Colliding balls

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Torn Apart By Dingos
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Colliding balls

Postby Torn Apart By Dingos » Sun Aug 08, 2010 11:07 pm UTC

Suppose you have n identical balls on a 1 meter long (one-dimensional) table, each with radius r and speed 1 m/s, but with different orientations (some go left, some go right). The table is frictionless and collisions are elastic.

1) At most how many collisions will there be before they all fall off the table?
2) At most how long will it take before they all fall off the table?
3) Suppose that there are stationary walls on both ends of the table (which the balls will bounce off). Will the balls necessarily return to their original positions with their original orientations, and if so, how long will this take at most?

aleph_one
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Re: Colliding balls

Postby aleph_one » Mon Aug 09, 2010 12:36 am UTC

1)
Spoiler:
It is equivalent to imagine two colliding balls as instantaneously passing through each other and keeping their directions. In this view, no virtual ball can catch up with a virtual ball moving the same direction as it. So, the most collisions are achieved with half the balls starting on the left moving right, and half on the right moving left (n^2/4 for n even, (n^2-1)/4 for n odd.

2)
Spoiler:
Each collision only hastens each virtual ball's progress, so a virtual ball can stay in the interval no longer than one second, the time it takes to roll across. So, within a second, all the balls fall off the table. This is achievable, as we can have all the balls moving the same way, with one of them starting on an edge.

3)
Spoiler:
It's easier to imagine the balls to be intervals of length 2r bouncing around on a line segment of length 1. Now, if we contract each interval to a point, we get an equivalent setup with n points bouncing around on a table of length 1 - 2nr. Since these can be imagined just to pass through each other, the repeats itself after each point has made a complete "circuit", which takes 2(1-2nr) seconds.


I'm being a bit sloppy here. Do these hold up?
Last edited by aleph_one on Mon Aug 09, 2010 1:04 am UTC, edited 1 time in total.

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Torn Apart By Dingos
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Re: Colliding balls

Postby Torn Apart By Dingos » Mon Aug 09, 2010 1:02 am UTC

Sure! I solved them the same way. I thought the trick needed for (3) was neat.

About your (2):
Spoiler:
Actually this will depend on the radius of the balls. Suppose you only have two balls, one on each edge of the table, precisely so large that they touch. They will both fall off instantaneously (not taking one second!). Fixing this just involves a trivial calculation though (which alas makes the answer uglier). There is another wording of this problem which uses ants on a stick instead. There we can assume the ants are point-shaped (and thus get the answer of 1 second), but I find that problem more artificial.
Last edited by Torn Apart By Dingos on Mon Aug 09, 2010 1:14 am UTC, edited 2 times in total.

aleph_one
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Re: Colliding balls

Postby aleph_one » Mon Aug 09, 2010 1:06 am UTC

Torn Apart by Dingos: I'm not understanding your technical correction to (2). If both balls roll right, does the right on not instantly fall off, and the one that starts on the left, take one second?

Edit: Oh, you're counting those as colliding if their radius is exactly a 1/2 meter. That is a strange corner case.
Last edited by aleph_one on Mon Aug 09, 2010 1:09 am UTC, edited 1 time in total.

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Torn Apart By Dingos
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Re: Colliding balls

Postby Torn Apart By Dingos » Mon Aug 09, 2010 1:09 am UTC

I mean the left one starts out by going right and the right one starts out by going left, so they will collide with eachother and both go off the table.

Spoiler:
You need to contract the interval in (2) just as you did in (3).


Yay for edits: Well, take the radius as r/2 minus epsilon, and you would still have to adjust the answer slightly. :)
Last edited by Torn Apart By Dingos on Mon Aug 09, 2010 1:15 am UTC, edited 3 times in total.

aleph_one
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Re: Colliding balls

Postby aleph_one » Mon Aug 09, 2010 1:10 am UTC

I still don't understand. Do the conditions require that at least one ball start travelling in each direction?

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Torn Apart By Dingos
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Re: Colliding balls

Postby Torn Apart By Dingos » Mon Aug 09, 2010 1:13 am UTC

Oh, you're right. Yes, that was what I was thinking of, but I realize now that's not what I wrote in the problem statement. :)

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jestingrabbit
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Re: Colliding balls

Postby jestingrabbit » Mon Aug 09, 2010 5:01 am UTC

Uses the sort of reasoning that you find here but it is different.
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