[MrForbes]

And you want to get it designed, built, and working in a week? That's going to be tough!

There isn't really enough information in your post to make specific recommendations, since we can't see what your catapult looks like. Generally, making something release at a certain position is a tough problem. We avoid it by making all our mechanisms have only two positions, and use pneumatics to move them. In past years we've used a pneumatic release on a winch, it worked so-so. And we've controlled the position of arms using potentiometer feedback and a PID loop in the program, with varying success. Both of these methods took several weeks to get working.