Brachistochrone Race

What is modelled

A point mass moves along three frictionless tracks from the same start to the same finish: a cycloid arc, a straight ramp and a circular arc. The brachistochrone problem asks for the curve of least descent time; Johann Bernoulli showed in 1696 that it is a cycloid. The ball starts from rest, so the cycloid first drops steeply to build speed, then flattens.

Equations

• Energy: v = √(2 g (y0 - y)) for a sliding point; rolling bodies carry rotational energy, a = g sinθ / (1 + κ), κ = 0.4 for a sphere, 2/3 for a shell.
• Cycloid x = a(θ - sinθ), y = a(1 - cosθ), descent time T = θend √(a/g).
• Straight ramp: T = √(2 L / (g sinα)).
• Circular arc: T = ∫ ds / v evaluated by quadrature.

Checks run in the page

• Simulated descent times against the three theoretical values.
• The cycloid is the fastest of the three.
• Arrival speed equals sqrt(2 g h) for all tracks.

Honest simplifications

The ball is a point on a curve (a bead on a wire); rolling and drag are simple loss terms, and the rolling modes only add the rotational inertia factor (no slipping assumed). Real balls on a real ramp leave the track if the curvature is too high. The cycloid is the optimum only for the frictionless sliding case; with friction or rolling the best shape changes slightly.

• Run Frictionless and watch the cycloid ball overtake the straight one even though its path is longer.
• Choose Moon gravity: all times scale as 1/sqrt(g) but the order does not change.
• Switch to the rolling sphere: every time grows by the same factor sqrt(1 + 0.4) = 1.18.