I created this project with some fellow teachers a few years ago for our Algebra 2 classes. Last year, with the switch to common core, it got moved into Algebra 1. They are doing well with it.
The lab itself is here: Catapult Quadratic Lab I assign the project to groups of 3 or 4 at the start of the chapter. They build it outside of class. We have a 20 minute part of class after lunch where they just bring them into test them. Then, they can bring them home to fix them if they need to. The lab itself takes 2 class periods but it is worth it. It happens at the end of the chapter and brings it all together. They need to put the equation into vertex form and find the value of 'a' when they shoot from ground to ground. On the second day of the lab, the catapult is moved up to a desk and shot for the target shooting that counts. They need to take the desk height into account and translate their equation up and find the new x-intercept. That is where the bowl goes. They haven't tested it or shot it from that point, so they are amazed that the math actually works.
Here are some pictures from day 1 when the kids are practicing shooting, timing, and then doing the actual measurements. They have to measure in meters, which is a challenge onto itself.
Another thing I stress is it isn't about the distance, it is about the accuracy. We are going for the shot to be consistent. I would also recommend no mouse traps - they snap too fast to get a good time recording.