Here's what we did with our Bank Holiday weekend - it has taken two weekends to construct, and today we took it to the school field for testing.
Although this scaled down version was built without too much assistance from advanced mathematicians (on the basis that C14th engineers very definitely didn't know the three simultaneous differential equations for the motion of the counterweight, arm and sling and even if they did they couldn't solve them) and on the assumption that I should be able to figure it out from first principles and a little bit of trial and error, "Trebuchet for Windows" was used to run some simulations before I went too far, and some background reading established that the arm ratio (the ratio of the length of the arm in front of the pivot point to the length behind it) should be somewhere between 3:1 and 4:1 for an ideal engine.
The scale was decided based on the scrap wood I had available. All of it was pine, with the exception of the main part of the arm which is ash for strength, as I didn't want to spend a lot of money on something which might not work.
See Youtube for some video evidence
The projectile was a small cobble, about the size of a golf ball.