## 2 Materials and Methods  
For a quantitative analysis of the effects of ball size and net height, a computational approach was followed. The basic element of the simulation was the solution of the equation of motion for table-tennis balls. The equation of motion needs a mathematical description of the acting forces. The gravitational force of the earth and aerodynamic forces determine the flight trajectory of a table-tennis ball. The gravitational force,  
\[\stackrel{{\rightarrow}}{{F}}_{G}=m\stackrel{{ \rightarrow}}{{g}} \tag{1}\]  
alone results in a parabolic trajectory. This force acts toward the center of the earth and depends on the mass \(m\) of the ball and the gravitational constant \(g\) (\(9.81\) m/s\({}^{2}\)). The aerodynamic forces modify the simple parabola by air drag and lift. Air drag acts as friction force against the direction of the movement of the ball. A simple example for this force is the back-pushing of a hand being held out of a driving car. A larger velocity gives stronger force acting against the direction of the car. This force also gets larger if one puts out not only a part of the hand, but the full hand. It scales with the cross-sectional area. The mathematical expression is as follows:  
\[\stackrel{{\rightarrow}}{{F}}_{D}=-\frac{1}{2}C_{D}\rho A \stackrel{{\rightarrow}}{{v}} \tag{2}\]  
where \(\rho\) is the density of air, \(A\) is the cross-sectional area for a ball with radius \(r\) (\(A=r^{2}\pi\)), \(v\) is the ball velocity, and \(C_{D}\) is an air drag coefficient, which can be measured, e.g., in wind-tunnel experiments. The second important aerodynamic force is the airlift. The so-called \"Magnus effect\", named after its discoverer [PERSON] (1802-1870), is the reason that a rotating ball experiences a deviation from its flight path. The Magnus effect is a surface effect, because, around the spinning ball, a co-rotating air layer is formed at the surface of the ball. The flying and spinning ball induces a pressure imbalance, because, on one side, the ball is rotating with the airflow created by the movement of the ball in the air, while the other side is opposite to it. On the side where counter-rotation exists, the total velocity of the airflow is reduced, because both velocities compensate for each other partly. On the co-rotation side, a larger flow velocity is created, because both velocities add up. Higher velocity in a flow means lower pressure, and the pressure differences of the two sides lead to the deviating Magnus force, mathematically expressed with an airlift coefficient \(C_{L}\) as  
\[\vec{F_{L}}=\frac{1}{2}C_{L}\rho Ave^{\rightarrow}_{\omega}\times\vec{v} \tag{3}\]  
The airlift force acts perpendicular to the axis of rotation \(\vec{e_{\omega}}\) and to the velocity \(\overrightarrow{v}\). Air drag and lift coefficients of a rotating ball (Figure 1) as a function of the ratio of spinning velocity to translational velocity were implemented into the computer code as a fit of experimental data [11; 12; 13; 14; 15]. During a topspin shot with forward rotation, the lift force acts downward; during a backspin with backward rotation, it acts upward. Swirling balls, often quoted in soccer and volleyball, can be created when the ball is hit with a critical velocity leading to access of the inverse Magnus effect. It shows up in Figure 1 for low spinning velocities as a negative value of the airlift coefficient. This can lead also to swirling balls in table tennis, because, during the flight path, the regime of positive and negative airlift coefficients can change, resulting in swirling. However, for table-tennis balls, negative airlift coefficients exist only where the coefficient itself is already quite small. Therefore, the effect exists, but gives only deviations of some millimeters. The frequently quoted swirling balls with long pimples are, therefore, more a psychological effect than one due to physics: the pre-programmed movement of the player anticipates a flight path of a strongly rotating ball from a normal rubber sponge.