Derivation of How Newton's Cradle Works

by Ron Kurtus (revised 6 August 2013)

Newton's Cradle usually consists of five metal balls of the same size and mass. They are suspended on wires and aligned such that they are in a row. If you lift and release a number of balls from one end, allowing them to swing to strike the remaining stationary balls, then an equal number of balls will move outward on the other end at the same initial velocity. Since the balls are hung from wires, they act as a form of pendulum, swinging back-and-forth.

An explanation is needed to verify that the same number of balls move outward as those that initially strike the row of balls. Also that the final velocity is the same as the initial velocity.

One derivation assumes that the balls are touching, using the laws of Conservation of Momentum and Conservation of Energy. Another explanation assumes the balls are slightly separated. A third—and more exact—derivation accounts for the slight deformation of the balls, causing them to act like linear springs, obeying Hooke's Law. This is a more complex view.

Questions you may have include:

• What is the derivation when the balls are touching?
• What is the derivation when the balls are separated?
• What happens when the balls deform?

Balls are touching

If the Newton's Cradle balls are perfectly stiff and are touching or in direct contact with each other, the momentum and energy of the initial ball or balls will be transferred to the end balls, resulting in the same number of balls moving outward at the velocity of the initial balls.

This is shown by applying the Law of the Conservation of Momentum and the Law of Conservation of Energy.

Conservation of Momentum

The Law of Conservation of Momentum states that the total linear momentum (P) of a closed system is constant. That means that the momentum of the balls on impact equals the momentum of the second group of balls after impact:

P = m = MV

where

• m is the total mass of the balls being released
• v is the velocity on impact
• M is the total mass of the balls moved on the other end
• V is the velocity of the second group of balls after impact

Conservation of Energy

The Law of the Conservation of Energy states that the total kinetic energy (KE) of a system with no external forces acting on it remains constant. That means that the kinetic energy of the moving ball or balls upon impact equals the kinetic energy of the balls leaving the other side of the row of balls.

Note: Although the force of gravity is an external force, it is only a factor in accelerating the first set of balls and in the swing of the second set. The force of gravity is not a factor at the point of impact and with the resulting motion.

KE = mv²/2 = MV²/2

Mass and velocity of balls moved

In the momentum equality, solve for v and square both sides of the equation:

mv = MV

v = MV/m

v² = M²V²/m²

Substitute v² in the energy equation mv²/2:

mv²/2 = MV²/2

mM²V²/2m² = MV²/2

M/m = 1

Thus:

M = m

This means the mass of the balls leaving equals the incoming mass. Since the balls are of equal mass, that means the same number of balls leave the series as those which impacted the group of balls.

Also, since M = m, then:

mv²/2 = mV²/2

and

V = v

The final velocity of the end balls is the same as the initial velocity.

Note: Some sources say that if the balls are touching on impact, then releasing several balls may result in just one last ball moving forward at a greater velocity instead of moving the same mass at the same velocity. However, that would violate the laws of Conservation of Momentum and Energy, as seen in the above derivation.

Balls slightly separated

Another view for explaining how Newton's Cradle works, assumes the balls are slightly separated, even if they seem in contact. Suppose the five metal balls are perfectly stiff, such that no kinetic energy is lost in a collision. Each is also the same size and mass.

The movement of the balls is according to a series of simple collisions of a moving ball and a stationary one.

(See Derivation of a Simple Collision for more information.)

Sequence for one ball

The sequence when releasing one ball in a Newton's Cradle goes:

1. Ball a moves forward at velocity v

2. Ball a strikes ball b, causing it to start moving at v

3. Ball a becomes stationary while ball b moves the slight separation to strike ball c

4. Ball b becomes stationary while ball c moves the slight separation to strike balld

5. Ball c becomes stationary while ball d moves the slight separation to strike ball e

6. Ball d becomes stationary while ball e moves outward at v

7. Ball e continues to move outward until its pendulum reverses its motion

Sequence with one ball in Newton's Cradle

Sequence for two balls

The sequence when releasing two balls in Newton's Cradle gets a little more complex:

1. Ball a and ball b are released at velocity v

2. Ball b strikes ball c and becomes stationary, moving ball c at velocity v

3. Ball c strikes ball d and becomes stationary, moving ball d at velocity v;
Also, ball a strikes ball b and becomes stationary, moving ball b at velocity v

4. Ball d strikes ball e and becomes stationary, moving ball e at velocity v;
Also, ball b strikes ball c and becomes stationary, moving ball c at velocity v

5. Ball c strikes ball d and becomes stationary, moving ball d at velocity v;
Also, ball e continues at velocity v;

6. Balls d and e move outward at v

7. Balls d and e continue to move outward at v

Sequence for two balls in Newton's Cradle

Spring collision

The above derivations consider the metal balls as perfectly stiff. However in reality, the balls deform slightly during a collision. They then bounce back to their original shape, pushing the two balls apart.

In this scenario, the balls can be considered as small linear springs, such that Hooke's Law for springs applies. This is a more realistic situation, but the derivation can get complex and is beyond the scope of this material.

The simpler derivations should be sufficient for more practical uses.

Summary

Derivations to explain how Newton's Cradle works depend on various assumptions.

One derivation assumes that the balls are touching, using the laws of Conservation of Momentum and Conservation of Energy. Another explanation assumes the balls are slightly separated. A third—and more exact—derivation accounts for the slight deformation of the balls, causing them to act like linear springs. This is a more complex view.

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