So you can build a hand-carried catapult on the small toy-scale moderately well. I think I've seen toys that launch things like bean-bags or small water balloons, and they fly 10-20 feet. They usually use surgical elastic tubing (the same thing commercial slingshots generally use).
I recommend you watch some videos of Pumpkin Chunkin', they can be extremely informative, as these people are trying to do a very similar thing to what you are doing: design and build an optimal device that launches a projectile at a very high velocity.
The trouble with a hand carried catapult is when you try to scale it up and get some serious velocity going. You see, a catapult isn't really designed to use its energy efficiently. They are designed to fling an object at high velocity in a high arc. To achieve this, a big heavy arm swings forward. After the projectile releases, momentum causes the arm to keep swinging. The result is that unless the base is ridiculous heavy, or the whole thing is heavily and securely bolted into the ground, it kicks like a mule. If it is a device held in your arms, you don't have the option to make it heavy or to secure it to the ground.