UNDERSTANDING THE CONCEPT OF BALLISTIC COEFFICEITNT

 

Understanding the Concept of Ballistic Co-efficient

Ballistic Coefficient (BC) is a critical concept in ballistics, particularly in understanding how projectiles behave when they travel through the air. It is a measure of projectiles ability to overcome air resistance in flight and less drop due to gravity. The higher ballistic coefficient, the more streamlined and efficient the projectiles are, meaning it will retain more of its velocity and energy over a given distance.

A.    Definition of the Ballistic Co-efficient: The Ballistic coefficient is a numerical value that describes a projectiles ability to overcome air resistance during flight. It is a measure of how well a projectile, such as a bullet, can maintain its velocity and trajectory when moving through the air. The higher the ballistic coefficient, the more efficiently the projectile cuts through the air, retaining speed and energy over longer distances.

Mathematically, it is the ratio of a projectile’s sectional density (its mass relative to its cross-sectional area) to its form factor (a measure of its shapes aerodynamic efficiency).

 

The ballistic coefficient is typically expressed as a dimensionless number, though it can have units depending on the form used G1 and G7, and Higher numbers indicate better ballistic performance.

G Standards of Projectiles:

G1 (flat base with blunt nose ogive)

G2 (Aberdeen J)

G5 (short 7.5° boat tail, 6.19 calibers long tangent ogive)

G6 (flat base, 6 calibers long secant ogive)

G7 (long 7.5° boat tail, 10 calibers tangent ogive)

G8 (flat base, 10 calibers long secant ogive)

GL (blunt lead nose)

The G1 shape looks like your standard handgun round’s FMJ. he G7 shape more accurately represents modern rifle bullets – especially ones that feature a boat tail (aka tapered base).

Typically,

BC ranges from 0.1 to 0.2 for piston bullets.

For Rifle Bullets 0.2 to 0.4 (Hunting Bullets) and streamlined bullets, like match grade or long-range bullets, can have BC ranging from 0.4 to 0.7 or even higher.

For High-performance long-range bullets have BC between 0.6 and 1.0 or even higher in some specialized cases.

For Extreme long-range bullets such as some cutting-edge designs like those use in precision shooting can achieve BC above 1.0

  Formula For Calculating the Ballistic co-efficient:

The formula for calculating the Ballistic Coefficient of a bullet is:

Where:

M – is the mass of the bullet in pounds. (To convert grains to pounds divide by 7,000)

A – is the cross-sectional area of the projectile.

I – is the form factor, which represents the bullets shape and aerodynamic efficiency relative to a standard model such as the G1 or G7 standard.

 

The BC can also be expressed using sectional density (SD), which is the mass divided by the cross-sectional area:

 

This formula indicates that a higher mass, smaller cross-sectional area, or a more aerodynamic shape (lower form factor) will result in a higher ballistic coefficient.

 Another way to calculating

BC=m/(d²((2/n)√((4n-1)/n))
(m, mass of bullet; d, diameter of bullet; n, number of calibers of the projectile’s ogive)

·        The ogive number n affects the calculation of the BC by influencing the term that accounts for the bullets aerodynamic shape.

·        And more streamlined ogive higher n typically leads to higher BC, meaning the bullet is more efficient at overcoming air resistance.

 

Understanding Ogive and Ogive Number:

 

·        Ogive: The ogive is the aerodynamic shape of the bullets nose, which affects how the bullet cuts through the air. Different bullets have different ogive shapes, which influence their ballistic performance.

·        Ogive Number (n): The ogive number (n) in the BC formula is a measure of how pointed or streamlined the ogive is. It represents the ratio of the bullet’s overall length (or nose length) to its diameter.

A higher ogive number indicates a more pointed or elongated ogive, which generally reduces air resistance.

Calculating Ogive Number:

Were,

R is the radius of the curve forming the ogive.

D is the diameter of the bullet.

 

Steps for calculating:

Measure the Bullet Diameter (D): This is typically the Caliber of the bullet.

Determine the radius of the ogive (R): This is the radius of the arc that forms the bullet nose.

Calculate (n): Using the formula

Interpretation:

n – Greater than 1: A more pointed or elongated nose (secant ogive)

n – less then 1: A blunter or shorter nose (tangent ogive)

 

Practical Application:

·        Tangent Ogive: If the ogive radius (R) is equal to the bullet diameter (D), the (n=2). This is a common design for standard bullets, providing a good balance between air resistance and manufacturing simplicity.

·        Secant Ogive: If (R) is larger than (D), (n) will be greater than 2, leading to a more streamlined bullet with potentially higher BC.

A lower ogive number corresponds to a blunter or shorter ogive.

C.    Formula for finding cross-sectional area:

The cross-sectional area of a boat tail projectile is the area of the projectiles profile as viewed from the front or rear. For boat tail design, this area is smaller at the base that is tail, than at the main body, which helps reduce air drag and improve aerodynamic efficiency. Essentially it is the size of the shadow the projectile would cast if light were shined directly as tis nose or tail.

Or

The cross-sectional area of a boat tail bullet is the size of its front, or base, as seen when looking straight at it. For boat tail the cross-sectional area is increasing the base up to end boat tail and constant up to the end of the body that is the starting position of cone shape after it will be reducing gradually up to the end of the nose. If you slice the bullet horizontally you might see the horizontal cross section and if you slice vertically, you might see the vertical cross section.

