Lecture 4 - External Ballistics

Projectile Motion

• Firearm projectiles do not follow true ‘ballistic arcs’ due to various external factors influencing their trajectory.

• The path of the projectile is dictated by:

  • Gravity: The constant force pulling the projectile downwards.

  • Angle of launch (elevation): The angle at which the projectile is initially fired, affecting range and height.

  • Velocity: The initial speed of the projectile, a critical factor in determining its range.

  • Air density, temperature, and humidity: These affect air resistance and thus the projectile's speed and trajectory.

  • Projectile shape (and drag coefficient): The shape influences how much air resistance the projectile encounters.

  • Projectile stability: How well the projectile maintains its orientation during flight.

Ideal vs Real Flight Paths

• Ideal (No Air Resistance):

  • Perfect ballistic arc: A symmetrical parabolic path.

  • Trajectory is even either side of the maximum height: The ascending and descending paths are mirror images.

• Real (With Air Resistance):

  • Second half is more 'truncated': The descending path is shorter and steeper than the ascending path.

  • Trajectory is uneven either side of the maximum height: Air resistance causes the projectile to slow more quickly on the way down.

Trajectory Basics (Absence of Air Resistance)

• Calculations can be conducted using the SUVAT equations, which apply to uniformly accelerated motion.

• For horizontal launch (zero-degree elevation):

  • s = ut + \frac{1}{2}at^2

  • with u = 0, so therefore

  • s = \frac{1}{2}at^2

• In the x-direction:

  • s = ut

Maximum Range (Absence of Air Resistance)

• The equation can only be used when the launch and impact sites are on the same horizontal plane to simplify calculations.

Important Concepts

• Aerodynamic forces and gravity are the two most important factors affecting projectile motion.

• Aerodynamic drag is typically proportional to the square of the velocity, meaning it increases exponentially with speed.

• Other properties contributing to drag:

  • Projectile profile or shape: Streamlined shapes reduce drag.

  • Cross-sectional area of the projectile: Larger areas increase drag.

  • Air density: Higher density increases drag.

• Center of Mass (CoM): The point where the bullet balances its weight (W = mass × gravity).

• Center of Pressure (CoP): The point where the aerodynamic forces act; its position relative to the CoM affects stability.

• For best flight stability, the CoP should be rearward of the CoM but very close to it to prevent tumbling.

• Fin stabilization facilitates this by creating additional aerodynamic forces at the back of the projectile, moving the CoP rearward.

• Normal ‘spitzer’ bullet shapes have the CoP significantly forward of the CoM, requiring spin for stability.

Drag Stabilization

• Achieved by adding fins to the projectile to increase stability.

• Brings the CoP rearward of the CoM, ensuring stable flight.

• Results in a stable flight path without gyroscopic stabilization, useful at lower velocities.

Drag Coefficient (C_d)

• The total drag force experienced by the projectile can be calculated using the following equation:

  • Fd = \frac{1}{2} \rho V^2 A C_d

  • Where:

    • Fd = drag force in N

    • C_d = drag coefficient (no units), a dimensionless quantity

    • V = flow velocity in m/s

    • A = cross-sectional area in m^2

    • \rho = air density @ sea level, approx. 1.2 kg/m^3

Calculating \mathbf{C_d} from Wind Tunnel Data

• Projectile cross-sectional area (A = \pi r^2)

• Flow velocity (V)

• Fluid density \rho

• Drag force (F_d)

• Rearrange the formula to calculate C_d

  • Cd = \frac{2Fd}{\rho V^2 A}

\mathbf{C_d} vs Mach Number (Velocity)

• C_d values vary with flow velocity, typically increasing as the projectile approaches and exceeds the speed of sound.

• C_d increases at the point of going supersonic (past Mach 1) due to the formation of shock waves.

• Mach number defines how many times the speed of sound (c. 343 m/s) the projectile is moving at, important for understanding compressibility effects.

Sectional Density (S)

• Bullets with a high sectional density carry more impact energy at a given range because they maintain velocity better.

• Ballistics definition: Mass of projectile divided by its maximum diameter squared, indicating how well it penetrates air.

Ballistic Coefficient (Cb)

• Measure of the aerodynamic forces exerted on a bullet relative to its mass and cross-sectional area.

