Projectile motion in a medium with quadratic drag at constant horizontal wind

TL;DR

This paper derives simple, universal analytical formulas for projectile motion with quadratic air resistance and constant horizontal wind, applicable to various initial conditions with good accuracy.

Contribution

It introduces elementary-function-based analytical approximations for projectile motion considering quadratic drag and wind, enhancing simplicity and universality over existing models.

Findings

Formulas match numerical solutions well

Applicable to different projectiles like golf and tennis balls

Accurate over a wide range of parameters

Abstract

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. The action of the wind is also taken into account, which is considered constant and horizontal. The projectile velocity hodograph equation is used to account for the effect of wind. Comparatively simple analytical approximations are proposed for the main variables of motion (cartesian projectile coordinates and time). All obtained formulas contain only elementary functions. The proposed formulas are universal, that is, they can be used for any initial conditions of throwing. In addition, they have acceptable accuracy over a wide range of the change of parameters. The motion of a golf ball, a tennis ball and shuttlecock of badminton are presented as examples. The calculation results show good agreement between the proposed analytical solutions and numerical solutions. The proposed analytical formulas can be useful for all researchers of this classical problem.