Tyler Crockett — Macdaddy4sure.ai

_AugmentedIntelligence v11.0 · Ballistics Subsystem (ShootingCore, ShootingELRCore)_

Abstract

We describe the external-ballistics firing-solution engine implemented in the

_AugmentedIntelligence runtime for the long-range target-shooting disciplines

(F-Class, PRS, benchrest, and extreme-long-range steel). The engine integrates a

point-mass trajectory under a tabulated G1/G7 drag law and a standard-atmosphere

density model to produce elevation and windage corrections, time of flight, and

retained velocity and energy. For extreme ranges it augments the point-mass core

with the deterministic secondary effects that dominate beyond ~1000 yd —

gyroscopic (spin) drift, Coriolis and Eötvös deflection, aerodynamic jump, and

atmospheric density altitude — and provides a Monte-Carlo uncertainty model that

reports a probability of hit and an input-sensitivity (error) budget. We give the

governing equations, the numerical method, validation against a reference load,

and an explicit statement of the model’s limitations. **This is a marksmanship

training and planning aid; all computed dope must be verified against live fire.**

1. Introduction and scope

A firing solution is the set of sight corrections a shooter dials (or holds) to

place a projectile on a target at a known distance under known conditions. For

the target-shooting sports the relevant outputs are the elevation come-up and

wind hold, expressed in the shooter’s scope units (milliradians or minutes of

arc), together with supporting quantities — bullet drop, wind drift, time of

flight, and retained velocity and energy.

The physics is a classical initial-value problem: projectile motion under gravity

and aerodynamic drag. It is deterministic and well understood; the engineering

difficulty lies in (i) an accurate drag description, (ii) the secondary effects

that become first-order at long range, and (iii) the uncertainty in the inputs,

which at extreme range dominates the achievable precision. This paper documents

how the engine addresses each.

The engine deliberately targets known-distance, static-target shooting. It is

a calculator, not a fire-control system.

2. Coordinate system and problem statement

We integrate in the vertical plane containing the line of sight (LOS) — the

axis from the shooter’s eye through the scope to the aim point — with a lateral

axis for wind and spin effects. Let:

• s — distance traveled along the LOS (down-range),
• h — height of the projectile relative to the LOS (positive up),
• z — lateral offset (positive right).

The sight sits a distance H (sight height) above the bore, so the projectile

begins at h = −H. The bore is elevated above the LOS by a small launch angle

φ (determined by the zero, §7). The reported drop is h at the target

range; the come-up is the angle that must be added to the LOS to place the

bore such that h = 0 at that range.

3. Governing equations

3.1 Drag deceleration

For a point-mass projectile the aerodynamic drag produces a deceleration

anti-parallel to the velocity vector v relative to the air:

a_drag = ρ · C_D(M) · |v|² / (2 · B)          [m/s²]

where ρ is air density, C_D(M) is the drag coefficient of the standard

reference projectile at Mach number M = |v|/c, and B is the ballistic

coefficient expressed as an SI areal density (kg/m²). This form folds the

projectile’s mass, diameter, and form factor into B, which is the shooter-known

quantity. The equivalence follows from the standard definition

B = m / (i · d²) and C_D,bullet = i · C_D,std, giving

a_drag = (π/8)·ρ·|v|²·C_D,std / B; absorbing the geometric constant into the

tabulated C_D and the unit conversion yields the expression above.

BC units. Shooters quote the G1 (or G7) BC in imperial lb/in². We convert

once at input:

B[kg/m²] = BC_imperial · 703.06957

3.2 Equations of motion

Decompose the velocity into along-LOS (v_s), perpendicular (v_h), and lateral

(v_z) components. With a crosswind w (positive from the left, §8), the

air-relative lateral velocity is v_rz = v_z − w, and `|v| = √(v_s² + v_h² +

v_rz²)`. The accelerations are:

a_s =  −a_drag · (v_s  / |v|)
a_h =  −a_drag · (v_h  / |v|)  −  g·cos(α)
a_z =  −a_drag · (v_rz / |v|)

where g = 9.80665 m/s² and α is the look (incline) angle; the factor cos(α)

is the component of gravity perpendicular to the LOS (the “rifleman’s” correction

for uphill/downhill shots). Wind enters through v_rz, so lateral drift emerges

naturally from the integration rather than a closed-form add-on.

3.3 Auxiliary quantities

Speed of sound (m/s) from air temperature T_F (°F), with T_C in Celsius:

c = 331.3 · √(1 + T_C/273.15)

Muzzle/retained energy (ft·lb) from bullet weight w_gr (grains) and velocity

v_fps:

E = w_gr · v_fps² / 450240

Angular conversions (small-angle sight corrections): `1 rad = 3437.75 MOA =

1000 mil. The come-up and wind hold are atan2(|drop|, R)` and

atan2(drift, R) scaled to the shooter’s units, where R is the slant range.

