Forces on a spinning baseball in flight. The drag force acts in the direction, the Magnus force acts in the direction, and the force of gravity acts downward.
Experimental results for . The closed circles are from the present experiment. Open circles are from Watts and Ferrer (Ref. 18), open triangles are from Briggs (Refs. 12 and 17), open diamonds and squares are from Alaways two- and four-seam (Ref. 23), respectively (Refs. 8 and 9), and closed triangles are from the pitching machine data of Jinji (Ref. 11). Also shown are the parametrizations of Ref. 5 (solid) and Eq. (3) (dashed), the latter calculated for a speed of .
Calculated ratio of the Magnus force to weight for . The solid and dashed curves utilize the parametrizations of Refs. 5 and 7, respectively, the latter essentially reproducing Fig. 2.2 of Adair (Ref. 7).
Trajectory data (top) for one of the pitches, where and are the coordinates of the dot on the ball in the coordinate system shown in the inset. The ball is projected at a slight upward angle to the direction and is spinning clockwise (topspin) about an axis perpendicular to the plane. Solid curves are least-square fits to the data using Eq. (4b), resulting in and . The long dashed curve is the center-of-mass trajectory for the coordinate, which is consistent with a downward acceleration of due to the combined effects of gravity and the Magnus force. The short dashed curves are the center-of-mass coordinates for both and with both and set to zero, indicating that the data are very sensitive to but not to . The fit residuals for the (points) and (curve) coordinates are shown in the bottom plot.
Results for from the present motion capture experiment.
Results for from the present experiment for in the range 0.15–0.25, demonstrating that does not depend strongly on (or Re) for fixed values of .
Calculated trajectories of a hit baseball with an initial speed of , angle of 30°, height of , and backspin of (solid), (long-dashed), and (short-dashed). The points indicate the location of the ball in intervals. These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki et al. (Ref. 5).
Calculated range of a hit baseball with an initial speed of , height , and backspin of (solid), (short-dashed), and (long-dashed), as a function of the initial angle . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki et al. (Ref. 5).
Calculated deflection of a pitched baseball thrown with an initial horizontal velocity , spin , and spin factor after traversing . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki et al. (Ref. 5). These deflections are in accord with experimental observations (Refs. 11 and 28).
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