I decided to make a trebuchet for the Brick Builders contest and here it is. This one is all Lego, no tape or string or anything but Lego pieces. If you like the project I would appreciate your vote.
Also, please excuse the mess- I'm in the process of moving and the whole place is a disaster.
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Step 1: Support Arms
Make two of each of these. The shorter ones will go in front.
Step 2: Throwing Arm
The sail is one part here that may be difficult to procure. I have them from when I was like 6.
Step 3: Weight
These Lego train/boat weights are not common pieces but they are Lego. I have them for a clock I am working on (based almost entirely on KEvronista's clock with Arnfield escapement).
If you can't find these perhaps some old lego motors or the old RCX 1.0 would work as weights if you have them. Or you could always cheat and use non-lego parts.
Step 4: Assembly
This trebuchet works reasonably well. The projectiles (lego wheels) travel a few meters if everything is set up correctly. The throw in the video was much shorter than usual but I couldn't seem to get video of a better throw.