A real artillery shell loses range to aerodynamic drag, opposing velocity and scaling with the square of speed. The orange trail integrates full drag physics; the dashed grey trail shows the same shot in a vacuum for comparison.
F_drag = ½·ρ·v²·Cd·A (opposing velocity)
m·dv/dt = F_gravity + F_drag
x,y integrated via semi-implicit Euler
- Muzzle velocity — initial shell speed v0.
- Launch angle — elevation angle above the ground.
- Drag coefficient Cd — shape-dependent drag factor (0 = perfectly streamlined, higher = blunter shell).
- Air density ρ — thinner air (higher altitude) reduces drag and extends range.
Real use: Modern fire-control computers solve this exact drag-augmented trajectory (plus Coriolis and wind terms) to compute artillery firing solutions.