Calculation of Minimum Speed of Projectiles Under Linear Resistance Using The Geometry of The Velocity Space

Abstract

We look at the problem of the minimum speed of projectiles in a constant gravitational field. In the absence of resistance, the problem may be studied in the frame of a high school curriculum. One needs only Newton’s laws and a minimum amount of analytic geometry to compute the orbit, which turns out to be parabolic. Furthermore, in case the projectile is launched upwards, employing the theorem of conservation of mechanical energy we conclude that the minimum speed occurs at the highest point.
In the presence of resistance, the system is dissipative and hence, the previous tools are not available. We focus at the case where the resisting force is linear in speed and opposing the velocity vector. It has been observed in numerical experiments that the minimum speed, if any, occurs when the projectile is on the way down, after having achieved the maximum height. We propose a presentation of the solution to this problem using the geometry of the velocity space. As a basic tool, we apply an old idea introduced by Hamilton and Maxwell, the hodograph of motion.
It turns out that the hodograph of motion in this case is a straight line. This fact allows us to describe the values of initial speed and launching angles that will result in an orbit with or without minimum speed. In the former case, the calculation of the value of minimum speed represents the distance of the origin to the hodograph line and is given by elementary manipulations. This approach in the study of physical problems, besides being elegant on its own right, helps college students feel the deep relation between physics and geometry.