What is the general formula for a trebuchet? What I'm really looking for is a formula for a trebuchet that I can input the desired initial velocity after launch and mass of the object, and from that figure out how long the long arm, short arm, sling and pivot need to be, and how much mass I need to have for a counterweight. Does this exist?
 A: The math is quite abstruse but if you want you can go to this calculator and punch in numbers to find what you need.
http://virtualtrebuchet.com/Trebuchet.aspx
It's the best simple trebuchet calculator I've seen.
A: To my knowledge such a thing does not exist.  A trebuchet is a rather complex instrument.  Just describing the path the payload makes as the trebuchet fires is complicated.
Some information can be gathered by using Google.  One interesting page is
http://www.algobeautytreb.com/
and the accompanying mechanics page.  Be warned that the analysis presented there is simplified.
---- Paul J. Gans
A: You could find the moment of inertia of the apparatus around the pivot as a function of three arguments (angle between sling and vertical, angle between arm and vertical, sling tension) and use x=cos(angle) and y=sin(angle) to get three equations and unknowns. Then evaluate the differential equation numerically. 
A word of caution: this took me several hours. Approximation with conservation of energy is the best way to go if you're not fond of differential equations.
Hope this helps.
A: If you assume essentially all of the potential energy goes into the projectile, then by setting the kinetic energy of the projectile $\frac{1}{2} (\mathrm{mass of projectile}) *(\mathrm{initial velocity of projectile})^2$ equal to the potential energy before launch $g*(\mathrm{height of weight})*(\mathrm{mass of weight})$, you get the sort of equation you desire.
A good estimate of the efficiency of such a mechanism would be quite difficult, but my impression is that they are rather good at what they do.
Another possible modification would be to include the mass and moment of the arm throwing the projectile, but this is also simple to do.
