Physics
Brachistochrone
Bead race: the cycloid beats the straight line.
What happens
Four beads race down four curves between the same two points: which track is fastest? Not the straight line. The cycloid wins every time, because diving steeply first buys speed early, and that is precisely the curve the calculus of variations predicts. Move the finish point and race again. The tautochrone mode is the other half of the discovery: four beads released from different heights of ONE cycloid arch all reach the bottom together, at T = π·√(R/g) - whatever the start.
Equations used in the module
Equations x = a(θ−sin θ); y = a(1−cos θ) T = √(a/g) θ_end v² = 2gy sin α/v = const
