## 3 Results and Discussion
### Part II: Case Study
#### 3.2.3 Take-Off and Early Flight Phase  
The PosEst system was used for the analysis of take-off and the early flight phase. The trajectories of the CoM of the two jumps for that section are shown in Figure 9.  
A difference in vertical position between the jumps started to emerge around \(10\,\mathrm{m}\) after the in-run edge, increasing to \(\sim\)0.17 m. The resultant, horizontal and vertical velocities of the CoM are shown in Figure 10.  
Figure 8: Braking force acting on the ski jumper during the in-run phase measured with the dGNSS. The horizontal length from the in-run edge is shown on the x-axis and velocity is shown on the y-axis. Figure 9: CoM trajectory during the take-off and early flight phase for the two jumps in the case analysis. The horizontal and vertical distance from the in-run edge are shown on the x-axis and y-axis, respectively. Similar velocity trends were observed to those seen by both [PERSON] et al. and [PERSON] et al. [17; 22]. The initial speed difference from the in-run is withheld during this phase. [PERSON] et al. [22] found a correlation between the horizontal speed and the jump length. A similar relationship was found in our comparison. Figure 11 shows the lift-to-drag ratio, the aerodynamic lift and the aerodynamic drag during the early flight phase. The forces are shown from +2 m after the in-run edge to ensure that the skis have left the in-run. The negative drag at onset of the flight phase (i.e., propulsion rather than resistance) cannot be explained. During take-off, a small component of the push-off force will appear as negative 'drag' according to Equation (7). This was indeed the case for the signal before +2 m (not shown). Thus the brief continuation of the negative value may be an outcome of inaccuracy of the PosEst measure. Due to rapid changes in \(F_{L}\) and \(F_{D}\) in this early flight phase the \(F_{L}/F_{D}\) ratio was unstable but increased during the last 5 m (15-20 m). The ratios of \(F_{L}/F_{D}\) for the jumps are in the range of what has been previously reported by [PERSON]\(\ddot{\rm o}\)[PERSON] and [PERSON] [19; 20]. Figure 12 shows the calculated angles and the angular velocity for the two jumps. For the knee abduction, in Figure 12a, Jump 1 had a steeper increase. The knee angle (\(\theta\)) was also kept at a larger angle in Jump 1 during the early flight phase. The evolution of the hip angle (\(\gamma\)) for the two jumps seems to be similar. The body angle of attack (\(\psi\)) was introduced for the early flight phase in accordance with earlier studies [20; 21; 22], and also plotted for the take-off phase for comparison (noting its interpretation is different than for the flight phase). Whilst \(\psi\) was of a similar value at the in-run edge immediately after take-off, a difference in \(\psi\) emerged, which remained constant after \(\sim\)10 m, where Jump 1 had a lower \(\psi\). Earlier studies have found that a low \(\psi\) is beneficial for flight performance [18; 21; 22; 46]. An explanation for why Jump 2 has a higher \(F_{L}/F_{D}\) ratio in the last part of this phase may involve the orientation of the skis, which was not included in the PosEst, but influences the aerodynamic forces in a flight [53]. Even if the angular results show that the athlete reaches a stable position in this phase, the \(F_{L}/F_{D}\) ratio varies  
Figure 11: External forces acting on the ski jumper during the early flight phase for the jumps in the case study. (**a**) shows the lift-to-drag ratio (\(F_{L}/F_{D}\)), (**b**) the aerodynamic lift (\(F_{L}\)) and (**c**) the aerodynamic drag (\(F_{D}\)). The horizontal length from the in-run edge is shown on the x-axis. On the vertical axis the unit less ratio is shown in (**a**) and the force [N] in (**b**,**c**). Figure 10: Velocity components of the CoM in the take-off and early flight phase for the jumps in the case study. (**a**) shows the resultant velocity \(v\), (**b**) the horizontal velocity \(v_{x}\) and (**c**) the vertical velocity \(v_{y}\).