Hackborn W.W. Motion through air: what a drag

Canadian applied mathematics quarterly, 2006. Volume 14, Number 3.This paper considers the motion of a projectile in a uniform, downward gravitational field for the realistic case in which the drag force on the projectile due to the air through which it moves varies as the square of its speed. Although the trajectory of the projectile cannot be expressed in elementary terms, a first integral of the projectile’s motion is constructed and used to rediscover an elementary expression that approximates the trajectory quite well for launch angles up to moderate size. It is shown that this approximate trajectory lies between the exact trajectory and Galileo’s parabolic trajectory for all non-vertical launch angles. Other elementary expressions for approximate trajectories are also found and investigated.