Is the circular arc the unique circle through the two points?

The arc shown is the circle through the origin and the endpoint whose center lies on the perpendicular bisector with the lower tangent geometry used in many textbook comparisons — it is a standard competitor, not the brachistochrone.

Does friction or rotation of the bead matter?

This lab uses a point mass sliding without friction and without rotational kinetic energy.

Brachistochrone race

Beads race down a line, a circular arc, and a cycloid with v = √(2gy); the cycloid wins the descent-time minimum.

About this model

Johann Bernoulli posed the brachistochrone: which curve between two points lets a bead sliding without friction under gravity arrive fastest? The cycloid wins against the straight chord and a circular arc through the same endpoints used here. Times are computed as ∫ ds/√(2gy) with y measured downward from the common start. The animation advances three beads at the same parameter pace for a race feel, not three separate physics clocks.

Who it's for: Calculus of variations introductions and classical mechanics problem sets.

Key terms

Brachistochrone
Cycloid
Energy conservation
Descent time
Johann Bernoulli

How it works

Three tracks join the same endpoints with zero initial speed; speed along the path follows v = √(2gy) with y measured downward from the start. The cycloid minimizes the descent time (Johann Bernoulli, 1696).

Frequently asked questions

Is the circular arc the unique circle through the two points?: The arc shown is the circle through the origin and the endpoint whose center lies on the perpendicular bisector with the lower tangent geometry used in many textbook comparisons — it is a standard competitor, not the brachistochrone.
Does friction or rotation of the bead matter?: This lab uses a point mass sliding without friction and without rotational kinetic energy.