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Suppose a ball is dropped vertically downward, but it also given some spin along a horizontal axis. Then when the ball bounces, it will presumably pick up some horizontal velocity.

Is it possible for the ball to pick up this horizontal velocity without slipping against the ground? It would seem that, as the ball bounces, it would necessarily slip against the floor, dissipating energy. This is because the horizontal velocity of the outside edge of the ball, relative to the ground is $\omega r$ on impact.

We could minimize the distance the ball slips by increasing the friction coefficient. However, this increases the frictional force proportionately, so that even in the limit of infinite friction coefficient, the energy dissipated by slipping does not go to zero. Thus, it would seem that "bouncing without slipping" is not a valid description of something a ball can do.

However, some physics problems, such as problem 11 in this PDF, suggest that such a thing can happen. Where is the discrepancy?

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  • $\begingroup$ I don't understand the concern. Why would slipping be necessary? What about pure rolling? $\endgroup$ – JMac Jun 26 at 17:49
  • $\begingroup$ @JMac Then the ball would need to instantly pick up a horizontal velocity when contacting the floor, right? $\endgroup$ – Aaron Stevens Jun 26 at 18:04
  • $\begingroup$ @AaronStevens If you treat the ball as free to deform you should be able to account for it without assuming it's instantly changing velocity. $\endgroup$ – JMac Jun 26 at 18:15
  • $\begingroup$ @JMac I rigidly only believe in rigid bodies :) $\endgroup$ – Aaron Stevens Jun 26 at 18:16
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    $\begingroup$ @AaronStevens Probably not the most accurate belief in this situation :P $\endgroup$ – JMac Jun 26 at 18:18
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The trick here would be to consider that the ball is able to deform relative to itself.

We can assume that the upon contact with the ground, the surface of the ball contacts the ground without slippage. The ball itself still has momentum though, which cannot instantly disappear. Instead, there will be internal forces in the ball acting between the points in contact with the ground, and the rest of the ball. The ball is essentially twisting itself up temporarily, storing that energy elastically; then when the friction between it and the surface reduces as the ball starts bouncing back upwards, that internal energy is released, and would contribute to the spin of the ball.

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