Two forces act on the projectile: constant gravity, and quadratic aerodynamic drag opposing its velocity relative to the air. Because drag depends on the square of speed, the equations of motion have no closed-form solution — this simulation numerically integrates them step by step (semi-implicit Euler, dt = 2 ms) instead of drawing an algebraic curve.
Drag force: F_d = ½ · ρ · |v_rel|² · Cd · A (opposing v_rel)
v_rel: velocity relative to wind (horizontal wind only)
Acceleration: a = g + F_d / m (F_d direction = -v_rel/|v_rel|)
Air density: ρ = 1.225 kg/m³ (sea level)
- Mass — heavier projectiles carry more momentum per unit of drag force, so they decelerate less and stay closer to the vacuum parabola.
- Drag coefficient (Cd) and area — both scale the drag force directly; a streamlined, small-area shape (low Cd·A) behaves almost like the vacuum case, a draggy, large-area one falls far short of it.
- Wind — shifts the relative-velocity vector the drag force reacts to, so a tailwind reduces effective drag (adds range) and a headwind increases it (cuts range), independent of gravity.
- The dashed amber curve is the same launch angle and speed evaluated with zero drag and zero wind — the textbook parabola — so you can see exactly how much distance and height air resistance costs at your current settings.
Real-world relevance: this drag-aware model, not the vacuum parabola, is what actually governs artillery fire-control tables, thrown or kicked balls, and long-range projectile ballistics — the vacuum case is only a useful first approximation.