Exact turning invariants for projectile motion under quadratic drag and a vertical-axis Magnus force

Abstract

A ball spinning about a vertical axis is pushed sideways by the Magnus force, so its ground track curves. We show that when the lift and drag coefficients are constant, the horizontal part of this problem is exactly solvable in elementary functions. The step that makes this work is to write the horizontal velocity as a complex number and to use path length rather than time as the independent variable. Both aerodynamic forces are quadratic in speed, so the speed cancels and what remains is a linear equation with constant coefficients in which gravity does not appear. Three exact results follow from it. The heading turns at a fixed rate per unit path length, so a half turn always costs the same distance whatever the launch conditions and whatever the drag. The horizontal speed decays exponentially in the turn angle at a rate fixed by the drag-to-lift ratio, which makes the horizontal hodograph a logarithmic spiral. The radius of curvature of the ground track equals the lift length times the cosine of the flight-path angle. The whole flight problem also collapses to one scalar differential equation controlled by two dimensionless numbers. We use this to settle a concrete question. Two identical balls thrown in opposite horizontal directions with opposite spin can meet head-on when drag is absent, on a one-parameter family we give in closed form, and they fail to meet once drag acts. Every analytic claim is checked against machine-precision numerics, and the code and data are openly available.