4. Drag models

The engine ships two standard drag functions, tabulated as C_D versus Mach and

linearly interpolated:

• G1 — the classic flat-base reference projectile. Appropriate for

  conventional bullets; the drag rises sharply through the transonic region and

  peaks near Mach 1.3.

• G7 — a boat-tail, secant-ogive reference projectile. A markedly better

  match for modern long, sleek match bullets, with correspondingly lower and

  flatter C_D.

The choice of model must match the BC the shooter supplies (a bullet’s G7 BC is

numerically ~half its G1 BC). Selecting the correct reference is the single

largest fidelity lever short of a radar-measured custom drag curve.

5. Standard atmosphere

Air density is expressed as a ratio to the G1 standard metro condition

(59 °F, 29.92 inHg, sea level), because the tabulated C_D is referenced to that

standard:

ρ/ρ₀ = (P / 29.92) · (518.67 / (T_F + 459.67))

with pressure P in inHg. When only altitude is known, ICAO pressure altitude

supplies P. For field use we also report density altitude (DA), the single

number that captures the combined temperature/pressure/altitude effect on drag:

PA   = (29.92 − P) · 1000                    [ft, pressure altitude]
DA   = PA + 118.8 · (T_F − (59 − 3.57·PA/1000))

Humidity is a small second-order effect and is neglected.

6. Numerical integration

The trajectory is advanced with a fixed-step semi-implicit (symplectic) Euler

scheme at Δt = 2×10⁻⁴ s. Velocities are updated from the accelerations, then

positions from the new velocities; this is stable and conserves the qualitative

structure of the motion far better than explicit Euler at the same step. At each

step the drag magnitude is re-evaluated at the current Mach number. When the

along-LOS coordinate first exceeds the target range, the solution (drop, drift,

time of flight, velocity) is linearly interpolated to the exact range so that

outputs are not quantized by the step size. Time of flight at extreme range is

5–10 s, so the small step is necessary to keep integrated error negligible.

7. Zeroing

A rifle is zeroed at a chosen range R₀ (commonly 100 yd): the shooter adjusts

the sight so the trajectory crosses the LOS at R₀. The engine solves for the

bore launch angle φ such that h(R₀) = 0 by bisection, exploiting the

monotonic dependence of h(R₀) on φ. All subsequent solutions integrate at the

fixed φ; the reported come-up at range R > R₀ is the additional elevation

needed beyond the zero.

8. Wind model

Wind is entered as a speed and a clock direction (the o’clock position the

wind blows from; 3 o’clock is a pure right-to-left crosswind). The crosswind

component driving lateral drift is

w = −W · sin(θ_clock),     θ_clock = (clock/12)·2π

so a 9-o’clock wind (from the left) pushes the bullet right, a 3-o’clock wind

pushes left, and 12/6-o’clock (head/tail) winds produce no lateral drift. Because

w enters the air-relative velocity, drift accumulates through the full flight

rather than via a lag-time approximation.

9. Extreme-long-range corrections

Beyond ~1000 yd several effects that are negligible at short range become

first-order (feet, not inches). These are deterministic given their inputs and

are therefore computed, not learned or guessed. They are added to the point-mass

solution.

9.1 Gyroscopic stability (Miller)

The Miller formula estimates the gyroscopic stability factor S_g from bullet

mass m (gr), diameter d (in), length in calibers l = L/d, and twist in

calibers t = T/d, corrected for velocity and atmosphere:

S_g = [ 30·m / (t²·d³·l·(1+l²)) ] · (v/2800)^(1/3) · (T_R/518.67) · (29.92/P)

S_g < 1.4 flags marginal stability — a warning that the bullet may destabilize

as it decelerates through the transonic region.

9.2 Spin drift (Litz approximation)

A gyroscopically stabilized bullet precesses, drifting laterally in the direction

of barrel twist. We use Litz’s empirical form (inches; positive right for

right-hand twist):

SD = 1.25 · (S_g + 1.2) · TOF^1.83

9.3 Coriolis and Eötvös deflection

Earth’s rotation deflects the long-flight-time projectile. With rotation rate

Ω = 7.2921×10⁻⁵ rad/s, latitude L, azimuth of fire A (0 = north, 90 = east),

range R, and time of flight T:

Horizontal (right in N. hemisphere):  Δz = Ω · R · T · sin(L)
Vertical (Eötvös; high shooting east): Δh = Ω · R · T · cos(L) · sin(A)

The horizontal term is azimuth-independent; the vertical term makes east-bound

shots print high and west-bound shots low.

