Lesson 2: Drag Forces and Ballistic Coefficients

Introduction

Air resistance is the dominant force affecting bullet trajectory. At typical rifle velocities, drag forces can be 100+ times stronger than gravity. Understanding drag is essential for accurate ballistic predictions.

The Drag Equation

The aerodynamic drag force on a projectile is given by:

$$F_d = \frac{1}{2} \rho v^2 C_d A$$ Where: $F_d$ = drag force (N) $\rho$ = air density (kg/m³) $v$ = velocity (m/s) $C_d$ = drag coefficient (dimensionless) $A$ = cross-sectional area (m²)

Key Insight: Drag force increases with the square of velocity. Doubling velocity quadruples drag!

The Drag Coefficient ($C_d$)

The drag coefficient varies with velocity, particularly around the speed of sound. We identify several velocity regimes:

Regime	Mach Number	Characteristics
Subsonic	M < 0.8	Relatively stable $C_d$
Transonic	0.8 < M < 1.2	Rapid $C_d$ increase
Supersonic	M > 1.2	Gradually decreasing $C_d$

$$M = \frac{v}{c}$$ Mach number, where $c$ is the speed of sound (~340 m/s at sea level)

Standard Drag Functions

Instead of measuring $C_d$ for every projectile, we use standard drag functions based on projectile shape:

Common Standard Models:

G1: Flat-base projectile with 2 caliber radius ogive nose
G7: Long, boat-tail projectile (better for modern rifle bullets)
G8: Flat-base projectile with 10 caliber radius ogive nose

Important: G7 is generally more accurate for modern long-range bullets with boat-tails, while G1 is traditional but less accurate for these projectiles.

The Ballistic Coefficient (BC)

The ballistic coefficient relates a projectile to a standard drag model:

$$BC = \frac{m}{i \cdot d^2} = \frac{SD}{i}$$ Where: $m$ = mass (kg or lb) $i$ = form factor (dimensionless) $d$ = diameter (m or in) $SD$ = sectional density

In practice, BC is often expressed as:

$$BC = \frac{m}{C_d \cdot A}$$ Higher BC means less drag relative to mass

Drag Acceleration

The deceleration due to drag is:

a_{drag} = -\frac{F_d}{m} = -\frac{\rho v^2 C_d A}{2m}

Using the ballistic coefficient:

$a_{drag} = -\frac{\rho v^2}{2 \cdot BC} \cdot C_{d,std}(M)$$ Where $C_{d,std}(M)$ is the standard drag function value at Mach $M

Velocity Decay

For subsonic velocities with constant $C_d$, velocity decay can be approximated:

$$v(x) = \frac{v_0}{1 + \frac{\rho C_d A}{2m} \cdot x}$$ Simplified velocity as function of distance

Practical Example: BC Comparison

Compare two .308 caliber bullets:

Bullet	Weight	G7 BC	Velocity at 1000 yards	Drop at 1000 yards
Sierra MatchKing	175 gr	0.243	1113 ft/s	-412 inches
Berger Hybrid	185 gr	0.283	1287 ft/s	-367 inches

Both bullets leave the muzzle at 2,650 ft/s, the same speed so that only the BC differs, with a 100-yard zero, a 1.5-inch sight height, a standard sea-level atmosphere (59 °F, 29.92 inHg) and no wind. Calculated with the open-source Ballistics Engine, v0.49.0.

The higher BC bullet retains more velocity and has less drop!

BC Variation with Velocity

Real-world BC is not constant—it varies with velocity:

$$BC(v) = BC_{ref} \cdot f(v)$$ Where $f(v)$ is a velocity-dependent correction factor

Advanced Concept: Modern ballistic solvers use "banded" BCs—different BC values for different velocity ranges—to improve accuracy.

Retardation Function

The retardation (deceleration) can be expressed as:

$$R(v) = \frac{F_d(v)}{m} = K \cdot G(v)$$ Where $K = \frac{\rho_0 \cdot \pi d^2}{8m \cdot BC}$ and $G(v)$ is the drag function

Computing Trajectory with Drag

With drag included, the equations of motion become:

$$\frac{dv_x}{dt} = -\frac{F_d}{m} \cdot \frac{v_x}{v}$$ $$\frac{dv_y}{dt} = -g - \frac{F_d}{m} \cdot \frac{v_y}{v}$$ $$\frac{dv_z}{dt} = -\frac{F_d}{m} \cdot \frac{v_z}{v}$$ Where $v = \sqrt{v_x^2 + v_y^2 + v_z^2}$ is total velocity

Note: These equations cannot be solved analytically—numerical integration is required (covered in Lesson 5).

Summary

Drag is the primary force affecting bullet trajectory, far exceeding gravity's effect. The ballistic coefficient provides a practical way to characterize a bullet's drag properties relative to standard models. Understanding BC and drag functions is essential for accurate long-range shooting predictions.

In the next lesson, we'll explore how environmental conditions affect air density and thus drag forces.

← Lesson 1: Fundamentals Lesson 3: Environmental Effects →