This particular kind of catapult is called a mangonel or [http://en.wikipedia.org/wiki/Onager_(siege_weapon) onager], named after a type of wild donkey because it kicks like one. It uses twisted nylon ropes for a spring, and can launch a nickel about twenty feet.
Since I already built the catapult, there are no pictures of the construction. I'll include dimensions and plans, though.
Step 1: Bottom Frame
2. Nail and glue four together to make a rectangle.
3. Nail and glue the third cross piece in the middle, five inches from each end.
Drill pilot holes to keep the wood from splitting, and use several nails and plenty of glue in each connection to keep everything rigid.
Make sure everything sits level and square. The three cross pieces sit on the bottom.