Paradoxical pop-ups: Why are they difficult to catch?

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. (c) 2008 American Association of Physics Teachers.