A fired shell obeys Newton's second law under two forces: constant gravity and quadratic aerodynamic drag opposing its velocity vector. The drag force depends on air density ρ, drag coefficient Cd, the shell's cross-sectional area A, and its mass m — this ratio m/(CdA) is the shell's ballistic coefficient, a dimensionally consistent measure (kg/m²) of how well it resists deceleration.
dv/dt = -g ĵ - (ρ·Cd·A)/(2m) · |v|·v
Vacuum range: R = v0² sin(2θ) / g
Ballistic coeff: BC = m / (Cd·A)
This sim integrates that equation numerically (RK4, fixed 20 ms physics step) rather than using the idealized formula, so drag, mass and air density all genuinely change the flown path — not just a cosmetic curve.
- Muzzle Velocity — initial speed v0 leaving the barrel.
- Elevation Angle — launch angle θ above horizontal.
- Shell Mass — heavier shells have higher ballistic coefficient and resist drag deceleration better, flying farther for the same v0.
- Atmosphere — sets air density ρ (0 = vacuum, 1.225 kg/m³ = sea-level standard, 2.4 = dense/humid extreme) used in the drag term.
Real proofing ranges use exactly this kind of drag-corrected exterior ballistics model, calibrated against radar-tracked test firings, to build the firing tables gun crews rely on.