Discussion
This study investigated the relationships between angular momentum variables and swing hand speed during jump serve aerial spiking, utilising correlation and regression analyses. Our results indicate significant correlations between the peak angular momentums of the attack arm, non-attack arm, non-attack leg, forearm, and hand with the arm swing speed at BI ( Table 1 ). Furthermore, the segmental angular momentums of the attack arm, the timing of the appearance of the peak values of the upper arm, forearm, and hand presents a distinct proximal-to-distal sequence ( Figure 4 ), thereby supporting the first hypothesis. Regression analysis, utilising three independent variables (peak non-attack leg angular momentum, peak forearm angular momentum, and peak hand angular momentum), explains 35% of the variation in predicted arm swing speed. This model successfully predicts 35% of the variation in arm swing speed, thereby supporting the second hypothesis.

During the arm-cocking phase, the decreases in the counter-clockwise (+) ( i.e ., spike-ball direction) angular momentums of the non-attack arm and non-attack leg compensated for the decreases in the clockwise (–) ( i.e ., opposite to the spike-ball direction) angular momentums of the attack arm and head–trunk ( Figure 3 ). Notably, the non-attack arm exhibited a larger counter-clockwise (+) angular momentum, with a significant correlation observed between its peak value (0.024 ± 0.008 s–1) ( Table 1 ). From a volleyball coaching perspective, players need to cock their attack arm and arch their back (‘cock the hammer’) to prepare to initiate the arm-acceleration phase of the arm swing ( Reeser & Bahr, 2017 ). Rotating their attack arm and head–trunk in the clockwise (–) direction as a counter-movement to eccentric pre-lengthening before explosive concentric shortening ( i.e ., stretch-shortening cycle) can effectively generate considerable momentum to spike the ball. However, these actions can cause the upper body to lean back, compromising balance. It is hypothesised that players initially raise their non-attack arm and then swing it downward ( Figure 3 ). This downward swing aims to produce significant counter-clockwise (+) angular momentum, counterbalancing the clockwise (–) angular momentum generated by the attack arm. Consequently, the entire body sustains counter-clockwise (+) angular momentum, aligning with the spike-ball direction. This facilitates the execution of the arm swings during the subsequent arm-acceleration phase.

During the arm-acceleration phase, the head–trunk angular momentum increased counter-clockwise (+), peaking near MSER, whereas that of the attack arm rapidly increased in the same direction and peaked near the 50% normalised time ( Figure 3 ). The peak angular momentum of the attack arm (0.062 ± 0.017 s –1 ) ( Table 1 ) was significantly correlated with the arm swing speed at BI ( Table 1 ). The peak angular momentum of the attack arm appeared considerably later than that of the head–trunk ( Figure 3 ). Studies on tennis serving ( Martin et al., 2013 ) and baseball pitching ( Ramsey & Crotin, 2016 ) have suggested that, owing to the shoulder being the terminal point of the trunk, the rotation of the trunk positively impacts the forward rotation of the shoulder, thereby enhancing the speed of the distal segment. Our results support this perspective, indicating that the generated head–trunk angular momentum is effectively transferred to the attack arm, contributing to an increased arm swing speed at BI.

In the later part of the arm-cocking phase, the angular momentums of the attack and non-attack legs simultaneously rotated clockwise (−). The peak angular momentum of the non-attack leg (−0.026 ± 0.012 s –1 ) exhibited negative correlations with the arm swing speed at BI ( Table 1 ). Our regression analysis indicates that an increase in arm swing speed by 1 m/s corresponds to an increase in clockwise non-attack leg angular momentum by 0.012 s −1 . The angular momentum generated by the movements of both legs, including hip flexion and knee extension, is consistent with the description provided by Reeser & Bahr (2017) . According to the principle of conservation of angular momentum, an increase in angular momentum in one body part leads to increases in the angular momentum of other body parts in the opposite direction ( Robertson et al., 2014 ). During the arm-acceleration phase, the angular momentums of both legs increased in the clockwise (–) direction to counteract the considerable counter-clockwise (+) angular momentums of the attack arm and head–trunk. Consequently, the counter-clockwise (+) angular momentum of the whole-body system could be reduced, and the body could be prevented from rotating excessively counter-clockwise (+), keeping the body relatively balanced in the air.

Figure 4 illustrates the segmental angular momentum of the attack arm, encompassing the upper arm, forearm, and hand. The peak counter-clockwise (+) angular momentum of the forearm (0.026 ± 0.007 s –1 ) and hand (0.022 ± 0.006 s –1 ) exhibited significant correlations with the arm swing speed at BI ( Table 1 ). Furthermore, the sequential emergence of peak angular momentum, progressing from the upper arm to the hand, demonstrates a proximal-to-distal sequencing, following the open kinetic chain principle. This sequence effectively enhances the end speed through the transmission of angular momentum from proximal to distal segments in rotational movements, such as tennis serves and baseball pitches ( Sprigings et al., 1994 ; Bahamonde, 2000 ; Martin et al., 2013 ; Ramsey & Crotin, 2016 ). Our regression analysis suggests that an increase in arm swing speed by 1 m/s corresponds to a decrease in forearm angular momentum by 0.003 s –1 and an increase in hand angular momentum by 0.001 s –1 . As previously mentioned, angular momentum is primarily generated by the moment of force exerted by each joint. Thus, we recommend that coaches be mindful of controlling both the magnitude and timing of muscle exertion at the elbow and wrist joints of the attack arm. This precise control is essential to optimise the transfer of angular momentum between the segments and to achieve faster arm swing speed.