Brachistochrone & tautochrone simulation

Of all the ramps between a high point and a lower one, which lets a frictionless bead slide between them fastest — and is it the straight line?

▶ Run the simulationSee the measured result

Units: s (least descent time A→B for B at the cycloid arch bottom; π√(R/g), Bernoulli 1697)

How the lab tests it

Race beads from the same A to the same B down a straight line, a circular arc, and a cycloid, stepping each by the exact tangential gravity (conserved energy ½v²+gy is the check), and time the arrivals; then drop beads from different heights on the cycloid.

What it checks

the BRACHISTOCHRONE: the cycloid is fastest (not the straight line) — and not just here: it is the optimum over ALL curves, proven by Johann Bernoulli in 1696 — reaching the minimum time π√(R/g) to the cycloid's bottom; and the TAUTOCHRONE: beads released from any height on that same cycloid reach the bottom simultaneously (Huygens' isochronism, 1659)