Paradoxical pop-ups: Why are they difficult to catch?
The forces on a baseball in flight with backspin, including gravity , drag , and the Magnus force . acts in the direction and acts in the direction.
Geometry of the ball-bat collision. The initial velocity of the ball and bat are and , respectively, and the pitched ball has backspin of magnitude . The bat-ball offset shown in the figure is denoted by , where and are the radii of the ball and bat, respectively. For the collisions discussed in the text, the entire picture should be rotated counterclockwise by , so that the initial angle of the ball is downward and the initial angle of the bat is upward.
Variation of the batted ball speed, initial angle above the horizontal, and spin with the offset .
Simulated trajectories of pop-ups, fly balls, and line drives with drag and spin-induced forces. These trajectories were produced when an fastball with backspin collided with the sweet spot of a bat moving at . Each trajectory was created by a different offset (in inches, indicated near the apex of each trajectory) between the bat and ball, as defined in Fig. 2.
Time dependence of the horizontal velocity, , and the horizontal forces for the trajectory, typical of a long fly ball. The drag, lift, and total forces are denoted by , , and , respectively, normalized to the weight. The vertical dashed line indicates the time that the apex is reached.
Time dependence of the horizontal velocity, , and the horizontal forces for the trajectory, typical of a high popup.
Optical acceleration cancellation (OAC) with moving fielders and realistically modeled trajectories as the interception control mechanism for moving fielders. The outfielder starts at a distance of from home plate. OAC directs the fielder to approach the desired destination along a smooth, monotonic running path. Shown are side views of the ball moving from left to right, and the fielder moving from the picture of eyeball at the right. The left diagram is the case of OAC directing fielder forward to catch a trajectory. The right diagram is the case of OAC directing fielder backward to catch a trajectory. In both cases, OAC produces a near constant running velocity along the path to the ball.
Side view of the horizontal and vertical trajectory of a pop-up being fielded by the third baseman as seen by the first baseman. The dashed lines show the third baseman’s gaze from his present position to the ball’s present position. The balls show the trajectory at half-second increments. The “eye” shows the fielder’s position at the start of the trajectory. The fielder moves in the direction shown by the arrows and exhibits little change in direction with a brief back turn near the end. This trajectory is for a pop-up.
Side view of a fielder using OAC to run up to a pop fly from the condition. In this case the fielder dramatically changes direction near the end.
Side view of a fielder using OAC to run up to a pop fly from the condition. The fielder changes direction twice, initially heading back, then forward, and finally back again.
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