Bouncing Cue Ball

Date: 10/29/96 at 17:14:24
From: Glenn & Nichole
Subject: Help!

First of all, thanks to all the doctors who wrote back with answers to 
my grains of wheat problem...you were great!  Now, what do you know 
about pool tables?

A cue ball was launched at an angle of 45 degrees from the lower left
hand corner of a pool table with dimensions 3x4, and ended up in the
lower right hand corner.  Using the grid of squares on the table you 
are supposed to investigate the ball's behavior under similar 
circumstances on tables of varying dimensions (3x5,3x6,4x8, etc.).  
Always launch the cue ball from the lower left corner from a 45 degree 
angle and follow its path until it hits a corner.

1. Develop a rule which will predict which corner the ball will hit.
2. What are some of the patterns involved?
	
Nichole

Date: 12/06/96 at 16:08:19
From: Doctor Lorenzo
Subject: Re: Help !

Hmmm.  Either you mean a 4x3 table (that is, 4 wide and 3 tall) or 
you'll wind up in the upper left corner, not the lower right.  

In my explanation, the bottom of the table will be the x-axis and the 
left side will be the y-axis. This means that you always start at 
(0,0), which is the lower left corner. If you have a 4x3 table, I get 
that the ball starts at (0,0), bounces at (3,3), (4,2), (2,0), (0,2), 
and (1,3), and finally hits the corner at (4,0). That is, the lower 
right.

What are the rules of motion?  A good concept to use when thinking 
about this problem is velocity. Velocity refers to a speed that has a 
direction (for instance, the ball has some speed that you give it by 
launching it, while its direction changes every time it hits a wall of 
the pool table.  For this problem we'll assume that its speed doesn't 
change).  When the ball hits the top or bottom edge (y=3 or y=0), it 
bounces down or up, which means its y-velocity switches sign (i.e., 
direction) while its x-velocity stays the same.  Similarly, when it 
hits the left or right edges (x=4 or x=0), its x-velocity switches 
sign and the y-velocity stays the same.  What this means is that THE 
X-VELOCITY IS AFFECTED ONLY BY THE X-MOTION AND THE Y-VELOCITY IS 
AFFECTED ONLY BY THE Y-MOTION.  It may LOOK as if you have a 2D 
problem on your hands, but you really have two, simultaneous, 1-D 
motions to figure out.

(There's an important lesson here. Problems in several dimensions are 
often solved by finding good coordinates that reduce the problem to 
several *independent* 1-dimensional problems. If the coordinates 
you're given already have that property, then you're in doubly good 
shape).

Let's consider the motion in the x direction only.  If we assume that 
the ball moves at 1 unit per second (the table is 4x3 units), then x 
does not hit the right side until 4 seconds have passed.  The ball 
does not hit the left side until another 4 seconds have passed.  
Again, it is another 4 seconds before the ball hits the lower right 
corner.  Every 4 seconds the ball hits a side and bounces.  It hits 
the right side at odd multiples of 4 seconds and the left side at even 
multiples of 4 seconds.

The motion in the y direction is similar, with a bounce every 3 
seconds, hitting the top at after 3 and 9 seconds and the bottom after 
0, 6, and 12 seconds.  So it hits the top at odd multiples of 3 and 
the bottom at even multiples of three.   

Now what is special about a corner?  A corner is a spot that's both a 
side and a top/bottom.  A corner is reached at the first moment when 
the ball is simultaneously at a side and either the top or bottom.  
That is, you reach corners when the time is simultaneously a multiple 
of 3 and a multiple of 4.  The first such time is after 12 seconds.  

By similar reasoning you should be able to figure out the rule for a 
general table (say, m by n).  As for the patterns, there's no 
substitute for drawing a few and getting a feel for them.  Which 
patterns correspond to winding up in the upper left or lower right or 
upper right corners?  Why aren't there any patterns that get you back 
to the lower left corner?

A few more things to think about.  What would happen if the 
dimensions, instead of being integers, were merely rational numbers?  
Can you rescale the dimensions to convert your answer in the (m,n) 
integer case to cover this possibility?  Finally, what if m is an 
integer but n is an irrational number?  Would you ever reach a corner?  
How long would it take you to get close?  

-Doctors Lorenzo and Rachel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/