9.4 Aerodynamic jump

A crosswind imparts a small vertical deflection at launch (angular, roughly

constant with range), modeled as ≈ 0.01 · S_g · W_cross (mil). It is small but

non-negligible at ELR and, unlike wind drift, acts vertically.

9.5 Powder-temperature velocity shift

Muzzle velocity varies with propellant temperature. Given a per-load sensitivity

k (fps/°F) and a reference (v₀, T₀), the corrected velocity is

v = v₀ + k·(T − T₀). At extreme range a 10 fps error is a foot or more of

vertical impact shift, so this correction is material.

10. Uncertainty quantification

At long range the precision of a firing solution is limited less by model

fidelity than by the uncertainty of its inputs — principally wind, then muzzle

velocity, then range. The engine therefore provides a Monte-Carlo dispersion

model. The mean shot is zeroed once; then N samples perturb the uncertain inputs

(muzzle velocity σ_v, wind σ_w, range σ_R, BC σ_BC) by independent Gaussians and

re-integrate at the fixed launch angle. The engine reports:

• Probability of hit (P_h) — the fraction of samples landing inside the target

  box;

• the dispersion (standard deviation of vertical and horizontal miss);
• an input-sensitivity budget — impact shift per 10 fps of MV, per mph of

  wind, and per yard of range error, computed by finite difference.

This reframes the extreme-range problem honestly: the output is not a single

number to dial but a cone, and the sensitivity budget shows which input

dominates it (almost always wind).

11. Truing

A generic solution is made rifle-specific by truing to observed impacts. Given

a measured drop at a known range, the engine solves (by bisection on velocity) for

the muzzle velocity that reproduces it — the first and most effective rung of

personalization. A learned residual model (a small neural network trained on a

shot log, confined to the observed envelope) is the natural next step for the

subsonic-drag error a single BC cannot capture; it is intentionally physics-anchored

so it corrects only what theory misses and never extrapolates.

12. Solution method: the _AugmentedIntelligence firing-solution pipeline

The preceding sections define the individual physical models; this section

describes how the engine composes them into a single firing solution, and the

design philosophy behind that composition. A solve is layered by order of trust:

a deterministic physics backbone first, then an honest uncertainty estimate, then

optional data-driven personalization — never the reverse.

Step 1 — Assemble the inputs. A solve takes a Shot (muzzle velocity, BC and

its drag model, bullet weight, sight height, zero range, temperature, pressure,

wind speed and clock, look angle) and, for long range, a Rifle (diameter,

length, twist rate and hand, shooter latitude and firing azimuth, and the scope’s

click value). These are the only quantities the shooter must supply.

Step 2 — Establish the atmosphere. From temperature and pressure the engine

computes the air-density ratio (§5) and the speed of sound (§3.3). The density

ratio scales the drag force; the speed of sound converts every velocity to a Mach

number so the correct point on the drag curve is used. This is done once and held

for the trajectory.

Step 3 — Zero the rifle (`solveZeroAngle`). Before any range solution the

engine finds the bore launch angle φ that puts the trajectory on the line of

sight at the zero range, by bisection (§7). This encodes the rifle’s actual

sight-in and is the reference against which all come-ups are reported.

Step 4 — Integrate the trajectory (`integrate`). Launching at φ, the engine

steps the point-mass equations of motion (§3.2) with the semi-implicit Euler

scheme (§6). At every step it re-evaluates the Mach number, looks up C_D from the

selected G1 or G7 table, and applies the drag, gravity-perpendicular, and wind

accelerations. When the down-range coordinate first passes the target range it

interpolates drop, drift, time of flight, and velocity to the exact distance.

This single routine yields the elevation, windage, TOF, and retained

velocity/energy — the core of the solution.

Step 5 — Convert to sight units. Drop and drift are converted from linear

offset to the shooter’s angular units — milliradians or MOA — via atan2, and

then to turret clicks using the scope’s click value. This is what the shooter

actually dials.

Step 6 — Apply the deterministic long-range corrections (`solveFull`). For

extreme range the engine adds the effects of §9 that are computable from the

inputs: it evaluates the Miller stability factor S_g, then adds Litz spin drift,

Coriolis (horizontal) and Eötvös (vertical) deflection, and aerodynamic jump, and

folds them into the total elevation and windage. It raises a transonic/stability

flag when S_g is marginal or the bullet has slowed below Mach 1.2. Crucially,

these are added analytically to the point-mass result rather than learned or

guessed — the engine computes everything theory can determine.

