The Great Brachistochrone Race

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Construct a smooth slide so that a particle can descend from one point to another positioned below it (but not directly below). The classic brachistochrone problem: what form should the slide take to minimize the time of descent?

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Naively, one might think the answer is a straight line, but in fact, a straight slide is slower than many other curves; instead, the slide should be made in the shape of an arc of a cycloid.

Move the lower point around. When you've selected the point, animate the descent to show that the cycloid always wins the race against the straight line.

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Contributed by: Phil Ramsden (May 2015)
Open content licensed under CC BY-NC-SA


Snapshots


Details

An arc of a cycloid has parametric equations , . The value of is calculated (usually numerically, as it is in this Demonstration) so that the cycloid passes through the lower point for some value of between and .

The cycloid may be derived by using the techniques of variational calculus to find that function of that minimizes the integral along the curve (where is the differential of arc length).

When the value of the scaling constant has been calculated, the parameter is found to be linearly related to the time by the equation .

In this Demonstration, you can drag the lower point freely in the fourth quadrant of the - plane, and you can animate the race down between a point on the straight line and a point on the cycloid.

The cycloid always wins, but its winning margin depends on the steepness of the descent. For shallow descents, the cycloid's advantage is clear and enormous; for steep ones, the result is closer to a photo finish.

The animation rate has been scaled so that the value of appears to be the same for all choices of the lower point.



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