Summary
The first version is a floating widget catapult (those patented plastic balls that you find in some black cans of beer).
Watch it in action:
http://vimeo.com/29575935

The second is an OpenSCAD catapult with one single parameter.
Just enter the diameter of your projectile and let it build arm and base accordingly. Short animation showing variation of "projectile_diameter" from 10 to 35 millimeters:
http://vimeo.com/29589013

If you want a longer arm catapult also adjust the "catapult_length" parameter.
Floating widget image from:
http://en.wikipedia.org/wiki/Floating_Widget

Instructions
If you're a beer drinker, impress your kids by not injuring yourself extracting the floating widget.
If not, impress them anyway by measuring the shadow of your projectile at a known distance from a light source and by doing the math to determine its diameter.
Throwing power determined by infill solidity at printing time.
0.3 in the demonstration above.
Thanks to Igor