Saramsha

No. 036  ·  22 October 2026

It Never Leaves Twice as Fast

Pull two balls back on a Newton's cradle and let go, and momentum alone would be just as satisfied if a single ball shot out the far side at double the speed — the total momentum comes out identical either way. That never happens, because doubling a ball's speed quadruples its share of kinetic energy, and the collisions simply don't have that much energy on hand.

Drag the cradle sideways to pull the left-hand balls back, then let go.

About this piece

Two conserved numbers, and only one lets this happen

Lift two balls on a Newton's cradle, let them fall, and two balls fly out the far end at the same speed you dropped them — never one ball at twice the speed, never four balls at half the speed. That specific outcome feels inevitable once you've seen it a hundred times, but momentum conservation on its own does not pick it. Lifting two balls of mass m to speed v gives the system momentum 2mv. Sending two balls out at v keeps that momentum exactly. So does sending one ball out at 2v — m times 2v is also 2mv. Momentum alone cannot tell these two futures apart.

Energy can. Kinetic energy grows with the square of speed, not in step with it, so doubling one ball's speed doesn't double its energy — it quadruples it. Two balls leaving at v carry a total kinetic energy of mv²; one ball leaving at 2v would carry 2mv², twice as much as the system actually has, because nothing here adds energy from outside. The collisions are elastic — no heat, no sound, no bending metal — so kinetic energy has to balance too, and it rules out every momentum-legal outcome except the one where the speed stays the same and only the headcount changes. That is the whole trick: it isn't that nature prefers tidy numbers, it's that the tidy outcome is the only one both ledgers agree on.

A chain of two-ball handshakes, nothing more exotic

No part of the cradle "knows" about this rule in advance — it falls out of a much smaller fact repeated down the line. When two balls of equal mass collide head-on and elastically, they don't share the speed between them: they swap it exactly, as if they'd traded places. Drop two lifted balls onto three resting ones, and the chain plays out one handshake at a time — each moving ball hands its speed to the next and comes to rest, all the way down a silent middle that never visibly moves, until there's no ball left to hand it to and the last two simply leave. Lift three instead of two, and the same handshake runs three times before it runs out of road. The n-for-n rule isn't programmed in anywhere in this page — it's the visible result of that one swap, repeated.

Change the mass of the last ball and the swap stops being an even trade. A heavier last ball barely moves on impact and mostly reflects the incoming speed back the way it came; a lighter one accelerates away faster than the ball that struck it. Either way, the clean "two balls at the same speed" picture breaks, even though momentum and energy are still both conserved exactly — they just no longer settle on the simple answer. The mass slider lets you watch that happen.

Each ball is a true pendulum, integrated with the real sine-of-the-angle restoring force rather than the small-angle shortcut, so large lift angles are handled honestly. Collisions resolve as genuine elastic impacts between neighbouring balls, several times a frame, which is what lets a chain of them propagate correctly down the row. There is no friction and no air resistance anywhere in the model — a real cradle loses a little energy to both every swing, which is why a real one eventually stops and this one, left alone, does not. Drag anywhere on the cradle to pull the left-hand group back by hand, or use Drop from here; changing any slider resets everything to rest. ?warm= fast-forwards past the opening drop for a representative first frame.