spinning-ball-spiral
2023 - 2024
Abstract
This project presents a comprehensive physics simulation investigating the aerodynamics of spinning spherical projectiles, with particular focus on the Magnus effect. Inspired by Roberto Carlos's legendary free-kick goal in 1997, where the ball curved dramatically in flight, we develop a mathematical model that captures the complex interplay between drag forces, lift forces, and the Magnus effect. The simulation allows for exploration of how spin rate, initial velocity, and environmental conditions affect the trajectory of a spinning ball.
Physical Model
The simulation models three main forces acting on a spinning ball: gravitational force, aerodynamic drag (proportional to velocity squared), and the Magnus force (perpendicular to both velocity and angular velocity vectors). The equations of motion are solved using numerical integration methods (Runge-Kutta 4th order) to compute the ball's trajectory over time. Parameters such as ball mass, radius, drag coefficient, and spin decay are configurable.
Visualization and Analysis
Using Matplotlib, the project generates 3D trajectory visualizations and 2D projections showing the ball's path from different angles. Analysis reveals how varying spin rates produce different curvature patterns, explaining phenomena observed in football, tennis, and baseball. The Roberto Carlos kick is recreated by setting appropriate initial conditions, demonstrating that the extreme curve was indeed physically achievable.