⚽ Football Kick Physics — Trajectory and Spin

Forces on the ball

Newton's second law with three forces: m dv/dt = m g + FD + FM, integrated with RK4 (step 1/480 s).

Drag FD = -½ ρ CD(Re) A |vrel| vrel, with Re = ρ |vrel| d / μ, ball diameter d = 0.22 m, A = πR². A football shows a drag crisis: when the boundary layer turns turbulent (Re about 2×10⁵, roughly 13 m/s at sea level) C_D falls from about 0.47 to about 0.2. That is why a hard-struck ball flies much farther than a slow one. The model is a smooth fit of that behaviour, not a measured table.

Magnus force FM = ½ ρ A CL |v|² · unit(ω × v), with the spin parameter S = R |ω⊥| / |v| and CL = 1/(2 + 1/S) (a Goff and Carré style fit). Only the spin component perpendicular to the velocity bends the path; axial spin (about the line of flight) does nothing to the trajectory. Backspin lifts the ball, topspin makes it dip, sidespin makes it curl.

Knuckle: with almost no spin, the seams make the side force fluctuate. It is modelled as a smooth pseudo-random lateral force of amplitude 0.1 · ½ ρ A v² that fades once real spin appears. It is a stand-in for a chaotic wake, not a CFD result.

Bounce, wall, post, net, keeper

Ground contact uses a restitution e = 0.76 and Coulomb friction (μ = 0.45) on the contact-point slip, with a hollow-sphere moment of inertia I = 2/3 m R², then rolling resistance. Posts and crossbar are cylinders (radius 6 cm, e = 0.62). The wall is a row of boxes (0.56 m × 0.4 m × 1.78 m) that can jump. The net is a drag-and-reflect volume. The keeper moves toward the predicted crossing point (plus your chosen misjudgement) after a reaction delay at a fixed speed; a save is a reach test of 0.8 m. This is a teaching model of a keeper, not a biomechanical one.

Honest simplifications

Rigid ball, uniform air, no ball deformation, constant spin-decay time, no wind gradient, and a point-like keeper reach. Presets were tuned to score in this model; a real result depends on many things we do not model.

Independent checks (run in the page)

1) With air off, the numerical range equals v² sin 2θ / g. 2) A long fall reaches the terminal speed that solves m g = ½ ρ C_D(Re) A v². 3) The kinetic + potential energy lost in a curling shot equals the integral of F_D·v dt (Magnus does no work).

• Press a preset, then slow the time speed to 0.1× and watch the ball spin and the Magnus arrow (gold) bend the path.
• Set air density to 0 and watch spin stop mattering: the path becomes a parabola.
• Change only the ball speed from 10 to 30 m/s: the drag coefficient graph shows the drag crisis around 13 m/s.
• Switch gravity to 1.62 m/s² (Moon) with air 0 for a floaty free kick.
• Add a headwind or a crosswind and see how much a 5 m/s breeze moves the ball at 25 m.
• Raise the altitude to 3000 m: thinner air means less drag and less Magnus curve.
• Try the knuckleball preset with different seeds: same speed, different wobble.