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Editing
Newton's cradle
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===Effect of pressure waves=== 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 they still converge at the last contact point, until the last ball is ejected βbut this is a lessening of a wavefront with each reflection.
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