Analytical construction of the projectile motion trajectory in midair

TL;DR

This paper develops an analytical method to accurately describe projectile motion with air resistance, providing universal solutions applicable over long periods without numerical simulation, demonstrated on sports projectiles.

Contribution

It introduces a universal analytical approach for projectile motion with quadratic air resistance, avoiding numerical methods and deriving explicit formulas for the vertical asymptote.

Findings

Analytical solutions match numerical results with high accuracy.

Explicit formulas for the vertical asymptote are derived from initial conditions.

Applications demonstrated on baseball, tennis ball, and badminton shuttlecock.

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. The quadratic law for the resistance force is used. An analytic approach applies for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal. As a limit case of motion, the vertical asymptote formula is obtained. The value of the vertical asymptote is calculated directly from the initial conditions of motion. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball, a tennis ball and shuttlecock of badminton are presented as examples.