Newton's cradle

The conservation of momentum (mass × velocity) and kinetic energy (¹/₂ × mass × velocity²) can be used to find the resulting velocities for two colliding perfectly elastic objects. These two equations are used to determine the resulting velocities of the two objects. For the case of two balls constrained to a straight path by the strings in the cradle, the velocities are a single number instead of a 3D vector for 3D space, so the math requires only two equations to solve for two unknowns. When the two objects have the same mass, the solution is simple: the moving object stops relative to the stationary one and the stationary one picks up all the other's initial velocity. This assumes perfectly elastic objects, so there is no need to account for heat and sound energy losses.

Steel does not compress much, but its elasticity is very efficient, so it does not cause much waste heat. The simple effect from two same-mass efficiently elastic colliding objects constrained to a straight path is the basis of the effect seen in the cradle and gives an approximate solution to all its activities.

For a sequence of same-mass elastic objects constrained to a straight path, the effect continues to each successive object. For example, when two balls are dropped to strike three stationary balls in a cradle, there is an unnoticed but crucial small distance between the two dropped balls, and the action is as follows: the first moving ball that strikes the first stationary ball (the second ball striking the third ball) transfers all of its momentum to the third ball and stops. The third ball then transfers the momentum to the fourth ball and stops, and then the fourth to the fifth ball.

Right behind this sequence, the second moving ball is transferring its momentum to the first moving ball that just stopped, and the sequence repeats immediately and imperceptibly behind the first sequence, ejecting the fourth ball right behind the fifth ball with the same small separation that was between the two initial striking balls. If they are simply touching when they strike the third ball, precision requires the more complete solution below.

The effect of the last ball ejecting with a velocity nearly equal to the first ball can be seen in sliding a coin on a table into a line of identical coins, as long as the striking coin and its twin targets are in a straight line. The effect can similarly be seen in billiard balls. The effect can also be seen when a sharp and strong pressure wave strikes a dense homogeneous material immersed in a less-dense medium. If the identical atoms, molecules, or larger-scale sub-volumes of the dense homogeneous material are at least partially elastically connected to each other by electrostatic forces, they can act as a sequence of colliding identical elastic balls.

The surrounding atoms, molecules, or sub-volumes experiencing the pressure wave act to constrain each other similarly to how the string constrains the cradle's balls to a straight line. As a medical example, lithotripsy shock waves can be sent through the skin and tissue without harm to burst kidney stones. The side of the stones opposite to the incoming pressure wave bursts, not the side receiving the initial strike. In the Indian game carrom, a striker stops after hitting a stationery playing piece, transferring all of its momentum into the piece that was hit.

For the simple solution to precisely predict the action, no pair in the midst of colliding may touch the third ball, because the presence of the third ball effectively makes the struck ball appear more massive. Applying the two conservation equations to solve the final velocities of three or more balls in a single collision results in many possible solutions, so these two principles are not enough to determine resulting action.

Even when there is a small initial separation, a third ball may become involved in the collision if the initial separation is not large enough. When this occurs, the complete solution method described below must be used.

Small steel balls work well because they remain efficiently elastic with little heat loss under strong strikes and do not compress much (up to about 30 μm in a small Newton's cradle). The small, stiff compressions mean they occur rapidly, less than 200 microseconds, so steel balls are more likely to complete a collision before touching a nearby third ball. Softer elastic balls require a larger separation to maximize the effect from pair-wise collisions.

Transfer of momentum in a Newton's cradle without balls touching when three balls are dropped (2 balls)

A cradle that best follows the simple solution needs to have an initial separation between the balls that measures at least twice the amount that any one ball compresses, but most do not. This section describes the action when the initial separation is not enough and in subsequent collisions that involve more than two balls even when there is an initial separation. This solution simplifies to the simple solution when only two balls touch during a collision. It applies to all perfectly elastic identical balls that have no energy losses due to friction and can be approximated by materials such as steel, glass, plastic, and rubber.

For two balls colliding, only the two equations for conservation of momentum and energy are needed to solve the two unknown resulting velocities. For three or more simultaneously colliding elastic balls, the relative compressibilities of the colliding surfaces are the additional variables that determine the outcome. For example, five balls have four colliding points and scaling (dividing) three of them by the fourth gives the three extra variables needed to solve for all five post-collision velocities.

