Newton's Cradle

Newton’s cradle is a demonstration device made of identical balls, suspended and touching. When one is released, it strikes the chain and a single ball swings out on the other side, at the same speed. It illustrates the conservation of momentum and of kinetic energy.

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How to measure it in class

The interest of Newton’s cradle is that both conservation laws at once - momentum and kinetic energy - can be verified from a single video. FizziQ’s kinematic analysis makes it possible to compare the speed of the incident ball just before the collision to that of the ejected ball just after.

Steps:

Film the device head-on, with a fixed camera and a graduated ruler in the frame, or use the video provided in the FizziQ activity
Track the released ball frame by frame over the few frames preceding contact, to obtain its speed v just before the collision
Track in the same way the ball ejected at the other end, over the few frames following contact, to obtain v′
Calculate the coefficient of restitution e = v′ / v, then the ratio of kinetic energies, which equals e²
Repeat by releasing two balls, then three, and check that the same number of balls swings out each time
Also compare the release and return heights: the ratio h′/h also equals e², and must lead to the same value of e
Film in slow motion if the phone allows it: at 30 frames per second the ball travels several centimeters between two frames, which limits the precision on the speeds

Scientific activities on this topic

To understand how the balls of a Newton’s cradle behave during the collision, we propose that the student carry out a kinematic analysis on a video they have made of a cradle, or using a Newton’s cradle video from our FizziQ site. This experiment makes it possible to test the hypothesis of an elastic collision and to measure the coefficient of restitution of the cradle.

To access the activity

Period of a pendulum: isochronism - precisely measure the oscillation period of a suspended ball, essential for understanding the rhythm of the device

Learn more

Newton’s cradle is often used to demonstrate basic physics concepts, such as conservation of energy and conservation of momentum. It is also used as a decorative object, and can be found in many toy and curiosity shops.

The operation of Newton’s cradle relies on a precise geometry. Each ball is suspended by two strings of equal length, which constrains its motion to a single plane, and the balls touch at rest, aligned and in contact. When the end ball is pulled aside and released, its gravitational potential energy is converted into kinetic energy; at the lowest point, it strikes the chain and the ball at the other end swings out alone, at the same speed, rising to the same height. The cycle repeats while gradually damping out.

The energy balance is simple to follow. Gravitational potential energy is maximal when a ball is raised aside, and minimal when all the balls hang at rest, vertically. At each passage through the lowest point, this potential energy has been entirely converted into kinetic energy. A small fraction is lost at each collision, as heat, vibrations and sound: this is why the rebound heights gradually decrease, until the device stops.

A single ball swings out: why momentum alone cannot explain it

This is the central point of the device, and the explanation most often read is incomplete. It is frequently said that “conservation of momentum requires that a single ball swing out”. This is wrong: on its own, it requires no such thing.

Take n identical balls of mass m, one ball launched at speed v. The initial momentum is mv and the initial kinetic energy ½mv². Suppose that k balls swing out together at a speed u.

Conservation of momentum gives k m u = m v, so u = v/k. This relation is satisfied for any k: two balls at v/2, three balls at v/3, everything is allowed. Momentum alone selects no solution.

It is the conservation of kinetic energy that decides. It requires k × ½m u² = ½m v². Substituting u = v/k, we get k × (v/k)² = v²/k, which must equal v². Therefore k = 1. A single ball, at speed v.

The case k = 2 is the most telling. Two balls leaving at v/2 would conserve momentum perfectly, but their kinetic energy would be 2 × ½m(v/2)² = ¼mv², that is only half of the initial energy. Half the energy would have disappeared for no reason. It is this solution that Newton’s cradle rules out, and both conservation laws simultaneously are needed to reach that conclusion. The same reasoning explains why releasing two balls makes exactly two balls swing out, and three balls exactly three.

How the disturbance really travels through the chain

The image of the intermediate balls “passing on the impact” one after the other is convenient but approximate. In reality, the impact generates a compression wave that propagates through the material of the balls, at the speed of sound in steel, on the order of 5,000 m·s⁻¹. The wave travels along the chain, reflects at the free end and sets the last ball in motion. The middle balls therefore do not remain strictly motionless: they deform very slightly and move by amounts too small to be seen. This is also why the contact must be perfect - a gap of a few tenths of a millimeter between two balls is enough to noticeably degrade the behavior of the device.

