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Letter - E

Page Updated: 06 March 2003

British Avoirdupois Weights The avoirdupois pound is the pound in general use today. As its name implies, it was intended to be used for weighing heavy goods. This pound is of 7000 grains, and is split into 16 ounces (each, therefore of 437.5 grains). Each ounce is divided into 16 drams (which my calculator makes of 27.34375 grains each - much more fun than metric isn't it?). British Avoirdupois Weight Table 16 drams = 1 ounce 16 ounces = 1 pound 7 pounds = 1 clove 14 pounds = 1 stone 28 pounds = 1 tod 112 pounds = 1 hundredweight 364 pounds = 1 sack 2240 pounds = 1 ton 2 stones = 1 quarter 4 quarters = 1 hundredweight 20 hundredweight = 1 ton   NB: The sack is not in common use. There was a 'Butchers stone' of 8lb until the end of 1939. The Troy pound was of 5760 grains, and was divided into 12 ounces, so a troy pound is lighter than an avoirdupois pound, but a troy ounce (at 480 grains) weighs more than an avoirdupois ounce. The troy pound was declared illegal in 1878, but the troy ounce continues in use today for weighing gold. The troy ounce is split into 480 grains, and you will see 1/2 ounce weights marked both '240 grains' and '0.5oz'. However, the apothecaries system also has an ounce weighing 480 grains, being divided into 8 drams (sometimes spelled drachms) of 60 grains, each dram being split into 3 scruples, of 20 grains. To make things more fun, a 2 dram weight would be marked '3ij' - I think that '3' means 'scruples' (there being 3 to the dram), and the 'ij' being an old-fashioned way of quoting the Roman numeral 'ii'. It doesn't end there - there are 20 penny-weights to the troy ounce, so the 1/2 ounce weight mentioned above could also be marked as '3iv' or '10dwt'.

"This article was originally published on Fr. Frog's Home Page & is used here with his express  written permission."   Fr. Frogs Home Page is at Web URL: http://home.sprynet.com/~frfrog.

A Word to the Wise on Ballistic Coefficient or BC Please Note:  Many manufacturer give rather generous BCs for their bullets because: a) they want to look good--high BCs sell bullets; b) they were derived by visual shape comparison rather than actual firing data; or c) they were derived from short range firings rather than long range firings (which are more difficult to do). You should confirm your calculations by actual firing if you require exact data. Several manufactures have recently "readjusted" some of their BCs to more closely conform to actual firing data. For a more in-depth discussion of ballistic coefficients see the section below.

NOTE: The bore's angle in relation to the line of sight above is exaggerated for clarity.

 

A Brief Discourse on Ballistic Coefficients

This is probably the best article I have read on ballistic coefficients. It was written by Jim Ristow of Recreational Software, Inc. and is reprinted here with his permission. It was designed to encourage a discussion about ballistic coefficients and to explain why good B.C.'s are crucial to getting accurate results from ballistic software. The illustrations were not part of the original article.

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 it's 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 B.C.'s.

 

War Advances Technology

By the middle of the last century (around 1945-1947) 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 remeasure 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 pre-determined velocity zones. The resulting rapid changes to ballistic coefficient do not duplicate "real world" conditions.

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 boat tailed bullet with a G1 based B.C. of .690 may actually have a G7 based B.C. of only .344, since the G7 drag model accurately describes its performance and everyone "knows" that .690 is "better" than .344. However, using the wrong drag model will yield trajectory data that indicates incorrect drop. Fortunately the differences only become important at very long range, but there is a difference. As an example the GI M80 Ball bullet (149 gr FMJ boat tail) has a verified G7 BC of .195. The commercial equivalents of this bullet are listed as having a G1 BC of between .393 and .395. You can see the differences in the plotted trajectories using both the G1 and G7 values and a program that handles both types.

Modern ballistics uses 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 fairly accurately from projectile dimensions and used to define custom drag models for unusual bullet shapes. (See caveat below.)

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 it's weight multiplied by it's frontal area.

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

You can see from the formula that a 1 inch diameter, 1 pound bullet (7,000 gr.) would produce a sectional density of 1. Indeed the standard projectile for all drag models can be viewed as weighing 1 pound and having a 1 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 pre-defined drag model's standard projectile.

