Kick Force x1 to x1000 (2D)

What is modelled

A thought experiment: what if the force of a football kick were k times larger, up to 1000 times? The kick is split into two parts. First the impact: a leg of effective mass mf moving at vf hits the ball (mass 0.43 kg) with restitution e, so the ball leaves at v0 = (1 + e) mf vf / (mf + m). Multiplying the contact force by k over the same contact time multiplies the impulse and so the launch speed by k. Second the flight: a point mass over a round Earth, integrated with RK4, under gravity GM/r², exponential air density ρ = 1.225 e-h/8.5 km, a Mach-dependent drag coefficient (0.25 subsonic, about 1 at Mach 1.1 to 3, 0.9 hypersonic), and an optional Magnus lift.

Equations

• Impact: impulse J = m v0; a half-sine force pulse of duration τ has peak Fpk = (π/2) J / τ (about 1.9 kN for a normal kick). The page also integrates a spring-dashpot contact with e = exp(-πζ/√(1-ζ²)) to draw the force curve.
• Drag: a = -½ ρ Cd(M) A v |v| / m. At constant Cd and density the speed decays as v = v0 e-x/L with L = 2 m / (ρ Cd A), about 10 to 50 m of sea-level air for this ball. This is the key number: to be thrown out of the atmosphere the ball would need to shed e-800 of its speed in the air above it, because the air column over each square metre weighs about 800 times what m/(CdA) weighs.
• Magnus (subsonic only): CL = S/(1 + 2.5 S) with spin ratio S = rω/v, a rough fit to measured football data.
• Stagnation temperature T0 = T (1 + 0.2 M²) shows why the model is a thought experiment: above about Mach 3 a real leather and polyurethane ball would burn through.
• In a vacuum the same integrator gives Kepler orbits: a launch speed of 11.19 km/s (about 397 times a normal kick) escapes the Earth.

Checks run in this page

• Vacuum range of a slow kick against v² sin 2α / g (the round-Earth integrator must reduce to the flat-Earth formula).
• Kepler: apogee of an 8.5 km/s launch against vis-viva, and a circular-orbit launch must keep its radius over one period.
• Impact: the integrated foot-ball contact must give the momentum-formula ball speed.
• Terminal speed of a dropped ball against √(2 m g / (ρ Cd A)), and constant-drag stopping against L = 2m/(ρ Cd A).

Honest simplifications

• The ball is an indestructible point mass: aerodynamic heating, ablation, shock waves and break-up are not modelled, and the foot and leg are assumed to survive a force k times larger. The model says what the physics of motion would do, not what the ball would survive.
• Drag uses a single exponential atmosphere and a table Cd(M) of a sphere; real football drag also depends on Reynolds number (the drag crisis) and on the panel pattern.
• Earth rotation, wind and the change of Earth radius with latitude are ignored.
• The kick is a dimensionless thought experiment about a ball, not instruction for any real activity.

The stadium, ball, goals and kicker mannequin are Blender-built meshes with baked PBR textures; the physics runs live in your browser.