The quickest way down is not a straight line

2026-06-03

You want to get a marble from a high point A to a lower point B, off to one side, as fast as possible. No motor, no push - just gravity and a frictionless track you get to bend into any shape you like. What shape do you bend it into?

The straight line is the shortest track. The obvious answer is almost always wrong.

Three beads, released at the same instant. The straight line is the shortest route and it loses badly. The winner is the cycloid - the curve traced by a point on the rim of a rolling wheel - and it wins despite being the longest track of the three, and despite dipping below the finish line before climbing back up to it.

Why the long way is faster

A bead sliding without friction has only one thing going for it: the speed it has built up. And the only thing that builds speed is dropping. At any point on any track, a bead released from rest is moving at exactly

v = √(2 g h)

where h is how far it has fallen so far - nothing else. Not the shape, not the distance travelled. Only the drop.

That is the whole secret. The cycloid plunges almost straight down at the start. In the first fraction of a second it has already fallen further than the other two tracks will let their beads fall in the same time, so it is already moving faster - and it carries that speed through the entire rest of the journey. The straight line, by contrast, spends the early moments barely dropping at all, creeping along at a crawl. By the time it finally picks up speed, the race is over.

The cycloid is making a bargain: a longer path, in exchange for getting fast early. It is the same instinct as a skier who drops into the steep part first to carry speed across the flat runout, rather than traversing gently and arriving slow.

A famous little contest

In 1696 Johann Bernoulli posed exactly this question to the mathematicians of Europe as a public challenge, and called the answer the brachistochrone - Greek for "shortest time". The replies came back from Leibniz, from l'Hopital, from Bernoulli's own brother Jakob, and - famously - one anonymous English solution that Bernoulli recognised at a glance as Isaac Newton's. "I know the lion by his claw," he is supposed to have said. Newton had received the problem after a long day at the Mint and solved it overnight.

Every one of them arrived at the same curve: the cycloid. There is no shape that beats it. Bend the track any way you like - the cycloid still wins.

The second surprise

The cycloid hides a second trick that is, if anything, stranger than the first. Start two beads from different heights on the same cycloid and release them together. They reach the bottom at exactly the same moment. The one that starts lower has less distance to cover; the one that starts higher builds more speed; the two effects cancel perfectly, every time. A cycloid is not just the fastest slide - it is a slide on which, no matter where you start, you always arrive in step. That is a diagram for another day.

Drag the slider to the very start and watch only the first half second. The straight line has barely moved. The cycloid is already gone.