Newton's Cradle: Why Only the End Ball Swings

The demo everyone has flicked

A Newton's cradle is a row of identical polished steel balls — almost always five — each hung by a two-string V so it can swing only in one vertical plane and always returns to rest touching its neighbours in a perfectly straight line. Pull the ball on the left aside and release it. It swings down, strikes the stationary four, and the ball on the far right kicks out to nearly the same height, arcs back, and passes the momentum straight back across the line. Left ball out, right ball out, left ball out — a metronomic clack…clack…clack that runs for a minute or more before friction and sound quietly bleed the motion away.

The behaviour is uncanny because the four middle balls barely move. The information — the momentum and the energy — seems to teleport through them. Release two balls together and precisely two emerge; three gives three. The device is, in effect, an analog computer that solves a collision problem in real time and displays the answer with a swinging ball. Sold as an "executive ball clicker" and a physics-classroom staple, it is one of the most-owned pieces of science apparatus in the world.

Two conservation laws, one collision

Every collision must conserve linear momentum p = mv — that is Newton's laws in disguise, a consequence of no net external horizontal force during the brief impact. So if a ball of mass m arrives at speed v, the total momentum leaving the far end must equal mv. That single equation, though, does not pick a unique outcome: one ball at speed v, two balls at v/2, or four balls at v/4 all carry the same momentum mv.

The tie-breaker is kinetic energy. The steel balls collide almost perfectly elastically, meaning kinetic energy ½mv² is (very nearly) conserved too. Now solve both books at once. For n identical balls that emerge at a common speed u:

• Momentum: n·m·u = m·v  →  u = v/n
• Energy: n·½·m·u² = ½·m·v²  →  n·u² = v²

Substitute the first into the second: n·(v/n)² = v², i.e. v²/n = v². That equation is only satisfied when n = 1. So the only outcome that pays both bills is a single ball leaving at the full incoming speed v. Two balls at v/2 would conserve momentum but throw away half of the energy — nature forbids it. Launch two balls, and the same algebra demands exactly two leave; the incoming and outgoing configurations must match.

The equation for one ball hitting one ball

The cleanest way to see the mechanism is the two-body elastic collision, then chain it. For a moving ball (mass m, speed v) striking an identical stationary ball, conserving both p and KE gives a famous result: the moving ball stops and the struck ball moves off at v. In general, for masses m₁, m₂ with m₁ moving at v₁ into a stationary m₂:

• v₁′ = v₁·(m₁ − m₂)/(m₁ + m₂)
• v₂′ = v₁·2m₁/(m₁ + m₂)

Set m₁ = m₂ = m and the first equation gives v₁′ = 0 (incoming ball stops), the second gives v₂′ = v₁ (target takes the full speed). The velocities are simply swapped. In the idealised cradle you can imagine this swap cascading: ball 1 stops and hands v to ball 2, which stops and hands v to ball 3, and so on down the line at the speed of a stress wave, until ball 5 has nowhere to pass it to and flies off. The coefficient of restitution e — the ratio of separation speed to approach speed — encodes how elastic the hit is; e = 1 is perfectly elastic. Hardened chrome steel reaches e ≈ 0.90–0.99, which is why the cradle runs so long.

The hidden truth: a compression wave, not billiard balls

The two-body swap story is a useful lie. The balls in a real cradle are touching, so the collision is not a sequence of separate impacts — it is a single event in which a compression pulse travels through the whole line. When the incoming ball strikes, it locally squashes the steel by a few micrometres and launches an elastic wave that propagates at roughly the speed of sound in steel, about 5,000 m/s. Across balls a few centimetres wide, the whole transaction is over in a fraction of a millisecond.

The contact between two spheres is not linear either: it follows Hertzian contact mechanics (Heinrich Hertz, 1882), in which the restoring force grows as the compression to the 3/2 power, F ∝ δ³/². This nonlinearity, plus the finite stiffness of the steel, is what actually determines that one clean ball emerges. In a perfectly rigid, dispersionless line the outcome would be ambiguous; it is the real elasticity of the balls that selects the single-ball answer and produces the crisp separation. Physicists have studied "Newton's cradle" as a chain of Hertzian contacts and even as a testbed for solitons — self-reinforcing solitary waves. Change the material (softer balls, or a graded chain of different sizes) and the tidy one-in-one-out rule breaks: you get multiple balls moving, exactly as a real, slightly dissipative chain should.

