Ballistic Coefficients Explained

by Jim Ristow

A Little History

In 1881 Krupp of Germany first accurately quantified the air drag influence on bullet travel by test firing large flat-based, blunt-nosed bullets. Within a few years Mayevski had devised a mathematical model to forecast the trajectory of a bullet and then Ingalls published his famous tables using Mayevski's formulas and the Krupp data. In those days most bullet shapes were similar and airplanes or missiles did not exist. Ingalls defined the Ballistic Coefficient (B.C.) of a bullet as its ability to overcome air resistance in flight indexed to Krupp's standard reference projectile. The work of Ingalls & Mayevski has been refined many times but it is still the foundation of small arms exterior ballistics including a reliance on BC's.

By the middle of this century rifle bullets had become more aerodynamic and there were better ways to measure air drag. After WWII the U.S. Army's Ballistic Research Lab ( BRL ) conducted experiments at their facility in Aberdeen , MD to measure the drag caused by air resistance on different bullet shapes. They discovered air drag on bullets increases substantially more just above the speed of sound than previously understood and that different shapes had different velocity erosion due to air drag. In 1965 Winchester-Western published several bullet drag functions based on this early BRL research. The so-called "G" functions for various shapes included an improved Ingalls model, designated "G1". Even though the BRL had demonstrated modern bullets would not parallel the flight of the "G1" standard projectile, the "G1" drag model was adopted by the shooting industry and is still used to generate most trajectory data and B.C.'s. Amazingly, the "G1" standard projectile is close to the shape of the old blunt-nosed, flat-based Krupp artillery round of 1881!

The firearms industry has developed myriad ways to compensate for this problem. Most bullet manufacturers properly measure velocity erosion then publish B.C.'s using an "average" of the calculated G1 based B.C.'s for "normal" velocities. In other words, the only spot on the G1 curve where the model is correct is at the so-called "normal" or average velocity. These B.C.'s are off slightly at other velocities unless the bullet has the same shape, and therefore the same drag as the standard G1 projectile.

Some ballistic programs adjust the B.C. for velocities above the speed of sound; others use several B.C.'s at different velocities in an effort to correct the model. While these approaches mitigate some of the problem, B.C.'s based on G1 still cannot be correct unless the bullet is of the same shape as the standard projectile. Also, the change to air drag, as a function of velocity does not happen abruptly. Drag change is continuous with only small variation immediately above or below any point along the trajectory. Programs that translate the Ingalls tables directly to computer or use multiple B.C.'s can produce velocity discontinuities when drag values change abruptly at predetermined velocity zones. The resulting rapid changes to velocity do not duplicate "real world" trajectories.

The Solution

Shooting software is finally appearing based on methods used in aerospace with drag models for different bullet shapes. Results are superior to traditional "G1 fits everything" thinking, but now shooters must learn B.C.'s are different for each model.

This is a scary proposition for most bullet companies who know many shooters pick bullets based only on their B.C.'s. For example, A VLD bullet with a 61 based B.C. of .690 will have a G7 based B.C. of only .344, even though it is a more aerodynamic shape with less drag. Modern ballistics use the coefficient of drag (C.D.) and speed of sound rather than traditional Ingalls/Mayevski/Sciacci s, t, a & i functions. This avoids velocity discontinuities and when combined with a proper drag model is far more accurate to distances beyond 1000 yards. A by-product of modern ballistics is that the C.D. can be estimated from projectile dimensions and used to define custom drag models for unusual bullet shapes.

What is a BC?

The Coefficient of Drag for a bullet is simply an aerodynamic factor that relates velocity erosion due to air drag to air density, cross-sectional area, velocity and mass. A simpler way to view C.D.'s are as the "generic indicator" of drag for any bullet of a particular shape. Sectional Density is then used to relate these "generic" drag coefficients to bullet size. The "Sectional Density" of a bullet is simply its weight multiplied by its frontal area.

Sectional Density = (Wt. in Grains/7,000) x (Dia.* Dia.)

You can see from the formula that a one inch diameter, one pound bullet (7,000 gr.) would produce a sectional density of one. Indeed the standard projectile for all drag models can be viewed as weighing one pound with a one-inch diameter.
Another term occasionally found in load manuals is a bullet's "Form Factor". The form factor is simply the C.D. of a bullet divided by the C.D. of a predefined drag model's standard projectile.

Form Factor = (C.D. of any bullet) I (C.D. of the Defined 'G' Model Std. Bullet).

Ballistic Coefficients are then just the ratio of velocity retardation due to air drag (or C.D.) for a particular bullet to that of its larger 'G' Model standard bullet. To relate the size of the bullet to that of the standard projectile we simply divide the bullet's sectional density by it's form factor.

Ballistic Coefficient = (Bullet Sectional Density) / (Bullet Form Factor)

From these short formulae it is evident that a bullet with the same shape as the 'G' standard bullet, weighing 1 lb. and one inch in diameter will have a B.C. of 1.000. If the bullet is the same shape, but smaller, it will have an identical C.D., but a form factor of 1.000 and a B.C. equal to it's sectional density.

Here are some current drag models used in small arms ballistics:

• G1.1-Standard model, Flat Based with two caliber (blunt) nose ogive.
• G5.1-For Moderate (low base) Boat Tails - 7 degree 30' Tail Taper with 6.19 caliber tangent nose ogive.
• G6.1-For flat based "Spire Point" type bullets - 6.09 caliber secant nose ogive.
• G7.1-For "VLD" type Boat Tails - long 7 degree 30' Tail Taper with 10 caliber tangent nose ogive.
• GS - For round ball - Based on measured 9/16" spherical projectiles as measured by the BRL .
• RA4 - For 22 Long Rifle, identical to 61 below 1400 fps.
• GL - Traditional model used for blunt nosed exposed lead bullets, identical to 61 below 1400 fps.
• GI - Converted from the original Ingalls tables.
  
  For Best Accuracy Calculate Your Own BCs
  
  Accurate B.C.'s are crucial to getting good data from your exterior ballistics software. A good ballistic program should be able to use two velocities and the distance between them to calculate an exact ballistic coefficient for any of the common drag models. This method of calculating a B.C. is preferred and can be used to duplicate published velocity tables for a bullet when the coefficient is unknown or to more accurately model trajectories achieved from your own firearm. A lot has changed in shooting software. If your software is more than two years old, chances are it does not employ the latest modeling techniques or calculate B.C.'s. You can find both PC & Macintosh software that use the latest exterior ballistic modeling techniques at commercial sites on the Internet.