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Thursday, January 15, 2009

How to Bounce a Ball Part 1 - The Problem

Rubber Band Ball of Death

This all started because I left my rubber band ball lying out. Inevitably, a boy picked it up and started bouncing it. First he was bouncing it against the floor. Then he was bouncing it against the wall, playing a little solitare game of catch. Then his friend arrived, and tried to catch it off the bounce. They starting counting how many times they could bounce it to each other without dropping it.

Then their math teacher noticed what they were doing. She pointed out that this game wasn't very hard, because they were standing right next to each other.

They moved away from each other laterally, maintaining their equal distance to the wall. "This is easy! See? We just have to throw it in the middle."

Then she asked one boy to take two steps forward. The game changed. They could still get it to each other, but it was trickier. They tried standing various distances from the wall. Classmates joined in. The teacher wondered out loud if someone could mark an X on the wall for any two people, such that throwing the ball to the X would guarantee a catch.

This is how my exponents unit was interrupted for a week, how 12 9th graders solved a problem and didn't care they were doing math, and why my north wall is sporting a galaxy of multicolored X's.

Give up? The solution is here.

6 comments:

  1. How wonderfully refreshing. I love it.

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  2. Well, it's a rather crude version of the problem (only 2D, only bounces on the ground), but I did get pretty close to a general solution for how hard they have to throw it:
    http://kennethfinnegan.blogspot.com/2009/01/bounce-ball.html

    Don't think its half bad for a sophomore undergrad mechanical engineering major. It probably isn't what you mean by a solution...

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  3. Kenneth, that's a good piece of work, but I think you're working too hard. Bizarre as it may seem for a mathematician to tell an engineer to use an approximation, that's exactly what I think you should be doing here.

    The students are throwing a ball with some significant initial velocity, and the total time in transit is minimal. Add in the fact that the ball is bouncing off the wall (not the floor) and the whole acceleration-due-to-gravity thing becomes negligible. The motion should all be in straight lines, except where it bounces off the wall.

    Now, I'm going to predict that Kate's solution will basically come down to similar triangles and setting up an algebra problem, but consider this Deep Fact: if the wall were a mirror, you could get the ball to your partner by throwing it at her reflection. That is, if you imagine an identical copy of the room on the other side of the wall (but reflected!), then the motion really is in a straight line that passes through the wall.

    From here you can imagine a billiard table, and flip out reflected copies of the table in all directions. You can predict the path of a billiard ball by drawing straight lines in this "flipped-out" space, and then seeing what points on the original table correspond to points on the straight line. And then you can consider tables of other shapes... there's a whole field of mathematics that way!

    Here's a simple question, though. Say you place a billiard ball in the center of a two foot by two foot billiard table with pockets in the corners. You strike the ball, and it caroms around until it eventually sinks into a corner pocket. Is the square of the distance the ball travelled an odd or an even number of feet?

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  4. Ah, it would be easier if it just bounced off the wall. I was envisioning it bouncing off the wall and the floor.

    As for the table, I would think the distance would have to be even:
    (2k+1)^2 + (2j+1)^2 with k and j being non-negative integers of the number of projections they travel across horz and vertically. Both terms inside the exponents would have to be odd, so added together, it'll always be even. The simplest case being (0,0), where the ball travels sqrt(2) feet.

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  5. This is a little late, but there is a mirrored wall in the blackbox theater and in the workout room. Kaz has a soft ball that bounces great and wouldn't break the mirror if you want.

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  6. I didn't know about the mirrored walls. Thanks!

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