# Magnus-Forces Analysis of Pitched-Baseball Trajectories Using YOLOv3-Tiny Deep Learning Algorithm
## 2 Materials and Methods  
### Dynamic Analysis of a Pitched-Baseball Trajectory Using Aerodynamic Theory  
A baseball in flight is mainly affected by the gravitational force \(\vec{F}_{g}\), force of air resistance \(\vec{F}_{D}\), and Magnus force \(\vec{F}_{M}\), as shown in Figure 1. To simplify the calculation, this study has ignored the influence of the wind direction on the baseball. The boundary-layer theory proposed by [PERSON] describes the interaction between a flying baseball and air [6]. When air flows across the surface of the baseball, friction occurs. As air is a viscous fluid, the velocity gradually decreases to zero as it approaches the surface of the baseball, and the area in which this happens is called the  
Figure 1: Force analysis during baseball flight. boundary layer. Due to the viscous force within the boundary layer, [PERSON]'s principle is not applicable. The flow outside the boundary layer is closer to the inviscid flow and can be regarded as an ideal fluid to which the [PERSON]'s principle can be applied. As shown in Figure 2, the green area is the boundary layer, and the line segment is the fluid streamline. According to [PERSON]'s principle, the fluid velocity from A to B (or A to D) increases outside the boundary layer, and the fluid velocity from B to C (or D to C) decreases. It can be seen that points A and C are high-pressure points, and points B and D are low-pressure points. When the pressure difference is sufficiently large, the boundary layer flows back and creates a separation point, and boundary-layer separation occurs from B to C (or D to C). As shown in Figure 2, the fluid instability after the separation point (points B and D) is called turbulent flow. Compared to the entire flow field, the turbulent flow is a low-pressure area. The pressure difference between the turbulent flow area and the rest of the flow field is the main cause of the air resistance. The air resistance is in the opposite direction of the baseball flight, and through dimensional analysis [4] the baseball air resistance formula is obtained as  
\[\overset{\rightarrow}{F}_{D}=-\frac{1}{2}C_{D}\rho A\overset{\rightarrow}{v }\left(\frac{\overset{\rightarrow}{v}}{v}\right), \tag{1}\]  
where \(C_{D}\) is the drag coefficient, \(\rho\) is the air density, \(A\) is the cross-sectional area of the baseball, and \(v\) is the ball speed. The value of the drag coefficient \(C_{D}\) is 0.3 from a wind tunnel experiment [28]. According to fluid mechanics [4], it can be assumed that the flying baseball is a rotating cylinder placed in a fluid field. As the real fluid is viscous, the fluid in contact with the rotating cylinder exhibits the same rotational speed as that of the cylinder, as shown in Figure 3. In Figure 3, according to [PERSON]'s principle, the flow velocity increases, and the pressure decreases above the ball and vice versa below the ball, assuming that the baseball is rotating clockwise. This pressure difference causes the ball to shift upwards, and this phenomenon is the Magnus force effect. For a flying baseball, a Magnus force effect is generated when the sphere rotates. Although the complicated interaction between the seams of the baseball and the boundary layer causes the Magnus force to change over time, it can be averaged according to the Magnus force formula as follows [29]:  
\[\overset{\rightarrow}{F}_{M}=-\frac{1}{2}C_{M}\rho A\overset{\rightarrow}{v }\left(\frac{\overset{\rightarrow}{\omega}\times\overset{\rightarrow}{v}}{ \omega v}\right), \tag{2}\]  
\[C_{M}=\left\{ \begin{array}{c}1.55\;,S\leq 0.1\\ 0.09+0.65\;,S>0.1^{\prime}\end{array} \right. \tag{3}\]  
Figure 2: Analysis of [PERSON]’s boundary-layer theory during baseball flight. where \(C_{M}\) is the Magnus coefficient, \(\omega\) is the angular velocity, \(S=\frac{R\omega}{\upsilon}\) is the spin parameter, and \(R\) is the radius of the baseball. The pitching image angle provided by the broadcasting unit camera makes the analyses of the horizontal displacement of the baseball flight and the baseball rotation axis very different.