A device that bears Newton’s name without coming from him

Newton did not invent this object. The laws of collisions it illustrates were established in 1668 and 1669 by John Wallis, Christopher Wren and Christiaan Huygens, who presented their work to the Royal Society: Wren and Huygens notably formulated there the conservation of momentum and that of kinetic energy in elastic collisions. Newton took up and cited these results in the Principia in 1687. The decorative version with chrome steel balls, as we know it, only appeared in the 20th century.

In the curriculum

Newton’s cradle belongs to lower secondary school for the qualitative aspect - conservation of energy, transfers - and to upper secondary physics for the quantitative study of collisions, momentum and the coefficient of restitution.

Formula

Speed of a ball at the lowest point, after release from a height h:

v = √(2 g h)

Conservation of momentum, k balls leaving at speed u:

k m u = m v, hence u = v / k

Conservation of kinetic energy, imposing the physical solution:

k × ½ m u² = ½ m v², hence v² / k = v², therefore k = 1

Coefficient of restitution measured on the device:

e = v′ / v = √(h′ / h)

Fraction of kinetic energy lost per cycle:

1 − e²

where:

v: speed of the incident ball just before the collision (m·s⁻¹)
v′: speed of the ejected ball just after the collision (m·s⁻¹)
h, h′: release and rise heights (m)
m: mass of one ball, all identical (kg)
k: number of balls leaving after the collision (dimensionless)
u: assumed common speed of these k balls (m·s⁻¹)
g: gravitational field strength, approximately 9.81 m·s⁻² (m·s⁻²)
e: coefficient of restitution, between 0 and 1 (dimensionless)

Application examples

A ball released from 5 cm reaches the lowest point at v = √(2 × 9.81 × 0.05) ≈ 0.99 m·s⁻¹, that is approximately 1 m·s⁻¹. The opposite ball leaves at roughly this speed and rises to a slightly lower height.
Measurable damping. With e ≈ 0.95 per collision, the rebound height equals e² ≈ 90% of the previous one. After ten cycles, approximately 35% of the initial height remains: the device stops much faster than one imagines.
Two balls released, two balls out. The result is far from obvious and is predicted by the same calculation: the only solution conserving both p and Ek is the departure of two balls at the initial speed.
The cradle that does not work. Deliberately separating the balls by a few millimeters, or suspending one by a single string, is enough to ruin the effect: the balls fly off in all directions. The demonstration only holds with aligned, nearly elastic collisions.
Analogy with billiards. A billiard ball striking an identical stationary ball head-on stops dead and transmits all its speed to it: this is Newton’s cradle reduced to two balls.

FAQ

Q: Why don’t the middle balls move? A: They do move, but very little. The disturbance passes through them as a compression wave, at approximately 5,000 m·s⁻¹ in steel: the crossing is too fast and the displacements too small to be visible. At the scale of observation, everything happens as if the motion “jumped” from one end to the other.

Q: Is it conservation of momentum that requires a single ball to swing out? A: No, not on its own, and this is the most widespread error. Two balls at v/2 would conserve momentum perfectly. Conservation of kinetic energy must be added, which eliminates that solution because two balls at v/2 would carry away only half the initial energy.

Q: What happens if the balls do not have the same mass? A: The beautiful effect disappears. The speeds are no longer simply exchanged, and the general elastic collision formulas must be applied. A heavier ball keeps moving forward after striking a lighter one; a lighter ball bounces backward off a heavier one.

Q: Why does the cradle eventually stop if energy is conserved? A: Total energy is conserved, but mechanical energy is not. The collisions are not perfectly elastic: each one dissipates a few percent as heat, vibrations and sound. The air and the attachment points also take their share.

Q: Does the sound of the device count in the losses? A: Yes, but very weakly. The emitted sound does represent lost kinetic energy, which proves that the collision is not perfectly elastic, but it constitutes only a tiny share of it. Most of the energy goes into heat in the metal.

Elastic Collision - Inelastic Collision - Momentum - Kinetic Energy - Conservation of Energy - Coefficient of Restitution - Pendulum - Potential Energy