Find a curve such that time travel is minimum

AI Thread Summary
The discussion focuses on finding a curve, y = f(x), that minimizes the travel time of a ball from point A to point B, referencing the brachistochrone problem. Participants emphasize the need to apply principles of physics, specifically Newton's laws and energy conservation, to derive the relationship between y and x. A hint is provided to consider kinetic and potential energy in the analysis. The conversation also suggests solving for velocity and separating variables for time calculation. The goal remains to identify the optimal curve for minimum travel time.
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Homework Statement


[PLAIN]http://vctson.com/temp/curve.jpeg
Find the curve y = f(x) so that the ball travels from A to B in minimum time?

Thanks!
 
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What have you worked out thus far? You have to show some effort.
 
I tried at some point on the curve, the gravity force goes down, then use 2nd Newton law. Thus, P = ma = my''. But, P = mg. It does not seem right and I can't find the relation y and x.
 
Any hint? Thanks!
 
this is the brachistochrone, so what is you kinetic and potential energy?

solve in terms of v then separate for ds/dt.

ds = sqrt(dx^2 + dy^2)
 
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