## 2 Calculation Method of Aerodynamic-Drag-Work by Athletes during Sliding  
### Overcoming Wind Resistance Energy Dissipation  
Athletes experience aerodynamic-drag-work during sliding [30; 31; 32], which is formed by the relative speed between the approaching wind and the athlete's sliding movement. Aerodynamic drag is as follows:  
\[F_{D}=\frac{1}{2}C_{D}A\rho U^{2} \tag{1}\]  
where \(F_{D}\) is the drag force experienced by the athlete [N]; \(C_{D}\) is the drag coefficient of the athlete's posture, with its value often determined by the athlete's posture and measured using wind tunnel experiments; \(\rho\) is the air density [kg/m\({}^{3}\)]; \(U\) is the relative speed between the approaching wind at the athlete's position and the athlete's movement [m/s]; and \(A\) is the frontal area of the athlete's sliding posture [m\({}^{2}\)], wherein the drag coefficient is not greatly affected by the relative wind speed within the athlete's sliding speed range. As shown in Figure 1, the athlete moves from point \(P_{i}\) to point \(P_{i+1}\) in period time \(\Delta t_{i}\) (times \(t_{i}\) to \(t_{i+1}\)). The angle between the sliding direction and the direction of the geographic coordinate system (\(x\)-axis) is \(\gamma_{i}\), and the field wind direction is \(\theta_{i+1}\) at point \(P_{i+1}\) and time \(t_{i+1}\). In a certain period of time \(\Delta t_{i}\), the relative speed between the approaching wind and the athlete's movement in the direction of the athletes' movement is expressed as follows:  
\[U_{i}=-V_{i}+\cos(\theta_{i+1}-\gamma_{i})\cdot u_{i+1} \tag{2}\]  
where \(V_{i}\) is the average sliding speed of the athlete during a certain period of time \(\Delta t_{i}\) [m/s], and \(u_{i+1}\), \(\theta_{i+1}\) is the wind speed and direction of the field wind where the athlete is located at time \(t_{i+1}\), which is obtained by the instantaneous field wind interpolation function. The specific calculation in the following ski field racetrack wind modelling is proposed as follows. In a period of time \(\Delta t_{i}\), the aerodynamic-drag-work to the athlete during ski is as follows:  
\[W_{aero\_i}=F_{D_{i}}\cdot\Delta s_{i} \tag{3}\]  
where \(\Delta s_{i}\) is the distance travelled by the athlete in the period of time \(\Delta t_{i}\),  
\[\Delta s_{i}=V_{i}\Delta t_{i} \tag{4}\]  
The magnitude and direction of the athlete's sliding speed, the field wind speed where the athlete is located, and the relative speed of the field wind and athlete all change with time. Therefore, the field wind may promote or hinder the skaters' sliding. Equations (1) and (2) are rearranged to determine the calculations for \(F_{D_{i}}\) as follows:  
\[F_{D_{i}}=\frac{1}{2}C_{D_{j}}A_{j}\rho|(-V_{i}+\cos(\theta_{i+1}-\gamma_{i}) \cdot u_{i+1})|(-V_{i}+\cos(\theta_{i+1}-\gamma_{i})\cdot u_{i+1}) \tag{5}\]  
In a certain stage \(S_{j}\), where the athlete's sport posture remains unchanged, \(C_{D_{j}}\) and \(A_{j}\) remain unchanged. Sport posture is the basis for dividing the entire track into different stages. Therefore, \(C_{D_{j}}\) and \(A_{j}\) on different stages are different. For \(S_{j}\), the skier has experienced \(n_{j}\) time periods \(\Delta t_{i}\). According to Formulas (3)-(5), the aerodynamic-drag-work to the athlete during the sliding can be expressed as follows:  
\[W_{aero\_j}=\sum_{i=1}^{n_{j}}\frac{1}{2}C_{D_{j}}A_{j}\rho|\left(-V_{j,i}+ \cos\left(\theta_{j,i+1}-\gamma_{i}\right)\cdot u_{j,i+1}\right)|\left(-V_{j,i }+\cos\left(\theta_{j,i+1}-\gamma_{i}\right)\cdot u_{j,i+1}\right)\ V_{i} \Delta t_{i} \tag{6}\]  
For the entire racetrack of the ski field, it can be divided into \(m\) stages according to different sports postures.