Introduction
Serving is an essential skill in volleyball, as it plays a significant role in determining the outcome of a match. It is the only volleyball movement unaffected by the tactics and behaviour of the opposing team ( Papadopoulou et al., 2013 ; Lima et al., 2021 ; Moras et al., 2008 ). An accurate and powerful serve not only leads to direct scoring opportunities but also exerts psychological pressure on opponents and disrupts their tactical layout ( Lima et al., 2021 ; Melnyk & Liakhova, 2022 ; Bari et al., 2023 ). The jump serve, a crucial serving technique, has a higher ball speed and higher direct scoring rate compared to other serving styles ( Moras et al., 2008 ; MacKenzie et al., 2012 ). Hence, to comprehend and enhance volleyball game performance, the techniques and mechanics of the jump serve must be studied, as they profoundly affect the game outcome.

The jump serve technique comprises five sequential movements: ball toss, run-up, jump take-off, spike, and landing. Biomechanical research on serving or spiking has mainly focused on the characteristics of motion and muscle exertion of the lower limbs in the last step before take-off ( Tilp, Wagner & Muller, 2008 ; Wagner et al., 2009 ; Fuchs et al., 2019a , 2021 ; Tai et al., 2021 ) and the motion of the arm swing during the spike ( Coleman, Benham & Northcott, 1993 ; Serrien et al., 2016 ; Giatsis & Tilp, 2022 ; Bari et al., 2023 ; Irawan et al., 2023 ). The results of these studies can aid volleyball players and coaches in designing training programmes. However, a need remains for a more comprehensive analysis of angular momentum in the jump serve technique.

The aerial spike of the jump serve encompasses multiple rotational movements of the trunk, arms, and legs. Compared to the traditional analysis of kinematic variables such as joint angle and angular velocity, angular momentum offers a more comprehensive understanding on the mechanics of this intricate movement. Angular momentum is a physical quantity that describes the rotational momentum of an object, and from a biomechanics perspective, it is mainly generated by the moment of force exerted by each joint ( Hinrichs, 1987 ; Bahamonde, 2000 ; Robertson et al., 2014 ). In addition, angular momentum is transferred between adjacent body segments and has been used to study some explosive hitting techniques in sports ( Bahamonde, 2000 ; Martin et al., 2013 ; Izumoto et al., 2020 ). For instance, Izumoto et al. (2020) studied the golf swing technique by quantifying angular momentum. Their findings revealed a significant correlation between the peak trunk angular momentum in the horizontal plane and the peak club head speed and indicated that increasing the angular momentum transferred to the swinging arms through trunk angular momentum is essential for achieving higher club head speeds. Martin et al. (2013) investigated the relationship between the segmental angular momentums and ball velocity in a tennis serve. They found that the segmental angular momentums of the trunk, upper arm, forearm, and hand–racket exhibiting a proximal-to-distal sequence were significantly correlated with the ball speed at various key time points. These scholars analysed the mechanisms of angular momentum transfer between adjacent segments or segment groups, offering valuable insights for athletes to optimise and enhance their overall performance.

The aerial spike motion of the jump serve resembles other explosive hitting techniques, such as the tennis serve, and follows the open kinetic chain principle ( Wagner et al., 2014 ; Reeser & Bahr, 2017 ). Angular momentum is sequentially increased as it is transferred from proximal to distal segments or from the upper arm to the forearm to the hand, ultimately achieving a high ball speed. Previous research conducted by Lima et al. (2021) on the jump serve showed a significant correlation between the arm swing speed and maximum ball speed. Therefore, increasing the arm swing hand speed is essential to impart greater speed to the ball. However, while executing a spike, the entire body is in the air and experiences no external force except for gravity. Therefore, a mechanism for the transfer of angular momentum to the hand of the attack arm must be determined. Other segments compensate for the angular momentum produced by the swing arm by reducing their angular momentum, as the angular momentum of the entire body is conserved in the air ( Reeser & Bahr, 2017 ). According to the principle of angular momentum conservation, to obtain a considerable angular momentum in one body part, the angular momentum of other body parts in the opposite direction must be increased ( Robertson et al., 2014 ). However, no studies have been conducted to quantify and analyse the angular momentum during the aerial phase of a jump serve.

This study investigates the relationships between angular momentum variables and swing hand speed during jump serve aerial spiking. We postulate two hypotheses: (1) Drawing from the open kinetic chain principle, we hypothesise that the peak segmental angular momentums of the upper arm, forearm, and hand of the attack arm manifest in a proximal-to-distal order and correlate significantly with arm swing speed at ball impact (BI). (2) We further hypothesise the existence of several crucial correlations that can accurately predict arm swing speed at BI. Clarifying these points can fill the gaps in the existing knowledge system and offer coaches and athletes valuable basic insights into the mechanics of the jump serve for teaching and training.