Here is an easier puzzle to get started. Imagine a frictionless, gravity-driven track between two points at the same height, separated by a horizontal distance X. The track is in the shape of a squared-off U, with a vertical drop, a horizontal traverse of length X, and a vertical rise. Suppose our car can negotiate the right-angle turns without losing energy. We start our car at one endpoint, at zero velocity, and we are interested in how long it takes the car to arrive at the other endpoint. If the verticals are very short, the car will only accelerate for a brief time, and thus it will move very slowly along the horizontal segment, so a sufficiently shallow track will take an arbitrarily long time to traverse. If, on the other hand, the verticals are very long, the car can take an arbitrary amount of time falling down and up the vertical segments, and so a sufficiently deep track will also take an arbitrarily long time to traverse. Somewhere between these two extremes is the optimal rectangular-U-shaped track. Calculus easily gives the optimal proportions for this rectangle. I was slightly surprised by the answer. I think this would be a fine exercise for elementary differential calculus. Is there a way to get the answer quickly without calculus? On Wed, May 16, 2012 at 9:56 AM, Simon Plouffe <simon.plouffe@gmail.com>wrote:
Hello,
I am not certain I understand correctly but isn't this curve the brachistochrone ?
http://en.wikipedia.org/wiki/Brachistochrone_curve
Which is the minimal time curve, is it what you are looking for ?
Or another one with constraint ? : http://en.wikipedia.org/wiki/Tautochrone_curve
Best regards, simon plouffe _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun