Pool Table Algebra

Date: 10/21/98 at 22:20:50
From: Roland
Subject: Algebra

The y-axis, the x-axis, the line x = 6, and the line y = 12 determine 
the four sides of a 6 by 12 rectangle in the first quadrant of the xy 
plane. Imagine that this rectangle is a pool table. There are pockets 
at the four corners and at the points (0,6) and (6,6) in the middle of 
each of the longer sides. When a ball bounces off one of the sides of 
the table, it obeys the "pool rule," the slope of the path after the 
bounce is the negative of the slope before the bounce.

Your pool ball is at (3,8). You hit it toward the y-axis, along the 
line with slope 2.

a. Where does it hit the y-axis?

b. If the ball is hit hard enough, where does it hit the side of the
   table next? And after that? And after that?

c. Does it ultimately return to (3,8)? Would it do this if the slope
   had been different from 2? What is special about the slope 2 for 
   this table?

Date: 10/22/98 at 12:34:14
From: Doctor Peterson
Subject: Re: Algebra

Hi, Roland. This problem is obviously meant to give you a chance to 
explore a bit and see how the problem works before you get to the final 
answer, so I'll just get you started. Here's a picture of the table and 
the ball:

      |           |
    12+-----------+---
      |           |
      |           |
      |           |
      |     *     |
      |           |
      |           |
      |           |
      |           |
      |           |
      |           |
      |           |
     0+-----------+---
      0           6

If you hit the ball toward the y axis with a slope of 2, it looks like 
this:

      |           |
    12+-----------+---
      |           |
      |           |
      |           |
      |     *     |
      |    /      |
      |   /       |
      |  /        |
      | /         |
      |/          |
      *           |
      |           |
     0+-----------+---
      0           6

To find where it hits, you have to write the equation of the line and 
find its y-intercept. (There are lots of other ways to do this, such 
as geometry, but I suspect you are supposed to work with lines and 
slopes.) Once you find that, you can write the equation of the next 
line the ball will roll on, using that intercept and the new slope, -2:

      |           |
    12+-----------+---
      |           |
      |           |
      |           |
      |     *     |
      |    /      |
      |   /       |
      |  /        |
      | /         |
      |/          |
      *           |
      |\          |
     0+-*---------+---
      0           6

Keep working this way. For the next two lines you will have to find 
where it intersects the lines x = 6 and y = 12, respectively, which 
will be just a little harder. Each time, check whether (3,8) is on the 
line, to see whether you will get back to the starting point. You may 
want to draw it on graph paper, which will help you see what's 
happening. Have fun with it!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/