-)
Then I think this should work:|dw:1347895555863:dw|
yep ... this is a physics problem actually.
Oh ya @estudier ........ But what if h and w are very large.... but some finite number
Will it still be that cycloid
yep .. it should be cycloid ... make by a huge circle.
OMG! This math stuff is out my level
Didnt get why cycloid........ I just know we need to maximise the accleration
Maybe at infinity, you have to go to some other mathematical model, maybe conformal, to analyze that (in conformal everything is a circle, point is circle with zero radius, line is infinite circle etc)
or sphere, perhaps..
I mean why cycloid is the answer to @experimentX question
It will only increase the length
well ... this is easily solved with calculus of variation. probably you need to be half way through undergrad to encounter calculus of variation.
When you solve the problem, that's what comes out, a deeper meaning, you mean?
oh
trust me ... things aren't that intuitive.
Ok..... I trust u. @experimentX
When did u learnt these stuff. @experimentX
i was wondering if anyone could solve it using purely calculus. I haven't tried it yet.
At university
I will think on this problem.... see If I can get to the solution
told you i'm old guy

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