2.0 The Ballistic Coefficient

“So, just what is a ballistic coefficient, and what does it do for a bullet’s trajectory?” These are questions we have been asked many, many times, and they are not easy questions to answer. We will try to answer the first question in this subsection and then proceed to the second question in the next subsection. In later subsections we will describe methods used to measure ballistic coefficients, and give some examples of measured BC values for Sierra’s bullets. Henceforth, ballistic coefficient will be abbreviated as BC.

2.1 The Ballistic Coefficient Explained

There are at least three ways to describe the BC. First, it is widely recognized as a figure of merit for a bullet’s ballistic efficiency. That is, if a bullet has a high BC, then it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will “shoot flatter.” But this description is qualitative, rather than quantitative. For example, if we compare two bullets and one has a BC 25% higher than the other, how much is the improvement in bullet ballistic performance? This question can be answered only by calculating the trajectories for the two bullets and then comparing velocity, wind deflection, and drop or bullet path height versus range from the muzzle. So, the figure of merit approach really gives only a qualitative insight into bullet performance, and sometimes this insight is not correct. It often happens that the bullet with the smaller BC is lighter than the bullet with the higher BC. The lighter bullet therefore can be fired at a higher muzzle velocity, and it can then deliver better ballistic performance just because it leaves the muzzle at a higher velocity. We will talk more about this later.

The second way to describe the BC is to use its precise mathematical definition. Mathematically, the BC defined as is the sectional density of the bullet divided by the form factor. This definition emerges from the physics of ballistics and is used in mathematical analysis of bullet trajectories. But in a practical sense, this definition is not satisfactory to most people for at least two reasons. The first is the question of a bullet’s form factor. The form factor is a property of the shape of the bullet design, but it is no easier to explain than the BC. The second reason is that this mathematical definition can lead to an erroneous conclusion. Assume for the moment that the form factor is just a constant property of the bullet design (not always true). The sectional density of a bullet is its weight divided by the square of its diameter. (The square of any number is the number multiplied by itself). So, to get a large BC we need a large sectional density. It appears from the mathematics that a bullet with a very small diameter should have a very large sectional density because its weight is divided by a very small number, and this should give it a very high BC. In other words, this line of reasoning would lead us to expect that small caliber bullets should have very large BC values. But this is not true because when the diameter of the bullet is small, the volume also is small. The weight of the bullet then is small, and the sectional density is necessarily small also. The net result is that small caliber bullets generally have lower BC values than larger caliber bullets.

The third way to describe the ballistic coefficient traces back to the historical development of the science of ballistics in the latter half of the 19th century. This explanation is lengthier, but it provides a better understanding of what the BC is and what its role is in trajectory calculations. The latter half of the 19th century and the early part of the 20th century was a period of very intensive and fruitful development in the science of ballistics. The developments in ballistics were driven by technological advances in guns, projectiles, propellant ignition, and propellants throughout the 19th century, and by warfare, particularly in Europe and America. Warfare was almost an international sport among the kings, emperors, Kaisers and tsars in Europe throughout the 1800’s. The United States experienced the War of 1812, the Mexican War, the Civil War, the Indian wars in the West, and the Spanish-American War within that same century. Governments were eager to fund research, development and manufacturing of improved guns and gunnery, because battles were generally won by the forces that had superior arms.

Percussion ignition was invented in 1807 by the Rev. Alexander Forsythe in Scotland. In 1814, Joshua Shaw, an artist in Philadelphia, invented the percussion cap. In 1842, the U.S. Army adopted the percussion lock for the Model 1842 Springfield Musket, replacing flintlock ignition in earlier shoulder arms. Rifled muskets and handguns began to replace smoothbore military weapons in the mid-1850’s after a French Army officer, Capt. Claude Minie, developed a means to expand a bullet upon firing to cause it to fit the grooves of a rifled barrel. This advancement combined the rapidity and ease of loading of round balls—which had been the standard military projectile for over a century—with the increased range and deadly accuracy of rifled arms. The range and precision of military weapons, for both small arms and artillery, was increasing dramatically.