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

So What Is A Ballistic Coefficient?

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 1 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.

For Best Accuracy, Calculate Your Own Coefficients!

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 for personal use 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 and even the newest software is not perfect as you can see from the next section.


Some Caveats

We mentioned that CD can be estimated fairly well from bullet dimensions. However, because of the effects of bullet wobble (precession due to rotation), nose tip radius or flatness, nose curvature and boat tail, boundary layer interaction from cannelures and land engraving, etc. (all of which affect the wave drag, base drag and friction drag of the bullet differently) it is really impossible to predict with total accuracy the actual CD vs. Mach number. Also, while a ballistic coefficient can be computed from velocity measurements at two points, differences in bullet wobble diminishes the validity of chronograph testing for BC change over separate series of different muzzle velocities--it needs to be done by separate measurements at different ranges for each shot. Why? Read on.

An elongated bullet, as opposed to a round ball, is inherently unstable aerodynamically. When made stable gyroscopically by spinning, its center-of-gravity will follow the flight path. However, the nose of the bullet stays above the flight path ever so little just because the bullet has a finite length and generates some lift. This causes the bullet to fly at a very small angle of attack with respect to the flight path. The angle of attack produces a small upward cross flow over the nose that results in a small lift force. The lift force normally would cause the nose to rise and the bullet to tumble as the nose rose even more. That is where the spin comes in and causes the rising nose to precess about the bullet axis. When the spin is close to being right for the bullet's length, the precessing is minimized and the bullet "goes to sleep" If it is too slow the bullet will not be as stable as it should. (That is why Jeff Cooper says it's wrong to shoot groups at 100 yards for accuracy testing and suggests 300 yards. If your twist isn't right for the bullet used your group size will be larger at long ranges than would be expected by extrapolation of 100 yard data due to bullet wobble.)

Of course, any other disturbing force such as a side wind gust could cause a difference in bullet nose precession but the effect would be quite small for a properly spin stabilized bullet. Most of the lift force is on the nose of the bullet and is proportional to the square of the bullet velocity as well as the nose shape and length. The new long-nosed bullets for long range match shooting can generate quite a bit more lift occurring farther ahead of the center- of-gravity and can produce a nasty pitch-up moment. That is why they require a faster than normal twist to stabilize them. Pistol bullets, being relatively short and with little taper to the nose, require a slower spin for stability.

Now consider a bullet chronographed at about 3000 f/s muzzle velocity fired from a rifle with say a 10" twist. It is rotating at around 3000 revolutions per second. Let the flight velocity decay to 2000 f/s. Now what is the bullet rotational speed? It doesn't fall off much because the only things slowing it down are inertia and skin friction drag which is pretty low, so the rotational velocity is only slightly slower than 3000 rps. Then chronograph an identical bullet from the same rifle, this time with a muzzle velocity of 2000 f/s. Its rotational velocity will be 2000 rps. Its stability will be different from the bullet fired at 3000 f/s and allowed to slow down to 2000 f/s. They will not have the same drag at 2000 f/s although the bullets are identical. Therefore, two identical bullets fired from the same rifle, will not have the same drag coefficient or ballistic coefficient just because of the way the measurements were taken. There are times when test data does not mean what you think it does. Again, radar range testing is the only way to fly for trustworthy bullet drag data. [I am indebted to Lew Kenner for this lucid description of bullet stability.]

Another factor is that it is not necessarily true that the drag coefficient of a particular bullet is proportional to that of another bullet of the same design across the Mach number range, but this is what a ballistic coefficient assumes.

Something else to worry about is the effect of the bullet tip shape/condition on the ballistic coefficient. Because modern bullet have soft points they are subject to damage and manufacturing tolerances that can alter the BC from bullet to bullet and across otherwise similar bullets.

For truly accurate results, individual bullet characteristics need to be measured on radar ranges as is done by the military--much too expensive a procedure for the commercial bullet industry--and the drag model from those measurements applied only to the particular bullet tested.

The good news is that for normal rifle ranges the drag coefficients and ballistic coefficients can work satisfactorily for most purposes--so let's proceed.

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