## 4 Discussion
### Analysis of Three Types of Jump Dynamics  
This study evaluated the kinetic parameters of three types of jumping movements: vertical jumps, lateral jumps, and horizontal (step-back) jumps. The analysis did not include forward horizontal jumps, because forward jumps are directly related to vertical jumps [45], and, therefore, were not separately explored in this study. All three types of jumps share the common characteristic of generating vertical force and utilizing the stretch-shortening cycle [46]. In particular, lateral and horizontal jumps have already been supported by previous research [47]. This study's PT supports the effectiveness of lateral and horizontal jumps, including movements such as the single-leg squat (MSLS), lunge split squat (LRSS), and bilateral squat (BS), in enhancing change of direction (COD) and. movements related to jumping. In fact, many basketball movements require the combination of lateral and horizontal jumps to complete a skill, including the Euro step, crossover, and step-back. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Test** & **Group** & **M \(\pm\) SD** & _df_ & _t-Value_ & _p-Value_ \\ \hline SIR (\%) & Pre & Experimental & 22 \(\pm\) 12.06 & 28 & \(-\)0.79 & 0.44 \\ SIR (\%) & Pre & Control & 15 \(\pm\) 3.80 & & & \\ SIR (\%) & Post & Experimental & 6 \(\pm\) 5.26 & 28 & \(-\)5.44 * & 0.00 \\ SIR (\%) & Post & Control & 25 \(\pm\) 12.20 & & & \\ \hline \hline \end{tabular} Abbreviations: Sports injury rate is presented as SIR (\%). The mean \(\pm\) standard deviation is presented as M \(\pm\) SD. The _t_-test values are presented as _t_-values. *_p_ < 0.05.  
\end{table} 
Table 7: Independent-sample _t_-test for pre-test and post-test sports injury rates between the two groups. The more precise the movements and the greater the take-off speed, the better the sports performance [48]. The height of a vertical jump is lower than the distances of lateral and horizontal jumps. This is because lateral and horizontal jumps involve horizontal force components and a horizontal impulse generated by the GRF, resulting in the vertical jump height being less than the distances of lateral and horizontal jumps. Scholars believe that lateral and horizontal jumps are more complex than vertical jumps [49]. This is because horizontal jumps also include vertical components. For example, in basketball, a player performing a step-back jump shot needs to jump backward (horizontally) and upward (vertically). Compared to athletes performing a countermovement jump (CMJ), horizontal jumps are more complex than vertical jumps [50]. Therefore, the results of this study are consistent with existing literature, as PT led to greater percentage improvements in lateral and horizontal jump shots compared to vertical jump shots. To clarify, this does not mean that vertical jump training was ineffective for improving horizontal jumps. On the contrary, incorporating horizontal jump training into PT was more effective than solely focusing on vertical jump training. An important finding of this study is that lateral jumps and horizontal (step-back) jumps both involve horizontal forces. The actual force generated was the R-GRF, and the force generated at take-off was the T-PRF, resulting in different jump distances. More specifically, the experimental results showed that the optimal angle range for the take-off foot in lateral jumps was between 66.1 and 66.8 degrees, while for step-back jumps, it was between 56.2 and 57.2 degrees. Additionally, the jump height provided by the flight time calculated for vertical jumps, as well as the distances of lateral and horizontal jumps, were influenced by the rate of force development (RFD) at take-off and the body's center of gravity [51]. Previous research has indicated that RFD is influenced by jump take-off momentum. Additionally, both jump momentum and sprint momentum contribute to the take-off momentum during a jump [52].