My Final Modeling Project was a multistage optimization model of team choice. My partner, Coleman Ellis, and I elected for a model using both Lagrangian and Newtonian mechanics to mathematically simulate the path of a load of fixed volume and mass launched from a trebuchet. We used hand analysis and Mathematica to develop our model, then translated our model into MATLAB and debugged until we had a working simulation. We then added animations.
After simulating, we ran arrays editing the counterweight mass, the lengths of each arm of the trebuchet, and the angle of release of the payload. We concluded that the counterweight should outweigh the payload by at least a factor of 10 and that the short arm (the part of the arm to the right of the fulcrum, carrying the counterweight) should be approximately 1/2-2/3 the length of the long arm (the part of the arm to the left of the fulcrum, carrying the sling). We also found that the sling should be equal or greater in length to the long arm, and that the arm or basket holding the counterweight should be as short as possible.
We then modeled travel and angle of release of the payload using initial positions and velocities from our trebuchet model and a kinematic model of travel accounting for drag. We found that the payload travels the farthest when released from a vertical or almost vertical sling.