## 3 Results and Discussion
### Part II: Case Study
#### 3.2.4 Flight Phase  
As soon as a ski jumper has reached a constant flight position, the head point will again be a valid representation of the skiers flight path as a point mass. Considering the angular data from 5-10 m after the take-off in Figure 12, the trajectories of the dGNSS in the flight phase of the two jumps are shown in Figure 13, where the early flight phase and landing are gray shaded. The take-off phase has been addressed earlier and the definition of where the landing started will be addressed below. The difference in the trajectories between the jumps at 20 m was 0.1 m, almost half of the difference measured with the CoM (PosEst) and head (dGNSS) at this point. In Jump 2, the athlete had a larger body angle of attack in the second part of the early flight phase, i.e., a more upright flight position. Hence, the position of the head was higher relative to the CoM for Jump 2, which might explain this difference. The resultant, horizontal and vertical velocities are shown in Figure 14.  
Figure 12: Knee angle (\(\theta\)), hip angle (\(\gamma\)) and body angle of attack (\(\psi\)) measured during the take-off phase for the jumps in the case analysis. (**a**) shows the angles and (**b**) the respective angular velocity. The horizontal length from the in-run edge is shown on the x-axis, angles in rad on the vertical axis in (**a**) and angular velocity in rad s\({}^{-1}\) in (**b**). Figure 13: Trajectory from the dGNSS in the flight phase for the two jumps. The gray shaded areas highlight the take-off and landing phases. The horizontal and vertical distances from the in-run edge are shown on the x-axis and y-axis, respectively. The landings of the two jumps were at 80 m and 76 m for Jump 1 and Jump 2, respectively. Here, both the horizontal and vertical velocity start to decrease. The jump distance is measured along the hill profile, so a 4 m horizontal distance corresponds to a total difference of 5 m, as shown in Table 3. Both Figures 13 and 14 show similar trends to another of the rare studies using dGNSS in ski jumping Blumenbach [33]. The resultant speed (Figure 14a) slightly decreased during the early flight phase, as already shown with the PosEst system, and started to increase after 20 m. The increase in resultant speed was caused by the increase in vertical velocity as a result of gravity acting on the ski jumper over time. The increase in the resultant speed was almost linear from 30 m until the ski jumper initiated the landing. The difference in resultant velocity between the jumps remained at 0.4 m s\({}^{-1}\) for most of the flight phase, emphasising the importance of the in-run speed. A sudden drop in horizontal velocity was seen right before the landing, assumed to be the ski jumper opening up his position to prepare for landing. Figure 15 shows the \(F_{L}/F_{D}\) ratio, \(F_{L}\) and the \(F_{D}\) of the flight phase. The \(F_{L}/F_{D}\) ratios are relatively constant from around 20 m and 24 m for Jump 1 and Jump 2, respectively, i.e., reaching a constant ratio after \(\sim\)1 s. This is in accordance with simulations performed by [PERSON]\(\ddot{\text{o}}\)lzer et al. [19]. The ski jumper maintains a ratio of 1.3-1.5, in the constant phase, which corresponds well to earlier simulations and experiments [18; 19; 20; 46; 47]. Jump 1 had a longer phase with a constant lift-to-drag ratio and a somewhat higher average for the lift-to-drag ratio. The overall trends of the lift and drag force of the two jumps were similar. In summary, the differences in speed and technique that were observed in the in-run, take-off and early flight phase led to differences in the flight kinetics during the flight phase. The main effect of the differences in drag and lift forces between the jumps was on the vertical position of the trajectory in the last part of the flight. Due to the squared lift-velocity relationship, the higher speed in Jump 1 led to more lift and consequently a higher vertical position and better performance.