In this paper, we investigate the correlation between the main physical characteristics of eight variants of football and hockey (such as field size, goal size, player velocity, ball velocity, player density, and game duration) and the resulting average numbers of goals scored per game. To do so, the Pi-theorem in physics is extended to sport science and a non-dimensional parameter of interest is defined. It is based on the ratio between the duration of the game and the order of magnitude of the time needed to cross the midfield, which depends on the average velocity of the ball and the players, the player density and the size of the goals. An excellent correlation is found between the proposed parameter and the average number of goals scored per game during recent international competitions. Using the derived correlation, the effect of any modification of the main characteristics of football and hockey (and their variants) on the scoring pace can be assessed. For instance, it can be predicted that decreasing the length of football fields by 20 m would raise the average number of goals scored to 3.6 (±0.6) per game, versus the 2.6 goals scored during the most recent men’s World Cup.

中文翻译：

---

在足球和曲棍球的变体中，场地大小，目标大小和球员人数对每场比赛平均进球数的影响：Pi定理适用于团体运动

在本文中，我们研究了橄榄球和曲棍球的8种变体的主要物理特征（例如场地大小，目标大小，球员速度，球速，球员密度和比赛时间）与所得平均进球数之间的相关性。每场比赛。为此，将物理学中的Pi定理扩展到运动科学，并定义了一个无量纲的感兴趣参数。它基于比赛的持续时间与穿越中场所需时间量级之间的比率，该比率取决于球和球员的平均速度，球员的密度和球门的大小。在最近的国际比赛中，建议的参数与每场比赛平均进球数之间发现了极好的相关性。使用导出的相关性，可以评估足球和曲棍球（及其变体）主要特征的任何修改对得分速度的影响。例如，可以预见的是，将足球场的长度减少20 m，每场比赛的平均进球数将提高到3.6（±0.6），而最近一届男子世界杯的进球数是2.6。