发表机构
Ibaraki University（茨城大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过线性化运动方程和扩展投射模型，定性分析空气动力（阻力与升力）及助跑对最佳投射角的影响，发现二者均降低最佳角且升力影响更强，助跑增大相对角但减小地面起飞角，为铅球出手角和跳远起跳角提供力学解释。
AI 中文摘要
我们考察了影响最佳投射角的两个主要因素：空气动力和助跑效应。在空气动力方面，我们不仅考虑了阻力，还考虑了飞行过程中旋转产生的升力。通过对包含这些力的运动方程进行线性化，我们推导出关于阻力和升力系数的摄动解，并阐明其定性效应。结果表明，阻力和升力都会降低最佳投射角，其中升力的影响更强。为了研究助跑效应，我们使用了一个扩展的投射模型，其中初始速度取决于初始角度。对该模型的分析表明，更强的助跑会增加相对投射角，但会减小从地面观察到的起飞角。这些发现为铅球中的出手角和跳远中的起跳角提供了力学解释。本研究建立了一个简单的理论框架，用于阐明空气动力和助跑效应在确定体育运动中最佳投射角方面各自的作用。
英文摘要
We examine two major factors that influence the optimum projection angle: aerodynamic forces and the effect of run-up. With respect to aerodynamics, we consider not only the drag but also the lift generated by spin during flight. By linearizing the equations of motion that include these forces, we derive perturbation solutions with respect to drag and lift coefficients and clarify their qualitative effects. The results show that both drag and lift reduce the optimum projection angle, with the latter exerting a stronger influence. To investigate the effect of run-up, we use an extended projection model in which the initial speed depends on the initial angle. Analysis of this model reveals that a stronger run-up increases the relative projection angle but decreases the launch angle observed from the ground. These findings provide a mechanical explanation for the release angle in shot put and the takeoff angle in long jump. The present study establishes a simple theoretical framework for clarifying the respective roles of aerodynamic and run-up effects in determining the optimum projection angles in sports.
Comments14 pages, 5 figures. This is an English translation of a conference paper presented at the Sports Engineering and Human Dynamics Conference 2025 (in Japanese)
