Projection Angles of Projectiles in Sports: Qualitative Assessment of the Effects of Aerodynamic Forces or Run-Up
Abstract
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.
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BibTeX
@misc{tsuboi2026projection,
title = {{Projection Angles of Projectiles in Sports: Qualitative Assessment of the Effects of Aerodynamic Forces or Run-Up}},
author = {Kazuhiro Tsuboi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08249},
eprint = {2609.08249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08249},
}
RIS
TY - UNPB
TI - Projection Angles of Projectiles in Sports: Qualitative Assessment of the Effects of Aerodynamic Forces or Run-Up
AU - Kazuhiro Tsuboi
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08249
UR - https://arxiv.org/abs/2609.08249
AB - 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.
ER -