JOSEF'S EXTERNAL BALLISTICS

Trajectory · Wind Drift · Energy · Hunt Classes · Phase 1+2+3 · G1/G7 · RK4

Stability ↗

Light mode

Input Parameters

Rifle & Bullet

Muzzle Velocity

▶ Chronograph details

Bullet Weight

BC Type

▶ Estimate G7 from G1

Sight Height

Vital Zone (radius)

Zero Range

▶ Estimate zero range

Optimize zero range for vital zone

Optimal zero range

◎ Optimal

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Estimate from measured offset

Offset

At distance

Estimated zero range

↗ Rising

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▲ Peak

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↘ Falling

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Zero Atmosphere (Phase 1)

Temp

Pressure

Rel. Humidity

Shooting Atmosphere (Phase 2)

Temp

°C

Pressure

hPa

Rel. Humidity

Propellant Temperature (Phase 3) EXPERIMENTAL

v₀ sensitivity

m/s/°C

Environment

Crosswind

Shooting Angle

Display

Max Range

Table Step

Ready — press Calculate

Summary at Zero & 100m

Energy

E@zero — J E₁₀₀ — J

Velocity

V@zero — m/s V@100m — m/s

Wind Drift

@zero — mm @100m — mm

Trajectory

Zero atmosphere Shooting atmosphere v₀ ±10 m/s band — — Vital zone ±60 mm

Ballistic Table

No data yet — press Calculate

Wind Drift

Sensitivity Analysis at 200 m (fixed zero angle)

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Model & Mathematics ▶ expand

Trajectory Model

Point-mass ODE integrated with RK4 (dt = 1 ms):
dvx/dt = −a_drag · (vx/v)
dvy/dt = −g·cos(α) − a_drag · (vy/v)

where:
a_drag = (π/8) · ρ · v² · C_d(M) / BC_SI
π/8 = 0.3927 (Siacci/Ingalls normalization, matches Gehtsoft G1 table)
BC_SI = BC_G1 × 703.07 [kg/m²]
ρ = (P_d · M_d + e · M_w) / (R · T) [kg/m³] (humid air, M_d=28.964, M_w=18.016 g/mol)
v_s = 331.3 · √(T/273.15) · √(1 + 0.16·e/P) [m/s] (humidity correction)
e = RH/100 · e_s, e_s = 6.1078 · 10^(7.5·T_C/(237.3+T_C)) [hPa] (Magnus formula)
M = v / v_s (Mach number)

G1/G7 drag function C_d(M): Gehtsoft G1 tabulation (79 Mach points, 0–5), π/8 normalization (Siacci/Ingalls convention).
Zero angle θ: found by bisection (60 iterations, ε < 10⁻¹⁰ rad).
Wind drift: lag-time method — z = w · (t_flight − x/v₀).
Shooting angle α: reduces effective gravity (g·cos α). Both uphill and downhill hit higher than level.

Three-Phase Model

Phase 1 — Zero: bisection solves for θ under zero-day atmosphere (T_zero, P_zero, RH_zero). θ is fixed thereafter — it represents the rifle's mechanical setting.
Phase 2 — Shooting day: same θ, new atmosphere (T_shoot, P_shoot, RH_shoot). Shows the pure external ballistic effect of weather change. Typically <2 mm at ≤200 m.
Phase 3 — Propellant temperature EXPERIMENTAL: same θ and shooting atmosphere, but v₀ is shifted by the propellant temperature effect: Δv₀ = k · (T_shoot − T_zero), where k is the user-measured sensitivity [m/s/°C] and propellant temperature is assumed equal to shooting temperature. This is the dominant real-world weather effect — a −30°C temperature swing with a sensitive powder (k = 0.5 m/s/°C) shifts v₀ by 15 m/s. Disabled when k = 0.

Validation vs JBM Ballistics & QuickTARGET

Test conditions: 165 gr Hornady InterBond · G1 BC 0.447 · v₀ = 785 m/s · sight 45 mm · zero 150 m · 15°C / 1013 hPa / 0% RH · no wind

Range This app JBM Δ traj. Δ vel. Δ energy 50 m +15 mm +15 mm 0 mm +1 m/s <1% 100 m +31 mm +30 mm +1 mm +1 m/s <1% 150 m 0 mm 0 mm 0 mm 0 m/s <1% 200 m −83 mm −84 mm +1 mm −1 m/s <1% 300 m −425 mm −424 mm −1 mm 0 m/s <1%

Validated against JBM Ballistics and Norma factory data. Trajectory error ≤1 mm and velocity error ≤0.1 m/s across all ranges. Drag model uses Gehtsoft G1 tabulation with π/8 (Siacci/Ingalls) normalization — no empirical correction factors. Humidity model: Magnus formula + partial-pressure density, validated at 15°C / 1013 hPa / 50% RH → ρ = 1.2211 kg/m³ (QuickTarget).

Model Notes

▸ Trajectory, velocity, and energy validated to <1% vs JBM — Gehtsoft G1 drag tabulation with π/8 (Siacci/Ingalls) normalization. No empirical correction factors.
▸ Reliable to ≤300 m — trajectory error <2 mm, velocity error <0.2 m/s across the full range.
▸ Relative humidity modelled — Magnus formula + partial-pressure density. Effect <0.5 mm at ≤200 m; included for physical correctness.
▸ G1 and G7 drag models supported — G7 recommended for VLD/boattail bullets.
▸ No spin drift, Coriolis, or aerodynamic jump — negligible at ≤300 m.
▸ Phase 3 (propellant temperature) is experimental — user-measured v₀ sensitivity [m/s/°C]. Disabled when k = 0.