Effect of Stage Coupling on Optimal Launch Angle in Multi-Stage Projectile Trajectories

A practical question is raised when designing Rube Goldberg-style launch system that traditional single-stage projectile formulas fail to answer well. How would the initial launch angle be selected when the flight is broken into many segments separated by lossy contacts? Prior studies on single stage motion, (including sports-tracking studies) indicated that the best angle depended on speed and context, while more complexity was added through speed losses and changes of the direction at each contact with multi-stage chains. This study specifically seeks to answer how initial launch angle interacts with coupling in stages to determine total horizontal range in a multi-stage chain. This study hypothesized that higher speed retention and more direction carryover in strong coupling would shift the optimal initial angle upward, while increasing achievable range. Using real Statcast distance and a large juggling dataset, the empirical coupling was estimated in terms of efficiency and angular redirection. These were combined in Monte Carlo simulations of three-stage trajectories. The single-stage baseline was averaged over observed exit speeds and turned out to be peaked near 26° with an average predicted distance of around 89 feet. In multi-stage analysis, the optimal initial angle rose as low from 29° to 41° as medium and to 48° as high coupling. Overall performance increased monotonically, with medians increasing from 106 to 215 and to 326 feet, respectively. The hypothesis in this study was supported that the high-minus-low shift in the optimal angle turned out to be positive and sizable in spite of broad variability in redirection. Practical implications are that designers would need to improve the weakest contact first and retune the initial angle after any coupling improvements.