Projectile with Drag and the Magnus Force in 3D

A struck ball does not follow the neat parabola of the textbook. Air drags on it, shortening every throw, and if it is spinning the air pushes it sideways: that is the Magnus force, the reason a free kick curls around a wall, a sliced drive peels off, and a backspun ball hangs in the air far longer than it should. The force is always perpendicular to both the velocity and the spin axis, so sidespin bends the path left or right, backspin lifts it and stretches the range, and topspin presses it down and cuts the range short. The scene flies the ball along its real trajectory over the ground, with a faint reference showing where the same ball would land with no spin, so the bend you see is the spin alone. The diagnostic sweeps the spin rate and plots how far the ball flies and how far it bends, the two numbers a coach, a golfer, and a ballistician all live by.

Figure 1. Projectile motion with quadratic drag and the Magnus force from spin. Top: the ball flying along its trajectory in pseudo-3D (x downrange, y lateral, z up), with the faint reference path it would follow with no spin. Bottom: the range and the lateral deflection at landing against the spin rate, with the live operating point. Method: RK4 on dv/dt = -g z - c|v|v + cM (omega x v).

spinsidespin

spin rate46

speed30

angle34°

WHAT TO TRY

Keep sidespin and push the spin rate up: the path bends further off the straight reference, a banana curve.
Switch to backspin: the ball lifts and flies past where it would land with no spin; topspin does the opposite, dropping it short.
Set spin to none: the path collapses onto the reference, just gravity and drag, no sideways push.
Watch the lower plot: sidespin grows the deflection while the range holds; backspin and topspin move the range instead.