My background:
Earlier in my life, I spent a lot of time studying for mathematics competitions. In my sophomore year of high school, I switched to studying physics, and that's what I've been doing since. I have participated in various physics competitions, and I plan to major in physics as an undergraduate!
Projects:
Here are some problems that I've looked at for fun. You might find some of the math to be not as interesting, but I have a lot of interesting pictures and plots that you might find interesting as a casual reader. But if you're really interested by it, don't be daunted by the math! Most of it is at a basic calculus level.
 Real sunsets (atmospheric refraction)
How much later does the sun set due to the refraction of light in the atmosphere?
And a generalization to times other than sunset
 The bucket question (Circular caustics)
What's that shape that light forms on the bottom of any reflective cylinder? Some various approaches to this problem, using computer and analytical methods

Fun with sand (Angle of Repose)
What is the slope of the sand that collects in the bottom of the hourglass? How does the shape of the sand grains affect the slope? Microscopes, Fourier analysis....

Pavement rainbows
An interesting phenomenon that you may have observed in the past. (Not to be confused with "rainbow" color patterns formed by oil films on pavement.)