Five balls, and the law that does not choose

A Newton’s cradle is a row of steel balls hanging in contact. Lift one, let it fall, and one ball leaves the far end at the speed the first arrived with. Lift two and two leave. The standard explanation is that momentum and kinetic energy are both conserved, and that only this outcome conserves both.

That explanation is wrong, and it is wrong in a way that is worth more than the correct one. Momentum and kinetic energy are two equations. Three balls have three unknown final velocities. Two equations in three unknowns do not have a solution; they have a curve of solutions.

The point marked at the top left is the familiar answer. There is nothing in the two laws that prefers it to any other point on that arc. If a cradle produced the outcome in which the middle ball comes back at the striker, every conservation law in mechanics would be satisfied and every textbook explanation of the toy would still apply. Something else decides, and the something else is not a conservation law at all.

The shape of the difficulty #

This is a familiar situation wearing an unfamiliar hat. Balancing a table on four legs is a problem the equations of statics cannot settle: three equations of equilibrium, four unknown leg forces, and a one-parameter family of load distributions all of which balance. The resolution there is that the legs and the floor are not rigid — the actual distribution is decided by how much each leg compresses, which is a property of the material and appears nowhere in the equilibrium equations.

The cradle is the dynamic version of exactly that. The conservation laws are the counterpart of the equilibrium equations: true, useful, and insufficient. What resolves the indeterminacy is the same thing in both cases — the bodies are not rigid, and the force between two of them depends on how far they have squashed into one another. A collision between rigid bodies is not an idealisation with a small correction; it is a problem with no answer, and every treatment of one has smuggled in a rule about deformation somewhere.

The rule that is usually smuggled in is a coefficient of restitution: a number saying what fraction of the approach speed is returned. For two bodies that is enough, because two bodies with two conservation laws and one restitution coefficient are fully determined. For three it is not, because there are now two contacts and the question of whether they are active at the same instant has no answer within a rigid-body model. That question is the whole of what follows.

When each contact is a separate event #

Start with the case where the difficulty does not arise. Separate the balls by a small gap, so that no two contacts can overlap in time, and every collision is a two-body collision with a determined outcome.

Nothing has been assumed about collisions here. The figure comes from integrating Newton’s second law with a force between neighbours that is zero when they are not touching and grows with their overlap when they are. What the picture shows is the sequential structure: five separate two-body events, each fully determined, and a result that really is one ball out at the striking speed.

The velocities in that figure are also a clean demonstration of why the two-body case is determined. Two equal masses, momentum $v = v_1 + v_2$ and energy $v^2 = v_1^2 + v_2^2$, give $v_1 v_2 = 0$: either nothing happened or the velocities were exchanged. There is no third possibility and no continuum of answers, which is why collisions between two bodies can be solved without ever mentioning a force. The moment there are three, the trick stops working.

A real cradle is not built this way. Its balls hang in contact, and the reason is practical: gaps make it noisy and make the balls swing out of line. So the toy that is used to demonstrate the determined case is built in the one configuration where the case does not apply.

What touching balls actually do #

The far ball does not get everything. It gets 98.9 per cent of the striking speed, and the balls behind it are left with small velocities that a real cradle shows as a faint jiggle in the middle of the row — which anyone who has watched one closely has seen and which the standard explanation says should not be there.

More interesting than the number is the mechanism. With the balls touching there is no sequence of collisions; there is a pulse, a region of compression that travels along the chain. That is a wave, and the medium it travels in is peculiar. The force between two spheres is not proportional to their overlap: the contact patch between them is a circle that grows as they press, so both the area under load and the pressure in it increase together, and the force goes as the overlap to the power three halves. A chain of spheres in contact but not squeezed therefore has no stiffness at all at zero amplitude, and a wave in it has no speed until it has an amplitude. The pulse’s speed depends on how hard it is — which is the defining property of a wave whose front steepens because its own height decides how fast it travels, and it is why the compression stays a compact lump instead of spreading, in the same way and for related reasons as a pulse that two failures keep alive.

The exponent is the answer #

The cleanest way to see that the contact law is doing the deciding is to change it and nothing else.