The numbers: speeds, forces, and where the energy goes

Take a garden-variety cradle: chrome-steel balls of diameter 2 cm (mass about 33 g), lifted so each rises h ≈ 3 cm before release. From energy conservation the impact speed is v = √(2gh) = √(2·9.81·0.03) ≈ 0.77 m/s — a gentle amble. The incoming kinetic energy is ½·0.033·0.77² ≈ 0.010 J, about a hundredth of a joule.

• Contact time. Each Hertzian impact lasts on the order of 100–200 μs. To change a 0.77 m/s ball's momentum (≈ 0.025 kg·m/s) in ~150 μs requires an average force F = Δp/Δt ≈ 170 N — roughly the weight of a 17 kg mass, delivered through a contact patch under a millimetre across, giving contact stresses of hundreds of MPa.
• Losses. With e ≈ 0.95, each collision keeps about e² ≈ 90% of the energy; the rest becomes the audible click (sound), a whisper of heat in the steel, and tiny elastic vibrations. That is why the rebound height creeps down by a few percent per swing and the toy eventually stops.
• Period. Between clicks each ball swings as a pendulum of length L. For strings of L ≈ 0.15 m the small-swing period is T = 2π√(L/g) ≈ 0.78 s, so the clicks land a little under once a second — the familiar office rhythm.

History, name, and where the physics shows up for real

The underlying collision experiments are far older than the toy. In 1662–1666 the physicists John Wallis, Christopher Wren, and Christiaan Huygens independently worked out the laws of elastic collision and presented them to the young Royal Society — this is the science Isaac Newton codified in the Principia (1687). The desktop device itself is modern: the English actor Simon Prebble is credited with coining the name "Newton's cradle" for a wooden version he marketed in 1967, and the chrome executive-toy form followed soon after. Newton never built one.

The same momentum-and-energy accounting governs serious engineering:

• Billiards and snooker. A dead-centre "stop shot" is the two-ball cradle collision — the cue ball halts and the object ball leaves at full speed.
• Pile driving and hammers. Momentum transfer through a chain of contacts is exactly how a drop-hammer drives energy into a pile or a nail.
• Crash safety. Cars are engineered to do the opposite — crumple zones deliberately make collisions inelastic so kinetic energy is absorbed rather than bounced back into the occupants.
• Granular chains and metamaterials. Chains of Hertzian spheres are an active research tool for shaping and filtering shock waves.

Misconceptions and the fine print

"The balls pass through each other untouched." No — they are always in contact, and momentum is handed on by a real elastic wave, not by teleportation. The middle balls do move, by micrometres, for microseconds.

"Momentum conservation alone explains it." It doesn't. Momentum permits two-balls-at-half-speed; it is the additional demand of energy conservation (and, in the fine print, the Hertzian nonlinearity of steel) that forces exactly one ball out. Drop this and the puzzle has infinitely many false answers.

"A perfect cradle would swing forever." Never — the click you hear is escaping energy. Sound radiation, internal friction (hysteresis) in the steel, air drag, and pivot friction guarantee decay. Even at e ≈ 0.99 the amplitude falls geometrically.

"Lifting two lifts them at half speed." Real cradles are imperfect: because the balls aren't perfectly identical or perfectly aligned, launching several often produces a slightly messy result — a lead ball plus a trailing straggler — which is itself a clue that the ideal swap picture is an approximation. On safety, the toy is harmless: forces are tens to a couple hundred newtons over a sub-millimetre patch, enough to click but not to chip a well-made hardened ball, and the swing energy is a fraction of a joule.

What conservation laws permit vs. what actually happens when 1 ball strikes 4 at rest

Outcome	Balls out	Speed out	Momentum conserved?	Kinetic energy conserved?
One ball at full speed (observed)	1	v	Yes (mv)	Yes (½mv²)
Two balls at half speed	2	v/2	Yes (2·m·v/2 = mv)	No (½·2m·(v/2)² = ¼mv²)
Four balls at quarter speed	4	v/4	Yes (4·m·v/4 = mv)	No (only ⅛mv²)
All balls stick, slide together	5	v/5	Yes (mv)	No (perfectly inelastic)
Two lifted → exactly two out	2	v	Yes (2mv)	Yes (mv²)