# The Big Project

The final two weeks of the course will be devoted to a Big Project. Each student or pair of students will choose a problem to solve via a computer program. At the end of the quarter, each student will submit a poster which describes both the problem and his solution.

#### The Problems

I have found several good sources for mathematical and physical puzzles that must be solved numerically.

• From the Earth to the Moon, version IV:
• Follow the motion of a projectile shot from a huge cannon on the Earth as it flies towards the Moon. Use a two-dimensional space. Include air resistance, gravitational forces from the Earth and the Moon, the rotation of the Earth and the orbital motion of the Moon.
• The Hundred Dollar, Hundred Digit Challenge :
• A good one is problem 2: Photons and mirrors

• The 2003 27th Annual ACM International Collegiate Programming Contest (this link calls up a PDF document with the problems)
• Problem D: Eurodiffusion, illustrates the process of diffusion

• Gravitational Lensing
If might help to read a little background material. The challenge here is to create a program which will take an (simple) image, in the form of an array of grid squares which are black or white, and calculate its appearance when viewed through a gravitational lens.

• The optimum trajectory with air resistance
You probably know that, in the absence of air resistance, you get the maximum distance when you throw a ball at 45 degrees above the horizontal. But what happens if there is air resistance? What's the best angle in this case? Does it matter how big the ball is?

• Predator-prey relationships
The population of certain small mammals runs in cycles; snowshow hares in the Arctic, for example, go from boom to bust and back to boom again every decade or so. Why? One possibility is the interaction between these animals and their predators. You can investigate how a very simple model of a predator species and its prey displays cyclic behavior.
• Look at section 28.2 in your textbook.
• Read The Ecology of Snowshoe Hares in Northern Boreal Forests for a good overview.
• Basics of the mathematical model
I have lots more references on this topic ...

• Ion-engine spacecraft trip to Mars

• The real pendulum
You know how to calculate the motion of an ideal pendulum, which is free of air resistance, friction, and swings only through small angles. Can you simulate a real pendulum, which suffers from all three of these complications?

• A melting spherical ice cube
A spherical ice cube (yes, it's an oxymoron) of radius 5 cm is taken from a freezer and suspended in air at ordinary room temperature. How long does it take to melt?
• What about a sphere of radius 10 cm? 20 cm? R cm?

• Periodic extinctions?

You may also choose some other problem, either from these items or from your own imagination ... but you must receive my approval to do so.

#### The Poster

When you have finished your project, you must create a poster on a stiff sheet of cardboard or foamboard. The poster should be very roughly 30x40 inches in size. We will hang them in the corridors of the COS, so your readers will be other students and faculty who have no particular knowledge of either the problem or numerical analysis. You must provide enough background for the reader to understand the problem, and also describe how you solved it. Graphs and figures are good. Printouts of source code are bad.

#### The Schedule

You are responsible for adhering to the schedule below. You will be graded on your ability to meet each landmark on time, as well as the quality of your work.
```   Tuesday,  May 8:   think about a project, discuss with instructor
Thursday, May 10:  receive approval from instructor to start project
Friday,   May 11:  progress report, written on paper, due by 5 PM.
Tuesday,  May 15:  progress report, written on paper, due at start of class
Thursday, May 17:  show final results to instructor in class
Monday,   May 21:  submit poster to instructor, due by 5 PM
```

Note that you must stop writing code by Wednesday, May 16. I encourage you strongly to try to complete all coding by Monday, May 14, in order to have time to analyze the results and check them over.

You are encouraged to reach each landmark ahead of schedule, of course.

#### Comparing Results

If you are working on the same problem as someone else, feel free to discuss your success or failure as you work on the problem. When you both have results, compare them, and make sure that they agree within the uncertainties. Exchanging ideas and algorithms is good; exchaning code is bad. Please write all your code with your own hands.