⚽ Football Free-Kick Flight Lab: Quadratic Drag and Magnus Curve with an RK4 Angle Experiment — 2D Cross-Section

A size-5 football is launched with adjustable speed, angle, backspin and sidespin; quadratic air drag and the Magnus lift force are integrated with fourth-order Runge-Kutta. A live experiment sweeps the launch angle and compares the simulated range with the vacuum formula v^2 sin(2a)/g.