Eight hundred revolutions a minute is worth 3.00 metres
A free kick is three forces and a differential equation. Solve it properly and the bend a commentator calls unplayable turns out to be an exactly predictable quantity.
The equation
m dv/dt = mg − ½ρC_dA|v|v + ½ρC_lA|v|²(ω̂ × v̂). Gravity, drag against the velocity, and the Magnus force perpendicular to both spin and velocity. Integrated with fourth-order Runge–Kutta at two-millisecond steps, which is the point at which the path stops changing.
Why the ball matters as much as the boot
C_d is not a constant. Below a critical speed the boundary layer is laminar and drag is high; above it the layer turns turbulent and drag collapses. Where that crisis sits depends on the seams, and a ball whose crisis lands near shooting speed is the one that wobbles.
RK4 at 2 ms with a speed-dependent drag coefficient fitted per ball model and lift from the spin ratio. Air density from altitude and temperature.
A deterministic solver gives the mean path. Near the drag crisis the real wake is unsteady, which is the knuckleball, and no smooth model reproduces the wobble.