• Relates the bullet’s sectional density to its drag coefficient, indicating its ability to overcome air resistance.

• Cb has units of kg/m^2.

Calculating \mathbf{C_b}

• Cb = \frac{m}{Cd \cdot \pi (d/2)^2 \cdot C_g}

  • Where:

    • C_b = Ballistic coefficient

    • m = Mass of the test bullet

    • C_d = Drag coefficient

    • C_g = Drag coefficient of the G1 standard projectile = 0.5191

    • d = Diameter of the test bullet.

    • S = Sectional density of the test bullet

Gyroscopic Stability – Bullet Spin Rate

• Spin rate affects aerodynamic stability, ensuring the bullet flies point-forward.

• Spin = \frac{V_m}{L}

  • Where:

    • V_m = Muzzle velocity in m/s

    • L = Rifling twist rate in meters

The Greenhill Formula – ‘Optimum’ Twist Rate

• L{optimum} = \frac{C \cdot d^2}{L{bullet}}

  • Where:

    • L_{optimum} = Optimum twist rate in meters

    • C = A constant (150 for muzzle velocities 860 m/s)

    • d = Bullet diameter in meters

    • L_{bullet} = Bullet length in meters

Aerodynamic Lift

• Occurs due to the 'boundary layer effect' over a curved surface, causing the bullet to rise or fall slightly.

• Relates to a spinning bullet in a crosswind, where one side experiences higher pressure, leading to lift.

Wind Deflection

• Two main types:

  • Aerodynamic: Caused by the interaction of the bullet's spin with the wind.

  • Windage: Direct deflection due to the wind's force on the bullet.

Wind Deflection Calculations

• Bullet wind deflection due to windage can be calculated using vector addition, considering wind speed and direction.

Causes of Abnormal Flight Characteristics

• Rifling-induced instabilities are caused by a low spin rate, leading to wobbling or tumbling.

• Main causes:

  • Low muzzle velocity: Insufficient spin.

  • High muzzle velocity: Over-stabilization or bullet disintegration.

  • Defective rifling: Inconsistent spin.

Projectile Instability

• Yaw: Lateral movement of the nose of the bullet away from the line of flight, increasing air resistance.

• Precession: Rotation of the bullet around the center of mass, causing a spiraling flight path.

• Nutation: Small circular movement at the bullet tip, a rapid and often unnoticeable oscillation.

Yaw

• Main causes:

  • Poorly cast bullet or bad loading: Imbalance causing initial deviation.

  • Irregular rifling or non-optimal spin rate: Inadequate

Drag Profiles

• Drag profiles describe how the drag coefficient (C_d) changes with Mach number.

• Different standard projectile shapes (G1, G7, etc.) have different drag profiles.

• G1 Profile:

  • Represents a short, flat-based bullet.

  • Generally higher drag compared to more streamlined shapes.

  • Used as a baseline for comparison.

• G7 Profile:

  • Represents a long, boat-tailed bullet.

  • Lower drag coefficient, especially at supersonic speeds.

  • Preferred for long-range shooting due to better ballistic performance.

• Other Profiles:

  • Other profiles exist for various bullet shapes, each with its own drag characteristics.

Factors Affecting Drag Profiles

• Bullet Shape:

  • Streamlined shapes (e.g., boat-tail) reduce drag by minimizing turbulence.

  • Sharp edges or flat bases increase drag.

• Surface Roughness:

  • Rough surfaces increase friction and turbulence, leading to higher drag.

  • Smooth surfaces reduce friction and drag.

• Mach Number:

  • As the bullet approaches the speed of sound (Mach 1), drag increases significantly due to the formation of shock waves.

  • Supersonic drag is typically higher than subsonic drag.

Using Drag Profiles in Ballistic Calculations

• Ballistic solvers use drag profiles to accurately predict bullet trajectory.

• By selecting the appropriate drag profile for a given bullet shape, the solver can account for the changing drag coefficient as the bullet's velocity changes.

• This allows for more precise calculations of bullet drop, wind drift, and time of flight, especially at long ranges.

Importance of Accurate Drag Profiles

• Accurate drag profiles are crucial for long-range shooting.

• Small errors in the drag coefficient can lead to significant deviations in bullet trajectory at long ranges.

• By using accurate drag profiles, shooters can improve their accuracy and consistency.