A linear spring between the balls is a perfectly ordinary elastic contact — it is what two flat faces give, or a helical spring, or a rubber pad in its linear range. It has the same conservation laws, the same masses, the same initial condition. It produces a visibly different answer, and worse: the pulse disperses, because a linear chain has a genuine sound speed and components of different wavelength travel at different speeds, which is the ordinary business of a medium with a dispersion relation. The Hertzian chain has no sound speed to disperse about, and that is why its pulse holds together.

The curve rises throughout. A contact whose stiffness grows faster with compression confines the pulse more tightly, and a more tightly confined pulse involves fewer balls at once, and fewer balls at once is closer to the sequential limit where the transfer is exact. In the limit of an infinitely stiffening contact, the touching chain behaves like the gapped one. Steel spheres are not at that limit but are usefully near it, which is the entire reason the toy works as well as it does.

That is a satisfying answer to a question the standard account cannot even ask: why does a cradle made of steel balls work so much better than one made of anything else? Not because steel is hard, exactly, but because the geometry of two spheres in contact produces a strongly nonlinear force law, and a strongly nonlinear force law localises the pulse. A cradle of cubes would fail, and it would fail while conserving momentum and energy perfectly.

Two in, two out — and the same gap in the argument #

The demonstration that is usually offered as decisive is the one with two balls. Lift two, release them, and two leave the far end at the arrival speed. The argument runs: if only one left, it would have to carry the momentum of two, so it would leave at twice the speed and carry twice the energy, which is forbidden. Therefore two must leave.

The argument is sound as far as it goes and it does not go far enough. It rules out one particular alternative and leaves every other. Two balls arriving at speed $v$ carry momentum $2v$ and energy $2v^2$; two leaving at $v$ satisfy both; so do a great many other assignments among the five, exactly as before. The demonstration proves that the outcome is not the single silly alternative somebody thought of. It does not prove it is the one observed.

What actually produces the two-out result is the same pulse argument as before, run twice. The two moving balls are themselves in contact, so the disturbance that crosses the chain is wider — it is launched by two balls rather than one — and what emerges at the far end is a disturbance of the same width. The number that comes out is set by the width of the pulse that goes in, which is a statement about the contact law and about nothing else. A cradle made with a linear contact law shows this plainly: lift two and what leaves is three balls at unequal speeds, which is a violation of nothing at all.

What the ancestors argued about #

The toy is not Newton’s, and the question it is used to settle was already the live one when it was invented. Marcus Marci described colliding balls in 1639; Huygens, Wallis and Wren each submitted rules for collision to the Royal Society in 1668, and the quantity that would come to be called kinetic energy appears in Huygens’ as a conserved thing in the elastic case. Mariotte built a cradle of suspended ivory balls in 1673 and used it to argue about exactly the point at issue here: what a row of touching bodies does when one end is struck, and whether the answer follows from the rules of two-body collision.

It does not, and the seventeenth century could not have known why, because the resolution needs a theory of how solids deform in contact — which arrived with Hertz’s paper of 1882, and arrived from a completely different problem. Hertz was thinking about the optical interference fringes seen between two pressed lenses, wanted to know how the contact area grew with the load, and solved the elastic problem to find out. The three-halves power in every figure here is his answer to a question about glass lenses in an optics laboratory.

Making it longer #

Length is a test of the mechanism, and the mechanism survives it. If the outcome were forced by the conservation laws, the number of balls would be irrelevant: one in, one out, at any length. If it is a pulse crossing a chain, then each contact is a chance for the pulse to shed a little, and a longer chain should transfer slightly less and leave slightly more behind. That is what the integration gives.

It also explains the practical fact that cradles are sold with five balls. A longer one looks worse: the residual motion in the middle accumulates, the balls fall out of line, and the demonstration that is supposed to show a clean exchange shows a shiver. The commercial choice and the physics agree, which is the kind of agreement worth noticing because nobody designed for it.

The medium with no sound speed #

The chain of touching spheres is worth one more paragraph on its own account, because it is a genuinely unusual medium and the cradle is its simplest instance.

An ordinary elastic solid has a sound speed: a small disturbance travels at $\sqrt{E/\rho}$ regardless of how small it is, and that number is a property of the material. A chain of spheres touching but not squeezed has none. The stiffness of a contact carrying no load is zero, and it only acquires one once the contact is compressed — so the speed of a disturbance depends on its own amplitude, vanishing as the amplitude does. A chain like that carries no sound at all in the ordinary sense, and the name it has been given is a sonic vacuum.

