Does the launch angle of a billiard ball determine whether it will eventually land in a corner pocket? You write a computer simulation in the Logo programming language to model a ball bouncing inside a rectangular table.
You test 63 different angles whose tangent is a rational number. For each angle you run 100 different table sizes. Every one of those 6,300 paths ends in a corner. Angles with irrational tangents, such as 30 or 60 degrees, never reach a corner.
The simulation supports a clear rule: a ball launched from a corner reaches another corner only when the tangent of its angle is rational.
Hypothesis
The hypothesis is that the path will terminate in a corner if and only if the tangent of its initial angle is rational.
A program can test thousands of cases that would take years to draw by hand. Using the Logo programming language, you write a simulation of a ball bouncing inside a rectangular table and run 6,300 paths. A clear rule emerges: a ball reaches a corner only when the tangent of its launch angle is rational.
Testing a mathematical idea about billiard ball paths on a real table would take months — you'd need to set up thousands of physical configurations, one by one. Instead, you write a program in the Logo programming language to simulate a ball bouncing inside a rectangular table. The simulation runs 63 different launch angles, each paired with 100 different table sizes, producing 6,300 complete paths in the time it would take to rack up a single real game.
Whether a billiard ball ever lands in a corner pocket turns out to depend on a single mathematical property of its launch angle. You write a computer simulation in the Logo programming language to model a ball bouncing inside a rectangular table, testing 63 different angles. When the tangent of the launch angle is rational — expressible as a simple fraction — every one of the 6,300 paths across 100 different table sizes ends in a corner. Angles with irrational tangents, such as 30 or 60 degrees, never reach a corner. The number type alone determines the ball's fate.
Method & Materials
You will use the Logo programming language and its turtle graphics facilities to write a computer program to simulate rectangular tables and billiard ball paths launched from a corner.
You will need the Logo programming language and its turtle graphics facilities.
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The results of the experiment showed that paths with rational tangent angles terminated in a corner, whereas those with irrational tangent angles did not. This sets the stage for confidently attempting to prove the mathematical statements of these outcomes.
Why do this project?
This science project is unique because it uses computer simulation to explore a mathematical concept.
Also Consider
Experiment variations to consider include testing different angles with irrational tangents, such as 45°, and testing different rectangles of different integral dimensions.
Full project details
Additional information and source material for this project are available below.
