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Take a system with two ideal, rigid spheres in vacuum, being under zero net external force. Now suppose I want to make the spheres collide, which I do by pushing one of them towards the other.

Now consider the distance between the two spheres: let it be, say, $x$. As you can now obviously deduce, that as the 2 spheres come closer, $x$ will keep getting smaller and smaller, approaching $0$, taking every real value between its initial value and $0$ at different times.

Now, well, I'm not sure if I should ask it here, but, if there are infinite real numbers before zero, and it's taking every one of them before getting equal to $0$, how are the balls colliding? What's really going on in there?

A little quirky and philosophical, I know. I also asked the same on Physics Stack Exchange.

Mauro ALLEGRANZA
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    A little like https://en.m.wikipedia.org/wiki/Zeno%27s_paradoxes – Deepak Sep 09 '20 at 12:37
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    Thus the axiom that to pass through an infinite number of different states must require an infinite amount of time is not consistent with reality. –  Sep 09 '20 at 14:38
  • Thus the axiom that to pass through an infinite number of different states must require an infinite amount of time is not consistent with reality. –  Sep 09 '20 at 14:39

2 Answers2

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The distance is finite.

There indeed are infinitely many positions (modulo physics) between the starting and ending position. But there is no contradiction since objects can move continuously through a finite segment of positions in finite time.

In particular an object moving at a nonzero speed $v$ can move a distance $d$ in finite time $d/v$. There are infinitely many positions along the path and it moves through all infinitely many in finite time.

It would be a different question if the infinitely many positions were not a segment. For example suppose position one is 1 mile away, position two is 2 miles away and so on. We would expect an object moving from position $1 \to 2 \to \ldots \to n$ to take say $cn$ seconds for some $c >0$ and so it cannot move through all the positions in finite time. Fortunetely I've never seen it happen. Have you?

If you are still wondering, then I suggest you continue wondering about how the spheres start moving in the first place. If the sphere moves at all it starts at some position 1 and moves to some position 2 one second later. But how did it get started at all since there are infinitely many locations in between?

Hmmm. . . that's a headscratcher.

Daron
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There is no problem with the distance taking infinitely many values, since there are also infinitely many times between the start of the movement and the collision for it to do that in.

Especially Lime
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