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http://en.wikipedia.org/wiki/Newton%27s_cradle wrote:

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<<Newton's cradle, named after Sir Isaac Newton, is a device that demonstrates conservation of momentum and energy via a series of swinging spheres. When one on the end is lifted and released, the resulting force travels through the line and pushes the last one upward. Christiaan Huygens used pendulums to study collisions. His work, De Motu Corporum ex Percussione (On the Motion of Bodies by Collision) published posthumously in 1703, contains a version of Newton's first law and discusses the collision of suspended bodies including two bodies of equal size with the motion of a moving body being transferred to one at rest.

Determining the velocities for the case of one ball striking four "touching" balls is found by modeling the balls as weights with non-traditional springs on their colliding surface. Steel is elastic and follows Hook's force law for springs, , but because the area of contact for a sphere increases as the force increases, colliding elastic balls will follow Hertz's adjustment to Hook's law, . This and Newton's law for motion () are applied to each ball, giving five simple but interdependent ("touching") differential equations that are 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, 40% to 50% of the kinetic energy of the initial ball is stored in the ball surfaces as potential energy for most of the collision process. 13% of the initial velocity 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, −7% in the first ball. This separates the balls, but they will come back together just before the fifth ball returns making a determination of "touching" during subsequent collisions complex. Stationary steel balls weighing 100 grams (with a strike speed of 1 m/s) need to be separated by at least 10 microns if they are to be modeled as simple independent collisions. The differential equations with the initial separations are needed if there is less than 10 micron separation, a higher strike speed, or heavier balls.

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. This is 2.03 times more kinetic energy in the fifth ball than the fourth ball, which means the fifth ball should swing twice as high as the fourth ball. But in a real Newton's cradle the fourth ball swings out as far as the fifth ball. In order to explain the difference between theory and experiment, the two striking balls must have at least 20 microns 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 may give a more accurate result. Gravity and the pendulum action influence the middle balls to return near the center positions at nearly the same time in subsequent collisions. This and heat and friction losses are influences that can be included in the Hertzian equations to make them more general and for subsequent collisions.>>

bystander wrote:

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Beyond wrote:Ahhh, a late morning nap is really refreshing.