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.
Step 2: Crossbeam and Uprights
2. I made a finger joint (aka box joint) on each end of the crossbeam to make it stronger. If you don't want to do this, cut the uprights to 3 inches each and skip steps 3 and 4.
3. Cut a 1/4" thick by 1" deep groove in the center of each end of the crossbeam.
4. Cut a 1/4" thick by 1" long finger on each of the uprights.
5. Glue and nail the uprights to the crossbeam.
6. Tie some chord or wrap some duct tape around the center of the crossbeam. This will absorb the impact of the arm and keep the crossbeam from splintering.
Step 3: Attach Uprights and Crossbeam to Frame
2. Cut two pieces of square wood 5 inches long and bevel the ends 45 degrees. Nail and glue these braces to the front of the uprights and to the front of the frame. I used deck screws instead of nails for this step for more strength.
3. Cut two pieces of the flat stakes to 6 1/4" and bevel the ends 45 degrees. Nail and glue these across the outside, attaching the uprights to the back of the frame. If you only have square sticks, repeat step 2 for the back of the uprights.
Step 4: Swingarm and Rope Spring
2. Drill two more holes with the same diameter through the long sections of the frame, 4 3/4" from the front or 6 inches from the back. They should be in line with each other and will be between the uprights and the middle crossbeam in the lower frame. I added washers on the outside to make winding the ropes easier.
3. Using two nails with the heads and points cut off to hold the rope on the outsides of the frame, thread the rope back and fourth four times: once through the center of the swingarm, once behind it, and twice in front. Make sure that the swingarm clears the bottom part of the frame, then tie off the ends of the rope and burn the ends to keep them from fraying.
4. Twist the nails in the direction that causes the swingarm to rest against the crossbeam, then twist them until tight. Drive in two smaller nails to hold the larger ones tight.
Step 5: Shoestring Sling
1. Cut a shoestring in half, or cut two lengths of thin string about 9 inches long.
2. Tie or staple one piece to the swingarm, a little lower than the top end. Tie the other end to a rectangular piece of cloth, about 3" by 2". Tie it through two holes in order to make a cup shape.
3. Repeat step 2 with the other piece of string and the other end of the cloth or leather. Tie a loop on the other end. This will go on the hook.
4. Bend the nail on the swingarm to make a hook, about 10 degrees forward at the base, then maybe two degrees at the tip. This takes some fine-tuning to get right. Cut off the head with some wire-cutters and file it smooth so the string can come off easily. Be careful when this swings, since it can really cut you when it's moving.