An Analytical Solution for the Motion of a Projectile Accounting for Drag in the Case of Vertical Launch

2024/08/28 by Fabian B. Wadsworth, Edward W. Llewellin, Jérémie Vasseur · 1 voice · 1 citation
Engineering · #Computational Fluid Dynamics and Aerodynamics #Guidance and Control Systems #Sports Dynamics and Biomechanics

paper · pdf · doi:10.1119/5.0161985

openalex publication_date 2024/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02

Abstract

We study the problem of projectile motion, subject to atmospheric drag, in the special case of a vertical launch (i.e., straight up). We find that there exists an analytical solution for the time evolution of the projectile position and for the maximum height it will reach, when we assume that the drag coefficient has a constant value. We compare our solution with a full numerical solution in which the drag coefficient is allowed to vary nonlinearly as the projectile changes velocity. We suggest that the analytical solution we present provides a useful pedagogical bridge between full solutions to projectile motion problems, which typically require numerical methods, and the very simplest solutions, which can be inaccurate in many real-world applications.

Cited by

• Projectile motion: experimental datasets and classroom exercises

Discussions

• In this new pedagogic paper we explore a neat solution to the projectile motion problem with air-drag included. We find that a fully analytical solution exists for the case where the projectile is lau [bsky, 7 points, 0 comments]

Related

• Vertical projectile motion taking into account the wind
• A More General Linear Projectile Problem
• Analytical construction of the projectile motion trajectory in midair
• Projectile motion in a medium with quadratic drag at constant horizontal wind
• Numerical-analytical Algorithm for Constructing the Envelope of the Projectile Trajectories in Midair