Physics and ODEs

Numerical Analysis · Axiom Academy

REAL WORLD Physics Applications of ODE Methods From projectiles to planets: See how numerical methods solve the differential equations that govern our physical world Simulating Reality with Differential Equations Nearly every physical system - from a thrown ball to orbiting planets - is governed by differential equations . These equations describe how quantities change over time, but most cannot be solved exactly. This is where numerical methods become essential. Engineers, physicists, and game developers use algorithms like Euler's method and Runge-Kutta to simulate everything from weather patterns to video game physics. Projectile Motion with Air Resistance A ball thrown through the air experiences both gravity pulling it down and air resistance (drag) opposing its motion. The drag force is proportional to velocity squared: Without air resistance, we have a simple parabola. With drag, the trajectory becomes asymmetric - the ball falls more steeply than it rises. Watch how different drag coefficients change the path: The Simple Pendulum (Nonlinear) The pendulum equation is deceptively simple but nonlinear . For small angles, we approximate sin(theta) as theta. But for large swings, this breaks down: Watch how a large-amplitude pendulum behaves differently from the idealized small-angle approximation. The period actually increases for larger swings!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.