Minimum and terminal velocities in projectile motion

TL;DR

This paper analytically investigates projectile motion with drag, revealing non-monotonous velocity behavior and conditions under which minimum and terminal velocities relate, supported by numerical simulations.

Contribution

It provides new analytical insights into the non-monotonous velocity behavior in projectile motion with drag, including the relationship between minimum and terminal velocities.

Findings

Minimum velocity can be lower than terminal velocity for V0 > Vterm.

Velocity is non-monotonous under certain initial conditions.

Numerical simulations confirm analytical results.

Abstract

The motion of a projectile with horizontal initial velocity V0, moving under the action of the gravitational field and a drag force is studied analytically. As it is well known, the projectile reaches a terminal velocity Vterm. There is a curious result concerning the minimum speed Vmin; it turns out that the minimum velocity is lower than the terminal one if V0 > Vterm and is lower than the initial one if V0 < Vterm. These results show that the velocity is not a monotonous function. If the initial speed is not horizontal, there is an angle range where the velocity shows the same behavior mentioned previously. Out of that range, the volocity is a monotonous function. These results come out from numerical simulations.