What is the brachistochrone?

It is the curve of fastest descent between two points under gravity. Johann Bernoulli posed this problem in 1696 and the solution is a cycloid. This simulator shows four balls racing on different curves to prove the cycloid always wins.

Why does the cycloid win even though it is longer?

The cycloid dips steeply at the start converting potential energy to speed early. The ball reaches high velocity quickly then traverses the longer path faster than a ball on the shorter straight line that accelerates slowly.

Brachistochrone — Curve of Fastest Descent Race

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