Brachistochrone Race (2D)

Four frictionless tracks between the same two points: which one is fastest? Same constrained-motion solver as the 3D page.

Live readout

Numerical checks

The science

For a frictionless bead the speed after a vertical drop d is v = sqrt(2 g d), so the travel time along a path is T = integral of ds / sqrt(2 g d). The cycloid x = a(t - sin t), y = a(1 - cos t) minimises it (Johann Bernoulli, 1696): T = t_end sqrt(a/g). The straight ramp has T = sqrt(2 (D^2 + H^2) / (g H)).

The integral is done numerically in a substitution that removes the start-point singularity; both closed forms are used as independent checks.
Simplifications: point bead, no friction, no air drag, no rolling inertia.

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