The concept of infinity is well known in Mathematics and I have no disagreement with this. But, in real world, something like number of species or number of water molecules or even number of stars, are anything exist that beyond finite?
Peter Breuer, in your example, the number of bounces is infinite if the ball, every time it hits the ground, loses a percentage, say p, of its energy that is strictly less than 1. But is this a realistic scenario? Clearly, p is not constant and depends on several factors.
By the way, there is a standard high-school physics exercise that asks to calculate the time until the ball stops bouncing if p is constant and strictly less than 1, and it involves the calculation of the (infinite) sum of a geometric progression with ratio 0
Ok, I have a question again in the example of 'bouncing ball'. If I start bouncing it and after some time it stops bouncing, that simply means it is a countable finite number of steps.
Now, according to the theory: the ball losses p (constant) unit energy in each steps, then I think the ball will be bouncing infinitely and never stop. It is not a real situation. If I am wrong then you can correct me.
Koushik Garain, the statement "If the ball stops bouncing after some time, then the number of bounces is finite" is false, bacause in our case, the time, say tn, from the n-th to the (n+1)-th time that the ball hits the ground - let us call this time the time of the n-th bounce - tends to zero as a geometric progression. As a result, the time of all but finitely many bounces is negligible; i.e., it is less than epsilon, for every epsilon>0.
Here is the simple high-school model. Consider a ball of mass m that falls freely from a height h0 and bounces on the ground. We assume that the ball loses a constant percentage 0
Here is my explanation, but it may be rejectable. I think this question derives from the problem that we live in third dimension (and we can still control everything in 3rd, 2nd, 1st dimensional space), but we can't reach something in 4th dimensional space or higher than that. I think that the definition of existence in 3-D and higher dimensions are different. That's why when one encounters some paradox like "Can God create another God bigger than himself?". This paradox is not suitable to be questioned in 3rd dimensional space. So i think the infiniteness itself exists (in some other definition), but it is unimaginable with respect to finite dimensional perspective. It is in another level. Cmiiw.
Enrico P. G. Cadeddu
, in the above simplified model, each bounce corresponds to exactly one term of a geometric series. How many terms does a geometric series have? How many terms does the series 1+1/2+1/4+1/8+... have? Are they finitely many?
Enrico P. G. Cadeddu
, the proposed simplified system is clearly a macroscopic system; classical mechanics apply, here. Infinity in physics appears because it appears in mathematics; it is not "taken into account".
Enrico P. G. Cadeddu
, the correspondence principle states that the quantum mechanics reduces to classical mechanics for macroscopic systems. If a system is macroscopic, classical mechanics is applied. If you want to calculate the velocity of a (macroscopic) ball that falls freely from a height h, you will use classical mechanics, you will not use quantum mechanics.
Enrico P. G. Cadeddu
, you are confused about quantum mechanics; the ball is a macroscopic system whether it is away or very close to the ground. Your statement "I studied quantum physics for years" does not prove your knowledge; it only shows that you are trying to speak "by authority".Once again, the proposed model is a simplified model to show how infinity can appear in a physical phenomenon. Nobody said that it is a real-world model. There are other such phenomena, too. For instance, the fall of an object into a black hole, as seen by a distant observer, is endless; it lasts infinite time, according to general relativity.
Regarding the last post, I should note for the readers that I have taken three university courses in quantum mechanics, two undergraduate and one post-graduate, and I have so far published four papers in quantum mechanics, where two of them have so far been cited by Scopus-indexed journals. I'm sorry for this off-topic post, but it was necessary.
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