For a boat tail or streamlined bullet, the cross-sectional area (A) used in calculating the Ballistic Co-efficient is typically taken at the base of the bullet, where the diameter is most relevant. Despite the boat tail or streamlined shape, the area calculation is still based on the diameter at this base.

 

 

The formula to calculate the cross-sectional area of the bullet is:

Where:

A – is the cross-sectional area at the base of the bullet.

Pi - is approximately 3.14159.

d - is the diameter of the bullet at the base.

 

In practice, even for boat tail bullets or streamlined bullets, the base diameter is used to approximate the cross-sectional area for BC calculations. This simplification is used because precise shape variations are complex to model but have minimal impact on the basic BC calculation.

D.    Formula For Calculating the Form Factor:

If you need to find the form factor (I) without knowing the ballistic coefficient (BC) directly but have information about the projectiles drag coefficient (Cd), reference area (A), and mass (m), you can use the following relationship.

 

Were,

Cd - is the drag coefficient

A – is the reference area of the projectile

M – is the mass of the projectile

 

This formula provides an estimate of the form factor based on aerodynamic properties rather than the ballistic coefficient directly.

 

E.     Formula For Calculating Drag Coefficient:

To calculate the drag coefficient (Cd) for use in determining the from factor (I) in the ballistic coefficient calculation, you can use the following formula:

 

Were,

FD – is the drag force experienced by the projectiles

Rho – is the air density

V – is the velocity of the projectile

A – is the reference area (usually the cross-sectional area of the projectile perpendicular to the flow.

 

F.     Formula For Calculating Air Density:

The air density Rho can be calculated using the Ideal Law. The formula is:

 

Were,

P - is the atmospheric pressure

R - is the specific gas constant for dry air (approximately 287.05)

T - is the absolute temperature in kelvin

 

For practical purpose, atmospheric pressure and temperature are often measured directly or obtained from weather data. The formula assumes that the air behaves ideally, which is a good approximation for many purposes.

 

G.    Formula For Calculating Atmospheric Pressure:

Atmospheric air pressure can be calculated using the barometric formula, which relates pressure to altitude. The most common form of this formula is:

 

Were,

P – is the pressure at altitude (h)

P₀ – is the standard atmospheric pressure at sea level approximately 101325

L – is the temperature lapse rate approximately 0.0065

h – is the altitude above the sea level (in meters)

T₀ – is the standard temperature at sea level approximately 288.15

g -is the acceleration due to gravity approximately 9.80655

M -is the molar mass of Earth’s air approximately 0.0289644

R- is the universal gas constant approximately 8.31447

 

H.    Formula For Calculating Drag Force:

·        Using Experimental Data: If you have experimental data, you can use the following method:

·        Measure the Deceleration: Determine the deceleration (a) of the projectile from its velocity data over time

·        Calculate the Drag Force: Use Newtons second law to calculate the drag force:

FD = m x a

FD – is the drag force

M – is the mass of the projectile

A – is the deceleration

·        Using Terminal Velocity Data:

If you know the terminal velocity, Vt of the projectile, you can estimate the drag force at that velocity. At terminal velocity, the drag force equals the gravitational force.

 FD = m.g

 G – is the acceleration due to gravity approximately 9.81

·        Using Simplified Models:

If you have a standard shape or if you can assume a typical drag coefficient, you can use known values for specific shapes. For example:

1.      For a sphere, CD is of the around 0.47 in many conditions

2.      For a streamlined body CD might be around 0.1 to 0.3

Estimate the FD using:

And use typical CD values to approximate the drag force.

Or

If possible, employ computational fluid dynamics CFD software to simulate the drag force and estimate CD based on the shape and flow conditions. This method provides a more accurate estimation if direct measurement is not feasible.

 

I.      Formula For Calculating Declaration:

The term deceleration of a projectile refers to the rate at which the velocity of the projectile decreases over time due to resistive forces, such as drag. It is the negative acceleration experienced by the projectiles as it moves through a medium like air.

 How to Calculate Deceleration:

1.      Measure Velocity Changes: Determine the projectiles velocity at different points in time.

2.      Calculate Deceleration: To calculate deceleration, you use the formula mentioned below:

a – is the acceleration which is a negative acceleration

Delta V – is the change in velocity VF initial – VI Finial.

Delta t -is the time interval over which the velocity change occurs.

 

Step to Calculate Deceleration:

·        Determine Initial Velocity (VI): Measure or obtain the initial velocity

·        Determine the Final Velocity (VF): Measure or obtain the final velocity after a time of interval.

·        Calculate the Change in Velocity: VF-VI

·        Measure the Time Interval: Measure or obtain the time over which the change in velocity occurs. And calculate the deceleration by using the above formula.

·        Formula For Calculating Time Interval: 

 

J.      Factors Affecting the BC:

The ballistic coefficient of a projectile measure how well it resists air drag relative to its mass and cross-sectional area. A higher BC means the projectile will retain velocity better, be less affected by wind, and have a better flatter trajectory.

·        Shape: Sleeker, more aerodynamic shapes (like a spitzer or boat tail) have higher BC

·        Mass: Heavier projectiles typically have higher BC

·        Diameter: Smaller diameter for the same mass usually increases BC

·        Material Density: Denser material such as lead vs copper can improve the BC if the shape is optimized.

·        Longer and Sleeker Projectiles: With a pointed tip and a boat tail base tend to have higher BC

·        Heavier: Projectiles with the same shape will also have a higher BC.

·        Smaller Diameter: Relative to the weight can increase BC, as it reduces air resistance.