V3I1P56

Numerical Implementation of Euler’s Method for Solving Ordinary Differential Equations in Projectile Motion Regimes

S Sai Teja^1*, M Devuja²,B Jagan Mohan³

Abstract

This study analyzes projectile motion, examining the shift from vacuum trajectories to paths affected by aerodynamic drag. While parabolic solutions describe vacuum motion, drag creates nonlinear differential equations without closed-form solutions. The Forward Euler method in Python provides computational simulation of ballistic motion. The evolution of ballistic theory progressed from Aristotelian concepts through Tartaglia’s curved trajectories to Newtonian mechanics. Euler’s method derives from Taylor series expansions, analyzing Truncation Errors and stability criteria. The study compares Python loops with NumPy vectorized operations for performance. Simulations test Euler’s accuracy against vacuum solutions and examine Stokes and Newtonian drag regimes. Reynolds number-dependent coefficients and the “drag crisis” were modeled, demonstrating complex aerodynamic effects. Sensitivity analysis revealed Euler’s limitations: coarse time steps reduced accuracy, whereas finer steps increased the computational cost. The conclusion underscores Euler’s pedagogical value but recommends higher-order solvers, such as Runge-Kutta, for precise ballistic modeling.

Keywords:

Projectile motion; Aerodynamic drag; Forward Euler method; Ballistic simulation; Numerical stability