Study of the asymptotic motion of a sporting projectile taking into account the Magnus force

TL;DR

This paper analytically investigates the long-term behavior of a projectile considering air resistance and Magnus force, deriving key motion characteristics and velocity hodographs without initial condition restrictions.

Contribution

It provides new analytical formulas for asymptotic motion characteristics of projectiles with Magnus force, validated by numerical calculations.

Findings

Derived analytical expressions for limiting trajectory angle and terminal velocity.

Obtained an approximate implicit formula for the velocity hodograph.

Numerical results confirm the accuracy of the analytical formulas.

Abstract

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force and Magnus force are taken into account with the use of the quadratic laws. We consider the asymptotic motion of the projectile, i.e., the motion on a sufficiently large time interval. The equations of motion are written both in Cartesian coordinates and in natural axes. The aim of the study is the analytical determination of the characteristics of asymptotic motion - the limiting angle of inclination of the trajectory to the horizontal and the terminal velocity, and obtaining an analytical representation for the velocity hodograph. No restrictions are imposed on the initial throwing conditions and other parameters. The equations of motion in Cartesian coordinates are used to find the angle of inclination of the trajectory and the terminal velocity. The equations of motion in natural axes are used to determine velocity hodograph. The velocity hodograph is defined in the form an approximate implicit analytical formula linking the trajectory angle of the projectile to its velocity. The motion of a golf ball, is presented as example. Numerical calculations show a complete coincidence of analytically found values of required quantities with numerically found values. The proposed analytical formulas can be useful for all researchers of this classical problem