Newtonian, Lagrangian, Hamiltonian, and stationary action are the different ways of mathematically expressing classical mechanics. They describe the same physics but must be solved by different methods. All enforce the conservation of energy and momentum. Newton's law has been used in research papers. It is applied to each ball and the sum of forces is made equal to zero. So there are five equations, one for each ball—and five unknowns, one for each velocity. If the balls are identical, the absolute compressibility of the surfaces becomes irrelevant, because it can be divided out of both sides of all five equations, producing zero.

Determining the velocities[4][5][6] for the case of one ball striking four initially touching balls is found by modeling the balls as weights with non-traditional springs on their colliding surfaces. Most materials, like steel, that are efficiently elastic approximately follow Hooke's force law for springs, ${\displaystyle F=k\cdot x}$, but because the area of contact for a sphere increases as the force increases, colliding elastic balls follow Hertz's adjustment to Hooke's law, ${\displaystyle F=k\cdot x^{1.5}}$. This and Newton's law for motion (${\displaystyle F=m\cdot a}$) are applied to each ball, giving five simple but interdependent differential equations that can be solved numerically.

When the fifth ball begins accelerating, it is receiving momentum and energy from the third and fourth balls through the spring action of their compressed surfaces. For identical elastic balls of any type with initially touching balls, the action is the same for the first strike, except the time to complete a collision increases in softer materials. Forty to fifty percent of the kinetic energy of the initial ball from a single-ball strike is stored in the ball surfaces as potential energy for most of the collision process. Of the initial velocity, 13% is imparted to the fourth ball (which can be seen as a 3.3-degree movement if the fifth ball moves out 25 degrees) and there is a slight reverse velocity in the first three balls, the first ball having the largest at −7% of the initial velocity. This separates the balls, but they come back together just before as the fifth ball returns. This is due to the pendulum phenomenon of different small angle disturbances having approximately the same time to return to the center.

The Hertzian differential equations predict that if two balls strike three, the fifth and fourth balls will leave with velocities of 1.14 and 0.80 times the initial velocity.[7] This is 2.03 times more kinetic energy in the fifth ball than the fourth ball, which means the fifth ball would swing twice as high in the vertical direction as the fourth ball. But in a real Newton's cradle, the fourth ball swings out as far as the fifth ball. To explain the difference between theory and experiment, the two striking balls must have at least ≈ 10 μm separation (given steel, 100 g, and 1 m/s). This shows that in the common case of steel balls, unnoticed separations can be important and must be included in the Hertzian differential equations, or the simple solution gives a more accurate result.

The forces in the Hertzian solution above were assumed to propagate in the balls immediately, which is not the case. Sudden changes in the force between the atoms of material build up to form a pressure wave. Pressure waves (sound) in steel travel about 5  cm in 10 microseconds, which is about 10 times faster than the time between the first ball striking and the last ball being ejected. The pressure waves reflect back and forth through all five balls about ten times, although dispersing to less of a wavefront with more reflections. This is fast enough for the Hertzian solution to not require a substantial modification to adjust for the delay in force propagation through the balls. In less-rigid but still very elastic balls such as rubber, the propagation speed is slower, but the duration of collisions is longer, so the Hertzian solution still applies. The error introduced by the limited speed of the force propagation biases the Hertzian solution towards the simple solution because the collisions are not affected as much by the inertia of the balls that are further away.

Identically shaped balls help the pressure waves converge on the contact point of the last ball: at the initial strike point one pressure wave goes forward to the other balls while another goes backward to reflect off the opposite side of the first ball, and then it follows the first wave, being exactly one ball's diameter behind. The two waves meet up at the last contact point because the first wave reflects off the opposite side of the last ball and it meets up at the last contact point with the second wave. Then they reverberate back and forth like this about 10 times until the first ball stops connecting with the second ball. Then the reverberations reflect off the contact point between the second and third balls, but still converge at the last contact point, until the last ball is ejected—but it is less of a wavefront with each reflection.

Using different types of material does not change the action as long as the material is efficiently elastic. The size of the spheres does not change the results unless the increased weight exceeds the elastic limit of the material. If the solid balls are too large, energy is being lost as heat, because the elastic limit increases with the radius raised to the power 1.5, but the energy which had to be absorbed and released increases as the cube of the radius. Making the contact surfaces flatter can overcome this to an extent by distributing the compression to a larger amount of material but it can introduce an alignment problem. Steel is better than most materials because it allows the simple solution to apply more often in collisions after the first strike, its elastic range for storing energy remains good despite the higher energy caused by its weight, and the higher weight decreases the effect of air resistance.