What it carries instead are compact pulses whose width is a few particles and whose speed grows as the fifth root of their amplitude. They pass through one another and re-emerge, which is the behaviour of a solitary wave rather than of a sound wave, and it is the reason the cradle’s disturbance arrives at the far end as a lump instead of as a spread-out rumble. It is also the reason granular chains are studied as shock protection: a medium with no sound speed cannot transmit a small steady vibration at all, and a large impact arrives as a pulse whose speed and width can be tuned by pre-squeezing the chain.

Pre-squeezing is the tuning knob and it is the same knob as the gap in the second figure, turned the other way. A gap makes the contacts sequential and the transfer perfect; a pre-compression gives the chain a genuine sound speed and makes it behave like an ordinary dispersive medium, spreading the pulse out. The cradle sits at the exact point between the two where the balls just touch, which is the one setting at which the sonic vacuum exists.

Where the model stops #

The balls here are point masses joined by a contact law, and a real ball is an elastic body. A Hertzian contact model treats the region near the touching point exactly and the rest of the sphere as rigid, which is excellent while the contact time is long compared with the time sound takes to cross a ball. For steel spheres a centimetre across, sound crosses in about two microseconds and the contact lasts tens of microseconds, so the model holds by an order of magnitude. Struck harder, or made of a softer material, the ball rings and the energy that goes into ringing does not come back into the translation.

Nothing here dissipates. The runs conserve energy to eight figures, which is why the discussion is entirely about how the energy is distributed rather than how much is lost. A real cradle loses a few per cent a swing to sound, to internal friction and to the strings, and after a few dozen swings the balls end up moving together — which is the perfectly inelastic outcome marked off the ellipse in the first figure. Dissipation moves the state inward, off the elastic solution set entirely, and it always moves it toward that point.

The chain is one-dimensional and the strings are ignored. Real balls hang on bifilar strings that constrain them to a plane, and the small out-of-plane motion is what eventually makes a cradle look untidy. None of that changes the transfer, and all of it changes how long the demonstration lasts.

And the gap in the gapped figure is a millimetre-scale choice, not a limit. What matters is only that a contact finishes before the next begins, so the requirement is a gap larger than the compression, which for these speeds is a few microns. A cradle whose balls are a hair’s breadth apart is already in the sequential regime, which is worth knowing before concluding that the touching case is what any real toy does.

What the pictures cannot show #

The velocity traces draw what each ball is doing and not what is happening between them. The compression itself — the pulse, which is the object the whole argument is about — is a region a fraction of a ball wide and lasting microseconds, and it appears in these figures only as the interval during which several curves are changing at once. A figure of the overlap against position and time would show it directly, and would also show the thing the velocity traces hide: that at the moment the pulse is in the middle of the chain, no ball is moving at the striking speed and the energy is in the deformation rather than in any motion at all.

The ellipse in the first figure is also less than the truth in one direction and more in another. It is less, because it is drawn for three balls and the full solution set for five is a surface in a five-dimensional space. It is more, because most of that set is unreachable by any contact law whatever — the arc marks the outcomes that respect the ordering of the balls, and the reachable set for a given law is a single point on it. The figure’s claim is only that the two conservation laws do not pick that point, and for that a curve is enough.

Where the ladder goes next #

The momentum ladder began with collisions being easier than forces, went through the point that keeps moving as though nothing had happened, the push that needs nothing to push against and the pile that lands heavier than it weighs. This rung asks what conservation does not settle. The rungs after it: the impulse–momentum theorem used where the force is genuinely unknown, which is what a restitution coefficient hides; granular chains, where the same Hertzian contact between many spheres produces a medium with no sound speed and pulses that survive collisions; and the momentum carried by fields rather than bodies, where the ledger only balances once the field is allowed to hold some.

The habit worth carrying away is a counting habit. Before believing that a conservation law explains an outcome, count the equations and count the unknowns. Two laws and two bodies is a determined problem and the explanation is sound. Two laws and three bodies is not, and an argument that produces a unique answer from it has used something it did not declare.