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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 it's 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.

P.S.: Didn't find an appropriate tag so I used kinematics.

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    $\begingroup$ I believe that this is Zeno's paradox $\endgroup$
    – garyp
    Commented Sep 9, 2020 at 11:45
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    $\begingroup$ This is essentially just zeno's paradox en.wikipedia.org/wiki/Zeno's_paradoxes $\endgroup$
    – Triatticus
    Commented Sep 9, 2020 at 11:46
  • $\begingroup$ There are an infinite amount of points in any distance, yet things can still have relative movement $\endgroup$
    – Adrian Howard
    Commented Sep 9, 2020 at 12:13
  • $\begingroup$ Maybe the time tag? $\endgroup$
    – jalex
    Commented Sep 9, 2020 at 13:13
  • $\begingroup$ More on Zeno's paradox. $\endgroup$
    – Qmechanic ♦
    Commented Sep 9, 2020 at 16:38

2 Answers 2

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The ball travels through an infinite number of positions in a finite amount of time.

The underlying confusion here seems to be that if the ball travels though an infinite number of positions, it must take an infinite amount of time, but this is not true. Suppose you measure the time it takes the ball to move half the distance to the goal, then half the remaining distance, then half that remaining distance, and so on. There are an infinite number of subdivisions, but the time taken between each one gets smaller and smaller, approaching zero as the ball approaches its final goal. While there are an infinite number of steps, the infinite sum of the time taken by those steps is, in fact, a finite number.

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  • $\begingroup$ There are infinite number of time slices, just as there are infinite number of distance slices. Why treat them differently? $\endgroup$
    – jalex
    Commented Sep 9, 2020 at 16:48
  • $\begingroup$ @JAlex There's no need to - if the object has fixed velocity, distance and time are effectively interchangeable. I focused on time here since the finite distance between the objects is a given, while the OP seems to have some confusion about how long it will take to cover a infinite number of arbitrarily small distances. $\endgroup$
    – Nuclear Hoagie
    Commented Sep 9, 2020 at 17:12
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Any moving object passes an infinite number of points in space during an infinite number of instants in time, but the velocity is defined as total displacement over total time (a ratio of two infinities). Or maybe not. It has been suggested that both space and time might be quantized.

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  • $\begingroup$ I don't get the "ratio of two infinities" part - both the total displacement and total time can be found as an infinite sum of arbitrarily small slices of the trip, but neither the total displacement nor total time are infinite. A well-defined velocity is a ratio of finite numbers. The velocity of something that travels an infinite distance in infinite time is undefined. $\endgroup$
    – Nuclear Hoagie
    Commented Sep 9, 2020 at 15:13
  • $\begingroup$ Re, "an infinite number of points," Yes, for some definition of "infinite." Saying "a continuum of points" would be more precise. When you say "...number of...," that makes it sound like you're talking about a countable infinity, but the points in a continuum are uncountable. $\endgroup$
    – Solomon Slow
    Commented Sep 9, 2020 at 15:30

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