Drag, gravity, Magnus

Three forces act on a ball in flight: gravity, drag against the velocity, and the Magnus force perpendicular to both spin and velocity.

m dv/dt = m g − ½ ρ C_d A |v| v + ½ ρ C_l A |v|² (ω̂ × v̂)

ρ is the air density (thin at altitude, so less of both), A the cross-section, C_d the drag coefficient and C_l the lift coefficient, which grows with the spin ratio S = rω/|v|:

C_l = 1 / (2 + 1/S)

The films integrate this with a fourth-order Runge–Kutta step every two milliseconds. Sidespin about the vertical axis bends the ball sideways; topspin pushes it down, which is why a kick serve can clear the tape by more and still land.

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Aim right of the wall, spin it back left, and watch how altitude and a crosswind change the bend. At 2,250 m in Mexico City the same kick bends less and carries longer.

The same strike three times, seen from above, aimed fourteen degrees outside the post. Without spin it misses by nearly two metres; with 400 rpm it is in the net.

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