Action cameras like the excellent GoPro Heros allow people to shoot their own action sequences, -and while their not terribly expensive (considering the high quality footage you can get from some rather heavy situations), wouldn't it be fun too see how far you could get with....say just 12 USD?
What you DO need, is a cheap "key chain spy camera", often referred to as the "808 camera". You can find versions e.g on eBay. I got mine from DealExtreme.com for about 11 USD, including shipping all the way from Hong Kong.
I already had a microSD memory card that I could use for this project.
I used a piece of packaging foam to wedge the 808 firmly inside the bottle. This also had the nice effect of getting the camera lense away from the plastic "tip" at the center of the bottle's base (remains from the plastic moulding process, I guess).
-Yes, we'll just be filming thru the transparent bottom of the bottle, thus the need for the colorless, transparent pee bottle. KEEP IT SIMPLE!