Result
Range (m)—
Apex (m)—
Flight (s)—
Impact |v| (m/s)—
Impact angle (°)—
Vacuum range (m)—
Drag loss (%)—
Terminal Re—
Range vs angle
Sweeping θ finds the drag-corrected optimal launch angle — below 45° once drag matters.
Monte-Carlo dispersion
Mean range (m)—
Std σ (m)—
CEP / P50 (m)—
Governing equations
m·a = mg − ½ρCdA·|v−w|·(v−w) + ½ρCLA·|v|·(ŝ×v)
k = ½ρCdA / m (drag param)
CL ≈ 1/(2+ v/(ωr)) (Magnus lift)
k = ½ρCdA / m (drag param)
CL ≈ 1/(2+ v/(ωr)) (Magnus lift)
A = πd²/4. With drag the ODE has no closed form — hence numerical integration. The vacuum limit ρ→0 recovers R = v₀²·sin2θ/g.