Details

The trebuchet model is similar to that of a pendulum assembly with three degrees of freedom:

are the angles with the vertical of the beam, the counterweight, and the sling; 


are the weights of the projectile and the counterweight;

are the lengths of the beam's long and short ends, the rope that holds the counterweight and the sling.

Drag, friction, and the weight of the beam are neglected in this Demonstration.

The Lagrangian of this system is obtained from the equilibrium of the potential and kinetic energies of its parts ( is the force of gravity):

This results in the three equations of motion:

Solving these with the "EventLocator" method of NDSolve, we catch the sling when it is at the desired angle to release the projectile.

Reference

[1] E. E. Viollet-le-Duc, "Diagram of a Trebuchet," Dictionnaire Raisonné de l'Architecture Française du XIe au XVe Siècle, Paris: Bance, 1858.