The Brachistochrone: Why the Fastest Path Isn't a Straight Line

Which ramp gets a bead from A to B fastest? Not the straight one. The fastest path dips steeply first to gain speed, then flattens: a cycloid. This is the brachistochrone, a problem that started the calculus of variations.

The video races the straight line, a circular arc and the cycloid, shows the surprising tautochrone property, and connects it to Snell's law of light.

In this video

• Why the straight line loses
• The cycloid as the fastest curve
• Tautochrone: same arrival time from any start
• The light-ray analogy and Snell's law