Newton's Cradle: Why Exactly One Ball Flies Out

A pendulum's swing depends on length, not mass

Each ball in a Newton's Cradle hangs as a pendulum, and Galileo noticed in 1602 that a pendulum's swing takes the same time regardless of how far you pull it back, at least for small angles — a property called isochronism that made pendulums the basis of accurate clocks for over 200 years, starting with Christiaan Huygens's first pendulum clock in 1656. The period T depends only on string length L and gravity g:

T = 2π × √(L / g)
1 m pendulum on Earth (g = 9.8 m/s²) → T ≈ 2.0 s

Mass cancels out of that formula completely: a heavy steel ball and a light wooden ball on identical-length strings swing in perfect sync. Galileo proved exactly this by releasing two different-mass pendulums together and watching them stay matched throughout the swing. As each ball swings, energy simply converts back and forth between potential energy at the top (PE = mgh) and kinetic energy at the bottom (KE = ½mv²), with the total staying constant.

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Two conservation laws must hold at once

Newton's Cradle uses hard steel balls, and collisions between hard, barely-deforming balls are elastic — kinetic energy is conserved along with momentum. When one ball at speed v strikes stationary balls of the same mass m, both of these must hold simultaneously:

Momentum:        m·v  = m·v₁ + m·v₂
Kinetic energy: ½m·v² = ½m·v₁² + ½m·v₂²

Two equations, two unknowns (v₁ and v₂) — and for equal masses, the only solution that satisfies both at once is v₁ = 0, v₂ = v. The striking ball stops dead, and the last ball in the row leaves at exactly the original speed. There is no other combination of exit velocities that solves both equations together.

Why "two slow balls" is mathematically forbidden

It's worth checking the tempting wrong answer directly. Suppose 1 ball hits at speed v, and 2 balls exit together at v/2 each instead of 1 ball exiting at v:

Momentum check: m·(v/2) + m·(v/2) = m·v            ✅ conserved
KE check:      ½m·(v/2)² + ½m·(v/2)² = ¼·m·v² ≠ ½·m·v²   ❌ half the energy vanished

Momentum alone is satisfied, but half the kinetic energy has gone missing with nowhere for it to go in an elastic collision — so physics forbids that outcome. The same logic scales up cleanly: pull back 2 balls and 2 balls exit at the same speed; pull back 3 and 3 exit. With 5 equal-mass balls, the middle balls barely seem to move at all, briefly compressing and acting purely as a relay for momentum and energy passing straight through the chain.

Why a real cradle eventually stops

A perfectly elastic, frictionless Newton's Cradle would swing forever. Real ones don't: every click of the balls converts a sliver of kinetic energy into sound and heat, and air resistance and string flex bleed away a little more each cycle. This gradual loss is called damping, and it's why the swings visibly shrink over time even though steel comes remarkably close to perfectly elastic — the underlying conservation laws are still obeyed at every single collision, it's just that "total energy" quietly includes the sound you hear with each click.

Frequently asked questions

Why does exactly one ball fly out of Newton's Cradle, not two slower ones?

A collision between equal-mass balls must satisfy both conservation of momentum (mv total) and conservation of kinetic energy (½mv² total) at the same time, since the balls are hard steel and barely deform. For one incoming ball hitting a row of stationary equal-mass balls, the only combination of exit velocities that satisfies both equations simultaneously is one ball leaving at the full original speed — two balls leaving at half speed each conserves momentum but only carries away half the kinetic energy, which is forbidden.

Does the mass of a pendulum bob affect its swing period?

No. The period of a simple pendulum is T = 2π√(L/g), which depends only on the string length L and gravitational acceleration g — mass cancels out of the equation entirely. A heavy steel ball and a light wooden ball on identical-length strings swing perfectly in sync, as Galileo demonstrated by releasing two different-mass pendulums together and watching them stay matched.

Why does a real Newton's Cradle eventually stop swinging?

No real collision or swing is perfectly elastic or frictionless. Each click of the balls converts a small amount of kinetic energy into sound and heat, and air resistance and string flex bleed away a little more each cycle. This gradual energy loss, called damping, is why the swings visibly shrink over time even though momentum and energy are very nearly conserved in any single collision.

Try it live

Everything above runs in your browser — open Newton's Cradle, lift 1, 2, 3 or 4 balls at once, and watch conservation of momentum and kinetic energy force exactly that many balls to exit every single time. Nothing is installed, nothing is uploaded.

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