The Brachistochrone Curve

The Curve of Fastest Descent

"Given two points at different heights, along which curve does a ball, pulled only by gravity, slide from one to the other in the shortest time?" — Johann Bernoulli, 1696

⌒ Cycloid The path traced by a point on the rim of a rolling wheel. Bernoulli's answer.
/ Straight line Shortest path — but not the fastest. Too shallow to build speed.
⌐ Steep-then-flat Drops fast to gain speed, but wastes that speed on a long flat run.
↗ The Orbital Analogy The same cycloid, scaled down, traces a rocket's ascent into a stable orbit — that's why rockets curve over instead of shooting straight up.

The Race

Three balls. Same start, same finish, same gravity. The cycloid wins — it trades a bit of extra distance for a steep early drop that builds speed fast, then levels off gracefully instead of wasting it on a flat run.

Why we don't shoot straight up

Altitude alone doesn't put you in orbit — sideways speed does. A rocket fired straight up spends all its fuel fighting gravity and arrives at the top with zero horizontal velocity, so it falls right back down. Real rockets begin vertical and then tilt over into a gravity turn that, in this diagram, traces the very same cycloid you saw win the race above — trading a little extra path for a lot more sideways speed, until it tangents into a stable circular orbit.

The same principle, two scales

Bernoulli's insight was that the geometry of a path shapes how efficiently gravity can be converted into speed. On a hillside this is a curiosity. In orbital mechanics it's the difference between falling back to Earth and circling it forever. Neither the cycloid nor the gravity turn is the shortest route — but both are the fastest, because they let gravity do work for you instead of against you.