The brachistochrone problem asks: what curve gives the fastest descent under gravity between two points?
The answer is a cycloid, not a straight line. This app compares three tracks between the same endpoints:
a straight line, a simple parabola-like curve, and the cycloid. Press play to race the balls.
You can drag the end point B on the canvas.
Drag point B. Then press Play Race. The green cycloid should arrive first.
For a falling bead without friction, the travel time along a path is approximated numerically by
T ≈ Σ ds / √(2 g y),
where y is vertical drop from the start and ds is a small arc segment.
Curve
Color
Approx. Time
Length
Current Progress
Result
The cycloid shown is chosen to pass through the same start and end points by solving for its generating radius and angle.
This is a numerical educational demo of the classical result by Johann Bernoulli.