## 3 Results and Discussion  
The drag coefficient for the non-spinning case is shown as a function of orientation (using a four-seam spin axis) for three new baseballs in Figure 5. Each ball was projected between 94 and 108 times, and each projection had a unique orientation. Given the geometric symmetry of the four-seam spin axis, only orientations between 0 and 90\({}^{\circ}\) were measured. The ball orientation shows a consistent dependence on drag, with a minimum at 0 and 90\({}^{\circ}\) and a maximum at 45\({}^{\circ}\). The peak drag at 45\({}^{\circ}\) is not unexpected, as the seams likely induce a flow separation location that maximizes the wake size at this orientation, as observed elsewhere [10,20]. This is confirmed by the velocity fields illustrated in Figure 6 for the orientations of 85\({}^{\circ}\) (Figure 6--left) and 45\({}^{\circ}\) (Figure 6--right). At \(\theta\) = 85\({}^{\circ}\), the flow remains attached to the baseball's surface up to \(\pm\)130\({}^{\circ}\) from the stagnation point, producing a narrow wake and low aerodynamic drag. Conversely, for the \(\theta\) = 45\({}^{\circ}\) orientation, flow separation occurs at \(\pm\)100\({}^{\circ}\) from the stagnation point, yielding a wide wake and high aerodynamic drag. The data in Figure 5 were averaged to allow comparison to the drag of balls projected with spin. Since the number of shots at each orientation was not evenly distributed, a simple average of the data would bias the drag toward the 0 and 90\({}^{\circ}\) orientations. Drag as a function of ball orientation was fit to a fourth-order polynomial. The average drag coefficient from this fit was 0.348. In order to obtain a spinning drag coefficient, balls 1, 2 and 3 were projected through the same drag measuring system with a 2500 \(\pm\) 250 rpm spin rate, using the cradle described in Figure 3. The average drag coefficient from these balls was 0.358, in agreement with drag found elsewhere [13,14,16,17]. The spinning and non-spinning average drag coefficients are in close agreement, as was observed in the comparison of non-rotating wind tunnel and rotating free flight drag measurements presented in the introduction. The spinning drag measurements (0.358) are slightly higher than the non-spinning results (0.348), consistent with the effects of spin on drag [18]. Figure 4: Image of tracking dots on baseball that were used to measure spin. The data in Figure 5 were smoothed using a moving five-point average, as shown in Figure 7. The dotted lines represent the maximum and minimum over each five-point averaging span. The 0-90\({}^{\circ}\) symmetry condition was applied to the wind tunnel results from Figure 1 (taken from [10]) and are plotted in Figure 7 using the same five-point averaging scheme applied to the free flight drag results. The wind tunnel and free flight drag measurements show a similar dependence on orientation, in that a peak drag coefficient is observed near 45\({}^{\circ}\), and minima near 0 and 90\({}^{\circ}\). The wind tunnel measurements appear to be more repeatable than the free flight measurements, which is attributed to the higher statistical convergence of the wind tunnel measurements and the different number of balls  
Figure 5: Scatter plot of the drag coefficient of three MLB baseballs without spin measured in free flight. Figure 6: Instantaneous velocity fields in the wake of non-spinning baseballs traveling left to right in free flight. (**Left**): \(\theta=85^{\circ}\); (**Right**): \(\theta=45^{\circ}\). The blank region at the bottom of the left figure is due to erroneous vectors from measurement noise, which were blanked out for clarity. used for each study (the wind tunnel tests used one ball, while the free flight tests used three balls). Drag from the wind tunnel tests was higher than that found from the free flight. The wind tunnel tests were performed at a slower speed than the free flight tests (30 vs. 36 m/s).