# A System for Analysing the Basketball Free Throw Trajectory Based on Particle Swarm Optimization
## 3 Results
### Using in Practice  
Table 2 shows the basic statistics of the analysed parameters (\(\pi\)--mean, \(sd\)--standard deviation, \(d\)--difference between the mean values of the missed and hit throws). The table presents data for all athletes grouped by body height. The parameters were calculated from 200 sequences including 155 hit and 45 missed throws. Additionally, to specify the statistical significance of the difference between hit and missed throws the \(t\)-Student's test was used (\(p\)). In the case of the parameter \(h_{1}\), measured from the lowest point of the altitude of the ball relative to the basket height, it may be observed that players who throw the ball into the basket reached the lower value of this parameter of 542 mm. The differences between hit and missed throws for this parameter had a statistical significance (\(p<0.001\)). This parameter reached the highest value among the shortest competitors (body height 171-180 cm; 695 cm for hit and 946 for missed throws). An analysis of the data showed that the maximum altitude of the ball for all participants (\(h_{2}\)) in hit and missed throws were at the same level (approximately 3900 mm). In addition, in the body height groups, there were no significant differences in \(h_{2}\) between hit and missed throws. The distance analysis of the first part of the ball trajectory \(l_{1}\) (measured from \(P_{1}\) to \(P_{2}\)--see Figure 3), showed that this parameter was higher for hit throws (2198 mm), but the difference was relatively small (13.3 mm). Considering the length \(l_{2}\) (measured from \(P_{2}\) to \(P_{3}\)), it may be seen that the hit throws had a longer distance (1648 mm) than the missed throws (1549 m). From analysing the values of \(l_{2}\) for individual height groups, it is noteworthy that, in each case, the value is higher for hit throws. Differences in the \(l_{2}\) parameters are  
Figure 7: Ball trajectory for selected sequences. statistically significant (\(p<0.001\)). The values of the initial angle \(\alpha_{1}\) for most players who throw the ball into the basket reaches 50.0\({}^{\circ}\). Since, according to earlier studies [42; 43], the initial angle of the ball should be about 52\({}^{\circ}\), it may be assumed that in such a case we are dealing with a correct throw. In missed throws, this parameter was at the level of 48.9\({}^{\circ}\). The analysis conducted also showed that the lowest \(\alpha_{1}\) in hit throws occurred for athletes with a body height in the range of 191-200 cm, while the largest angle (55.8\({}^{\circ}\)) during the throw was obtained by athletes belonging to a group of people with a body height of 181-190 cm. In the case of missed throws, this parameter was placed at the lowest level (46.2\({}^{\circ}\)) among the tallest athletes. Another analysed parameter is the angle \(\alpha_{2}\) (the angle of ball trajectory at the highest point of the trajectory). The angle \(\alpha_{2}\) measured during hit throws assumes similar values for each research group with a slight advantage among athletes whose body height is between 191-200 cm. Analysing \(\alpha_{2}\) for a missed throw shows that the situation is different. In the group of the shortest athletes, this angle assumes the lowest value at the level of 8.8\({}^{\circ}\), and, among the tallest athletes, it reaches the highest value (12.1\({}^{\circ}\)). For the whole group parameter, \(\alpha_{2}\) was slightly higher for hit throws (\(\alpha_{2}=11.2^{\circ}\), \(p<0.05\)). In the case of the angle \(\alpha_{3}\), the values of this parameter are larger in the hit throws than in the missed throws (\(p<0.001\)). However, we should pay attention to players with a body height of 191-200 cm. This group produced the lowest \(\alpha_{3}\) values during hits (44.5\({}^{\circ}\)) and missed (38.3\({}^{\circ}\)) throws. The last considered parameters are the throw velocities. According to [PERSON] and [PERSON] [8], the optimum speed \(v_{1}\), which is achieved when the ball is thrown, is approximately 7.3 m/s.