On the ascent and descent times of a projectile in a resistant medium

Abstract

The motion of a projectile in a uniform gravitational field loses its symmetry when a resisting force is present, essentially because Newton's law loses its reversibility. Allowing the magnitude of the resistance to depend arbitrarily on speed, the motion is governed by two coupled first-order non-linear differential equations. Though intractable to solve explicitly, these equations can be made to yield much qualitative information about the trajectory. The present paper focuses on the ascent and descent times of the projectile, providing a proof that the ascent and descent times are bounded, above and below respectively, by the corresponding (equal) times of a projectile reaching the same height without resistance. Additionally, the difference is shown to increase with the velocity of projection. Direct corollaries are two well known observed features of the motion: The time of ascent is always less than the time of descent, and the difference increases with the velocity of projection. For the bounds themselves, the resistance is only assumed to be positive, but to show that the difference increases with the velocity of projection requires the additional assumption that the resistance increases at least linearly with speed.