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Math Help - "Never-returning billiard balls"

  1. #1
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    "Never-returning billiard balls"

    Here's a problem I'm strugling with dealing with bouncing billiard balls.

    A ball is placed in a position P inside a rectangular billiard of size a x b (a < b).
    Here we discount any kind of friction and using standard reflection rule:
    a) Show that there exists at least one direction such that if the ball starts along such direction and continue bouncing from the walls of the billiard, then it will never return to its initial position.
    b) Let Q be any point inside the billiard. Find one direction such that the ball goes from P to Q, hitting the boundary just once; is this direction unique?
    c) Find one direction such that the ball goes from P to Q, hitting the boundary twice; is this direction unique?
    d) Generalize the results of b) and c) to the case of n bounces (n is a positive integer)

    However I'm not sure wether it's best to solve this problem by euclidian geometry or some kind of analytical geometry (or any another method).

    Thank you very much for help!
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  2. #2
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    Let A=b/a, launch the ball so that the slope of its trajectory is k*A, where k is an irrational number. Then this ball won't get back forever.

    It's quite easy to prove. Reflect the trajectory with respect to the side when it hits a side, you will get a straight line, as if the ball penetrate through the side. If the ball gets back in the real world, this imaginary straight line will end up at a point where m*a units are travelled horizontally while n*b units vertically, where both m and n are integers. Then the slope of this straight line is n*b/(m*a)=(n/m)*A, that is, a rational multiple of A.
    Last edited by xxp9; October 21st 2010 at 06:05 PM.
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  3. #3
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    Thank you for your reply! But actually I don't really understand your reasoning, so I have a couple of questions for you.
    1) What do you mean by saying that m and n are integrals?
    2) And how does your reasoning actually show that the billiard ball never gets back to the same place where it started from (P)?
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  4. #4
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    Oh, now I think I see what you mean (m and n are integers). But your your proof only holds for the cases where the ball has bounced 4*n times (where n is an integer), or am I wrong? (It's a little bit difficult to understand exactly what you mean without a picture) For example, what about the case when the ball has bounced 4*n+1 times
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  5. #5
    A Plied Mathematician
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    To understand xxp9's solution, go to this book. Click the book to look inside. Search for "billiard". Examine pages 45-47, starting with Section 2.7.
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