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.

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.

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 Angles and Corner Pockets

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

<h1><strong>1000 Science Fair Project Ideas by Subject</strong></h1><p>Looking for a science fair project idea? Each of our science fair project idea has complete instructions with video. Get ideas for your science projects, find extra resources for your class' scientific projects, or try a fun experiment at home! It doesn't matter what grade you're in, there's something for everyone!</p><p>You can find your perfect science fair project from the different science subjects below or from our <a href="https://www.all-science-fair-projects.com/category0.html#bottom">science fair project idea lists!</a></p>

<h2>Science Fair Project Idea Lists</h2><p>Looking for other ways to find good ideas for science fair projects? Check out these lists of science projects!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>Easy Science Fair Projects</h3><p><img src="/images2/flex/176_popular 2 (2).png" data-size="100" data-float="left" style="width:100px; float:left">Explore the exciting world of science with an easy science fair project that you can do at home using simple materials. Any one of these easy project ideas will teach you important scientific concepts and help you have fun while learning. An easy science fair project is a great way to introduce the wonders of science to kids and adults alike!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Changing the Color of Flowers</p></li><li><p>Baking Soda Volcano</p></li><li><p>Swimming Raisins</p></li><li><p>Egg in Vinegar</p></li><li><p>Color of Skittles Experiment</p></li><li><p>Jumping Rice Krispies</p></li><li><p>Mixing Colors of Light</p></li><li><p>Camouflage and Hide</p></li><li><p>Light and Dark on Seed Germination</p></li><li><p>Make a Parachute</p></li></ul><p>More science project details at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/12">Easy Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>The Top 10 Best Science Fair Projects</h3><p><img src="/images2/flex/177_popular 2 (3).png" data-size="100" data-float="left" style="width:100px; float:left">To find the best science fair project for you, we've picked the best science projects from 10 different science categories, all with very interesting topics. 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With just a few simple things, you can be fascinated by science and have tons of fun with these popular science projects!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Explosive Elephant Toothpaste</p></li><li><p>Egg in Vinegar</p></li><li><p>Electrifying Lemon Battery</p></li><li><p>Making Oobleck: Liquid or Solid?</p></li><li><p>Egg Drop Project</p></li><li><p>Colors of Skittles Experiment</p></li><li><p>Cloud in a Bottle</p></li><li><p>Baking Soda Volcano</p></li><li><p>Egg in a Bottle</p></li><li><p>Plastic Milk</p></li></ul><p>Project details and more cool science projects at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/1/Popular_Science_Experiments_At_Home">Popular Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>One-Day Science Fair Project Ideas</h3><p><img src="/images2/flex/172_popular 2 (1).png" data-size="100" data-float="left" style="width:100px; float:left">Quick experiments can be a great option for a science fair project. 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We'll help you find a science fair project idea and guide you through the process of doing and presenting your science fair project with <a href="https://www.all-science-fair-projects.com/science_fair_projects_coach/science-project-coach.html">Science Fair Coach's Science Fair Project Guide</a>.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>What should I do after I have a science fair project idea?</h3><p><img src="/images2/flex/143_intro science fair.png" data-size="100" data-float="left" style="width:100px; float:left">If you have a science fair project idea, Science Fair Coach can help you turn it into a full science fair project! Starting at the <a href="https://www.all-science-fair-projects.com/pgpage/4">Introduction to Science Fairs</a>, check out the section on <strong>How to do a science fair project</strong> for a step-by-step guide!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p><img src="/images2/flex/203_compared to scientific method.png" data-size="100" data-float="left" style="width:100px; float:left">We'll guide you through every step of the <a href="https://www.all-science-fair-projects.com/pgpage/5">scientific method</a>, from designing and conducting your experiment to collecting and analyzing your findings.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p><img src="/images2/flex/202_engineering design process.png" data-size="100" data-float="left" style="width:100px; float:left">If you're working on an engineering project, you can head over to <a href="https://www.all-science-fair-projects.com/pgpage/9">The Guide to the Engineering Design Process</a>.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><p> <img src="/images2/flex/150_science fair poster.png" data-size="100" data-float="left" style="width:100px; float:left">Finally, we'll help you showcase your results on a <a href="https://www.all-science-fair-projects.com/pgpage/7">science fair board</a> for others to learn from and enjoy.</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3 style="text-align: start"><strong>How do I make a science fair board?</strong></h3><p style="text-align: start"><img src="/images2/flex/150_science fair poster.png" data-size="100" data-float="left" style="width:100px; float:left">Your science fair board is where you show off your science fair project for everyone to see and learn from. 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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 Angles and Corner Pockets

Hypothesis

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