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If we bounce a ball in ideal conditions, so after each bounce it retains only 1/4 of the original kinetic energy, then the time between bounces diminishes in half after each bounce. This will predict that the ball will be at rest at time=2, after enduring an infinite number of bounces (a variation of the zeno paradox).

But contrary to zeno paradox, here the bounces can in principle be physically measured. If we limit our discussion to the newtonian framework, does it make sense that the ball will bounce an infinite number of times? or is this an example of the failure of Newtonian mechanics because it predicts infinities?

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    $\begingroup$ Maybe it’s a failure of the model to have enough detail to represent reality? The ball isn’t a point, the speed of sound in the ball isn’t infinite, etc. $\endgroup$ Dec 22, 2019 at 16:05
  • $\begingroup$ @BobJacobsen what you said makes sense, but I was thinking ideal newtonian mechanics in which those issues should not appear. Let us say it is a point particle. Well, still the bouncing cannot happen at unlimited short times I guess $\endgroup$
    – user65081
    Dec 22, 2019 at 16:38
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    $\begingroup$ @BobJacobsen: That should be an answer. $\endgroup$
    – user4552
    Dec 22, 2019 at 16:42
  • $\begingroup$ if you assume "ideal conditions" in a "newtonian framework" what would cause in each bounce to have the kinetic energy to be 1/4 of the previous one? Do you consider dissipation to be part of the ideal conditions within a newtonian framework? $\endgroup$
    – hyportnex
    Dec 22, 2019 at 17:39
  • $\begingroup$ @hyportnex yes, they are inelastic bounces $\endgroup$
    – user65081
    Dec 22, 2019 at 18:33

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Following the comments of @hyportnex and Bob, I believe that the problem cannot be idealized properly without introducing inconsistent forces during the bouncing, that is, increased forces as the bounce becomes smaller. The question was motived by the thompson lamp supertask which some claimed to have solved by using and ball connecting circuits that require halving times between bounces.https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp

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    $\begingroup$ I do not believe that a real lamp behaves that way, you cannot turn off or turn on a real lamp within an infinitely short time, so the Zeno paradox cannot be applied that way. I am not sure about this and just thinking but this kind of paradoxes reminds me of the paradoxes one may get with singular perturbation where small quantitative changes create completely different qualitative changes in behavior. $\endgroup$
    – hyportnex
    Dec 22, 2019 at 19:08

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