Fastest slide

draw a slide. try to catch the gold marble.

draw a slide between two rings and race it against the straight line and a hidden gold path

The shortest path between two points is a straight line, but it is not the fastest one for something falling under gravity. A slide that drops steeply first gets the marble moving early, and it keeps that speed for the rest of the trip. The fastest slide of all is a cycloid, the curve a point on the rim of a rolling wheel traces, and when the finish is far enough across it even dips below the finish and climbs back up, and still wins. Johann Bernoulli set this puzzle in 1696, and Newton solved it in a single night. The race here has no friction; a real rolling marble is slower on every slide by the same factor, so the winning shape does not change.

Concepts: gravity, energy, optimization, geometry.

An interactive you play with one thumb, on Wundr.