Computer Science is the study of how computers work and how we can use them to solve problems. By studying computer science, we can use technology to make our lives easier and more efficient.

My objective was to determine how the initial angle of an idealized billiard ball path, starting from a corner of a rectangular table with integral dimensions, affects whether the path will eventually end in a corner. Based on some mathematical background research, I hypothesized that the path will terminate in a corner if and only if the tangent of its initial angle is rational.

Ever wanted to know what happens when you hit a billiard ball from a corner? This project uses computer simulation to explore the relationship between the initial angle of an idealized billiard ball path and the potential termination of the path in a corner.

Title: Simulated Billiard Ball Paths

Objectives/Goals

My objective was to determine how the initial angle of an idealized billiard
ball path, starting from a
corner of a rectangular table with integral dimensions, affects whether the path
will eventually end in a
corner. Based on some mathematical background research, I hypothesized that the
path will terminate in a
corner if and only if the tangent of its initial angle is rational.

Methods/Materials

Using the Logo programming language and its turtle graphics facilities, I wrote
a computer program to
simulate rectangular tables and billiard ball paths launched from a corner. A
test harness executed the
basic simulation program for many different rational tangent angles and integral
rectangle dimensions,
recording the results in a file. This automated technique was made possible by
the fact that all such paths
did, in fact, terminate. Irrational tangent angle paths, on the other hand,
required manual execution
because such paths appeared never to terminate, making them unsuitable for
automatic scheduling.

Results

I tested 63 different angles with rational tangents (systematically generated
and with elimination of
duplicates), and for each angle 100 rectangles of different integral dimensions,
for a total of 6300 tests. In
all of these, the paths terminated in a corner. I also tested angles with
irrational tangents, such as 60°, 30°,
and 50°, and found that such paths did not terminate in a corner. Because the
tests in this second group
never terminated, I could not run as great an abundance of them as in the
rational tangent case.

Conclusions/Discussion

My hypothesis was correct. The paths with rational tangent angles terminated in
a corner, whereas those
with irrational tangent angles did not. While the simulation could only test a
finite number of cases and
was subject to the usual issues of numerical precision, my experiment sets the
stage for confidently
attempting to prove the mathematical statements of these outcomes.

Summary Statement

My project investigated, via computer simulation, the relationship between the
initial angle of an idealized
billiard ball path and the potential termination of the path in a corner.

Help Received

My father provided mathematical background that helped me to formulate my
hypotheses. He also
commented on my program and report. My mother helped with the layout of the
display.

Billiard Ball Paths

Design a computer program to simulate the travel-path of billiard balls.

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Billiard Ball Paths

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