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

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.