ABSTRACT
Professional baseball players occasionally find it difficult to gracefully approach seemingly routine pop-ups. We describe a set of towering pop-ups with trajectories that exhibit cusps and loops near the apex. For a normal fly ball the horizontal velocity continuously decreases due to drag caused by air resistance. For pop-ups the Magnus force is larger than the drag force. In these cases the horizontal velocity initially decreases like a normal fly ball, but after the apex, the Magnus force accelerates the horizontal motion. We refer to this class of pop-ups as paradoxical because they appear to misinform the typically robust optical control strategies used by fielders and lead to systematic vacillation in running paths, especially when a trajectory terminates near the fielder. Former major league infielders confirm that our model agrees with their experiences.
ACKNOWLEDGMENTS
The authors are grateful to former major league players Clete Boyer, Jim French, Norm Gigon, Bill Heath, Dave Hirtz, and Wayne Terwilliger for their valuable comments and advice. Also, we thank David W. Smith and Stephen D. Boren for information they provided about pop-ups in the major leagues. Finally, we thank Bob Adair for sharing his own unpublished work with us on judging fly balls and for the insight regarding the final backward movement.
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