Step 7 — Quantify the uncertainty (`monteCarlo`). Recognizing that at long

range the limiting factor is input uncertainty, not model form, the engine zeroes

once and then re-integrates the trajectory hundreds of times with the muzzle

velocity, wind, range, and BC perturbed by their standard deviations. It returns a

probability of hit for the target, the dispersion ellipse, and a sensitivity

budget. The firing “solution” is thus delivered honestly as a *center plus a

cone*, with the sensitivity budget naming the dominant error (typically wind).

Step 8 — Personalize by truing (`trueVelocity`). Finally, the generic solution

is made rifle-specific: from an observed drop at a known range the engine solves

for the muzzle velocity that reproduces it, shifting the whole trajectory to match

the shooter’s actual dope. A physics-anchored learned residual (§11) is the

optional next layer for the subsonic-drag error a single BC cannot express.

Design principle. The pipeline embodies one rule: *compute what is knowable,

learn only the irreducible residual, and never present a point estimate where a

distribution is honest.* Steps 1–6 are pure, deterministic physics; step 7 makes

the residual uncertainty explicit; step 8 closes the loop with the shooter’s own

data. Deterministic effects (Coriolis, spin drift, stability) are never delegated

to a model — they are derived — so any learned component is confined to the small,

genuinely uncertain part of the problem and is barred from extrapolating outside

the data that trained it. This is what makes the solution both physically faithful

and trustworthy at the ranges where naïve models quietly fail.

13. Validation

The engine is exercised by an automated behavioural test suite

(suite_shooting.cpp, suite_shootingelr.cpp) covering unit conversions, the

atmosphere model, drag tables, monotonicity properties, and a reference load. For

a **.308 Winchester, 175 gr SMK, G1 BC 0.505, 2600 fps, 1.9 in sight height,

100 yd zero, standard conditions**, the engine produces:

Range (yd)	Come-up (mil)	Retained V (fps)	TOF (s)	Spin drift (in)	Coriolis H (in)
100	0.00	2456	—	—	—
1000	10.05	1349	1.61	10.7	3.0

with S_g = 2.40 (stable) and, under a Monte-Carlo model (σ_v = 10 fps, σ_w =

2 mph, σ_R = 3 yd), a horizontal dispersion (15.7 in) that substantially exceeds

the vertical (5.1 in) — correctly reproducing the field truth that **wind is the

dominant error source**. Velocity truing recovers a synthetic reference muzzle

velocity to within the test tolerance.

The come-up and retained velocity are in the expected range for this load; the

model is slightly optimistic at extreme range (§14).

14. Limitations

1. Single flat BC. The point-mass core uses one BC per drag model. Real BC

   degrades below Mach ~1.2, so the model under-predicts drag transonically and is

   optimistic at extreme range. G7 selection and, ideally, a banded or

   radar-measured custom drag curve reduce this.

• Point-mass, not 6-DOF. Spin drift and aerodynamic jump are empirical

   approximations, not a full rigid-body (epicyclic) solution. Yaw-of-repose and

   dynamic-stability transitions are not modeled.

• Single-layer atmosphere and wind. One density and one wind value are used;

   at ELR both vary appreciably over the trajectory’s tens-of-feet apex.

• Transonic dispersion is physical, not modeled. As a bullet decelerates

   through Mach 1 it may lose dynamic stability and disperse unpredictably; no

   solver overcomes an unstable load.

• Estimates, not guarantees. All outputs are planning estimates. **Verified

   live-fire dope always supersedes the model.**

15. Implementation notes

The physics is implemented as dependency-free, header-only C++ cores

(ShootingCore.hpp, ShootingELRCore.hpp) that are unit-tested in isolation, with

a thin typed-command layer (shoot …) and Lua bindings for the host runtime. A

companion MySQL schema (sql/shooting_schema.sql) logs profiles, the dope diary,

computed solutions, chronograph strings, groups, and truing history — the dataset

that feeds the truing and residual-learning workflow.

References

• R. L. McCoy, *Modern Exterior Ballistics: The Launch and Flight Dynamics of

   Symmetric Projectiles*, 2nd ed., Schiffer, 2012.

• B. Litz, Applied Ballistics for Long Range Shooting, 3rd ed., Applied

   Ballistics LLC, 2015.

• D. Miller, “A New Rule for Estimating Rifling Twist,” Precision Shooting,

   March 2005 (gyroscopic stability formula).

• A. J. Pejsa, Modern Practical Ballistics, 2nd ed., 1991.
• Ballistics standard reference drag functions (G1, G7), derived from the

   Gâvre/Ingalls and BRL/ARDC projectile drag measurements.

1. E. Eötvös, on the latitude/azimuth dependence of apparent vertical

   acceleration for east–west motion (Eötvös effect).

*Prepared as internal technical documentation for the _AugmentedIntelligence

ballistics subsystem. For target-shooting sport and educational use.*

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Filed under: Uncategorized - @ July 18, 2026 6:33 am