Projectile motion with quadratic drag

Two-dimensional coupled nonlinear equations of projectile motion with air resistance in the form of quadratic drag are often treated as inseparable and solvable only numerically. However, when they are recast in terms of the angle between the projectile velocity and the horizontal, they become completely uncoupled and possess analytic solutions for projectile velocities as a function of that angle. The equations relating the time and position coordinates to this angle are not integrable in terms of elementary functions but are easy to integrate numerically. Additionally, energy equations explicitly including dissipation terms can be developed as integrals of the equations of motion. One-dimensional numerical integrations can be treated in a pedagogically straightforward way using numerical analysis software or even within a spreadsheet, making this topic accessible to undergraduates. We present this approach with sample numerical results for velocity components, trajectories, and energy-balance of a baseball-sized projectile.