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Some Geometry on a Pool Table
One strategy that beginners use when playing pool is to the hit the cue ball very hard in hopes that they will get lucky and something good will happen. On the other hand, if you hit the ball so it hits a bumper making a 90 degree angle, then the ball will bounce directly back, hit the opposite bumper at 90 degrees and never vary from this back and forth path. A small change in the direction of the shot will, over time, make a big change in the position of
the ball. If we ignore spin, then the path of the ball depends
only on how it hits the bumper. We always use the usual rule
"angle of incidence equals angle of reflection", that is, the ball bounces
off the bumper so that the incoming angle of the path with the bumper is
the same as the angle of the outgoing path with the bumper.
Getting Started
- What are some other possible paths of a ball bouncing around a table?
Are there other paths that return to the same initial position?
- Can you design a path so that the cue ball hits all four bumpers
before ending up in a pocket?
- Ignoring spin, how does the path of the ball change if the angle
of hit (that is, the direction the ball starts out moving) changes
a little bit. How quickly does this difference grow? (This is why pool is
a hard game...)
- A billiard table is like a pool table except there are no corners.
What happens to the cue ball if it is hit directly into a corner?
Going Deeper
The path the ball follows after being hit depends on the direction
the ball is hit. Does the path of the ball follow any pattern? How is
this pattern related to the original direction of hit?
What happens on other shaped tables, (e.g., an equilateral triangle table
table or a table
with differnt corner angles? What happens if the ball is hit
directly into one of the small corners?
Each of these questions can be answered with very careful drawing.
You should look for deeper patterns as well.
Communicated by the Chelsea High School Mathematics Department.
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