A paradoxical problem in Classical Mechanics

Question

In Kibble-Berkshire Classical Mechanics there is a problem which says:

A ball is dropped from height $h$ and bounces. The coefficient of restitution at each bounce is $e$. Find the velocity immediately after the first bounce, and immediately after the $n$th bounce. Show that the ball finally comes to rest after a time $$\frac{1+e}{1-e}\sqrt{\frac{2h}{g}}.$$

Even though I did not face any problem proving what I was asked to, the whole problem seems to me irrational. The velocity after the $n$th collision equals $u_n=e^nu_0$. Since the ball never stops how can we find that the ball comes to rest after a certain amount of time?. Any help to my misunderstanding is really appreciated.

(1st Edit: If we could reproduce such an experiment in lab exactly as the conditions of the problem predict, supposing that from the equations above the ball has to stop at time $t_1$ then at time $t_2$: $t_2>t_1$ what would we see? Would we find the ball being between $k$th and $(k+1)$th collision? The maths however of this collision problem would predict that both $k$th and $(k+1)$th collision happened before $t_1$.)

(2nd Edit: Even if we argued that the idealization of physical reality (neglecting the deformation of the ball, neglecting the time interval for which the ball touches the ground, etc) is responsible for the paradox, we would observe the same paradox if we could run a PC simulation with a virtual ball hitting a virtual ground changing its velocity in the way suggested by the excercise.The total time that we would predict mathematicaly that the ball is moving would be finite but an observer of such a simulation would see the ball moving infinitely)

Answer 1

Since the ball never stops...

Using that model, the ball bounces an infinite number of times in a finite amount of time, then it stops.

But what is really happening here is that the model is only partially accurate*. That model is for an infinitely rigid ball bouncing off of an infinitely rigid surface, that takes zero time to actually bounce, yet loses a finite amount of energy with each bounce.

Basically, that model ignores the fact that the ball and surface are 3D objects made of real elastic material. Because they are elastic, the bounce itself (i.e., the time the ball is in contact with the surface) takes finite time, and during that time the ball and the surface will deform.

As long as the mechanics of the ball bouncing are not significantly affected by the details of the dynamics of the materials of the ball and the surface it's bouncing upon, the model is accurate. Once the bounces get small enough that the period of time during which the ball stays in contact with the surface it's bouncing on significantly affects the outcome, then that model can't be used accurately.

What really happens, for any real materials, is that at some time- and distance-scales, the ball touches the surface, then both the ball and the surface deform, turning the kinetic energy of the ball into potential energy in the deformation, as well as heat and sound (and maybe EM radiation or other, weirder things). Then both the ball and the surface rebound -- generating even more heat and sound -- and the ball comes out of contact with the surface.

Eventually there will be a "last bounce" where the ball will hit the surface, there won't be enough potential energy in the deformation to make it lose contact, and the ball and surface will transition to decaying vibratory motion -- at least until the "ball and surface are uniform elastic material" model is no longer valid and one has to resort to either quantum mechanics, or the observation that the vibration of the ball is smaller than random thermal motion so who cares anyway?

That "quivers without bouncing" part is explained here: What happens when a ball stops bouncing?

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* All models are only partially accurate. There is never One Model to Rule them All -- this is the challenge to modeling physical phenomenon. You need to choose the right model for the problem at hand -- and that may not be the same model as you used last week, or the model that your boss wants you to use.

Choose a model without the right details, and your results are too inaccurate to be useful.

Choose a model with too many extraneous details, and your computation time goes up, as well as your opportunities for numerical errors or outright bugs in the model to corrupt your results.

Answer 2

After infinite number of bounces the ball attains a fixed position and what we are seeing is the value which time approaches after infinite number of collisions.$$\Sigma_{n=0}^{\infty} T_n=1+1/2+1/4...$$ After infinite collisons,Time will approach a finite value. In case of real ball there is certain loss of energy due to deformations or loss of energy to air resistance,this also causes the ball to attain a fixed time span. The above case has coefficient of restitution e<1 which confirms the loss of energy.