Penalty Kick Reaction-Time Lab

Magnus curve and drag crisis against a goalkeeper with a finite reaction time. Drag the goal on the 3D view to set the aim point.

Presets

Kick

Aim x

Aim height

Ball speed

Sidespin

Back/top spin

Aim solver compensates drag and curve

Sidespin: + curls the ball to the kicker's left (rev/s). Backspin: + gives lift, - is topspin.

Goalkeeper

Reaction time

Reach (hand)

Dive speed

Strategy

Air and planet

Environment

Panel roughness

Drag crisis (Cd falls with Reynolds number)

View

Trajectory preview Force vectors Ghost of last kick High quality

Slow motion

Live readouts

Keeper gap vs time

Top view: ball path vs straight line

Experiments

beats keepersavedno goal (miss)

Pick an experiment. They use the current speed, spin, environment and keeper settings.

The science

Flight. m dv/dt = m g - ½ρA C_d|v|v + ½ρA C_L|v|² (ω̂ × v̂), integrated with RK4 at 2 ms. Ball: 0.43 kg, radius 0.11 m.

Magnus lift. Spin parameter S = rω/v, lift coefficient C_L = 1/(2 + 1/S) (empirical fit; a few spin revolutions per second give S about 0.15 and C_L about 0.12, a lateral acceleration of 3-5 m/s²).

Drag crisis. Re = ρ v D / μ. Below the critical Re (about 2-4e5, earlier for rougher panels) C_d is near 0.47; above it the turbulent boundary layer delays separation and C_d drops to about 0.2. A 20 m/s ball is near the crisis, a 28 m/s ball well past it.

Goalkeeper. At t = t_r (reaction time, default 0.20 s) the keeper commits to a point on the goal line. His centre of mass must move until a hand (reach R) can touch it: distance max(0, |T - C0| - R). Each axis is accelerated from rest up to a speed limit: t = sqrt(2d/a) or v/a + (d - v²/2a)/v. Vertical dive: jump 9 m/s² up to 3.5 m/s, fall at g.

Unsaveable means t_flight < t_r + t_dive. The analytic comparison uses vacuum: launch angle tanθ = (v² - sqrt(v⁴ - g(gL² + 2Δy v²)))/(gL), flight time L/(v cosθ).

Assumptions. Rigid sphere, constant spin axis with slow decay (14 s), no knuckle-ball wobble, no wind, keeper moves in a 2D plane with fixed kinematics (not a biomechanics model), point-contact post collisions with restitution 0.7, deflected shots counted as saved.