Editing Newton's cradle (section)

===Simple solution===
The conservation of momentum {{nowrap|(mass × velocity)}} and kinetic energy {{nowrap|(<sup>1</sup>/<sub>2</sub> × mass × velocity<sup>2</sup>)}} can be used to find the resulting velocities for [[elastic collision|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.

====Other examples of this effect====
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 [[spallation|sharp and strong pressure wave strikes a dense homogeneous material immersed in a less-dense medium]]. If the identical [[atom]]s, [[molecule]]s, 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 [[Kidney stone disease|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.

====When the simple solution applies====
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&nbsp;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.

[[File:newton_cradle_wave_propagation_3_balls.svg|thumb|Transfer of momentum in a Newton's cradle without balls touching when three balls are dropped [[:File:newton_cradle_wave_propagation_2_balls.svg|(2 balls)]]]]