The period between 1855 and about 1870 witnessed much research and development in breech loading rifles and handguns. The first metallic self-contained, internally primed cartridge (the 22 Short rimfire cartridge) was introduced by Smith & Wesson in 1857 in their Model No. 1 breech-loading revolver. Breech-loading rifles firing self-contained cartridges appeared in the 1860’s, and some were used during the U.S. Civil War (e.g., the Spencer carbine and the Henry rifle). In 1866 in the United States, Hiram Berdan obtained a patent on a primer that was suitable for centerfire cartridges. That same year in England, Col. Edward Boxer patented a full cartridge for the British Snider Enfield rifle, which was a centerfire cartridge utilizing the Boxer primer. (It is interesting to note that later the Berdan primer was widely adopted on the European continent, while the Boxer primer became standard in the United States.) In 1873, the U.S. military adopted the Model 1873 Trapdoor Springfield rifle with the 45-70 centerfire cartridge. In the space of just 31 years the U.S. Army changed from smoothbore muskets with flintlock ignition to rifles with self-contained metallic centerfire cartridges.

Just 11 years later in 1884, a French physicist named Paul Vielle developed the first smokeless propellant that was stable and loadable for military purposes. Earlier powder developments had led up to Vielle’s discovery, but they were useful only for sporting purposes. The French Army loaded Vielle’s smokeless propellant in the 8mm Lebel cartridge for the Model 1886 Lebel rifle, the very first military rifle firing a smokeless propellant cartridge.

Smokeless propellant was quickly adopted by other nations, including the U.S., and caused significant advancements in bullet performance and design. Muzzle velocity in military rifles, which was less than 1400 fps in the 45-70 and most other black powder cartridges, increased to more than 2000 fps in the earliest smokeless propellant cartridges. This led to the development of jacketed bullets of smaller caliber and lighter weights, i.e., 7mm, 30, and 8mm calibers, which could be fired at even higher velocities and not deposit lead in the barrels at those velocities. Before the end of the 19th century, pointed bullets and boat-tail bullets were also developed to significantly improve bullet ballistic performance.

With all these developments in guns and ammunition, the need to understand the ballistics of projectiles became more acute. It was no longer sufficient to target a gun by hit-and-miss methods. Of course, graduated sights had existed on both smoothbore and rifled muskets for many years, but the elevation marks on the sights had been determined by firing tests of these weapons with a specific projectile at a specific muzzle velocity, at a specific altitude, and with a specific set of weather conditions. As warfare grew in intensity and mobility, it became vitally necessary to understand the physics of bullet motion. In other words, it was necessary to find a way to calculate bullet trajectories as well as the changes in those trajectories caused by changes in bullets, muzzle velocities and firing conditions.

An immense problem thwarted this objective for many years. This problem was understanding the physics and mathematically describing the aerodynamic drag force on a projectile. The invention of the ballistic pendulum by the English ballistician Benjamin Robins in 1740 had led to the astounding discovery (at that time) that the drag force on a bullet was many times more powerful than the force due to gravity, and that it changed markedly with bullet velocity. That event started a chain of firing tests, instrumentation developments, and theoretical investigations that lasted at least 200 years. Progress was slow because aerodynamic drag is a very complex physical process, and mathematics had to be developed to make accurate computation of trajectories possible long before the age of computers.

An early observation was that the drag force was different on every type of projectile, so that measurements of drag deceleration seemed to be necessary on each type of projectile over the full velocity range between muzzle velocity and impact velocity. However, around 1850 Francis Bashforth in England proposed a practical idea that greatly simplified things and is used in the present day. He proposed a model bullet, or “standard” bullet, on which comprehensive measurements of drag deceleration versus velocity could be made. Then, for other bullets this “standard” drag deceleration could be scaled by some means, so that exhaustive drag measurements could be avoided for those bullets.

Bashforth could not have known how successful his suggestion would be. Ballisticians and physicists were working intensively to mathematically describe the aerodynamic drag force and derive the equations of motion of bullet flight. They had recognized in the equations of motion a theoretical scale factor for aerodynamic drag that would adjust the standard drag model to fit a nonstandard bullet. This scale factor turned out to be the form factor of the nonstandard bullet divided by the sectional density, that is, the reciprocal of the BC. The form factor was a number that accounted for the different shape of the nonstandard bullet compared to the standard bullet.

Bashforth’s suggested standard bullet had a weight of 1.0 pound, a caliber of 1.0 inch, and a point with a 1.5 caliber ogive. Firing tests on projectiles of approximately this shape and weight were conducted in England and Russia between about 1865 and 1880. However, the definitive drag deceleration tests were performed by Krupp at their test range in Meppen, Germany, between 1875 and 1881. In 1883 Col. (later General) Mayevski in Russia formulated a mathematical representation of the drag force for the standard bullet. In the 1880’s, an Italian Army team led by Col. F. Siacci formulated an analytical approach and found analytical closed form solutions to the equations of motion of bullet flight for level-fire trajectories. This meant that trajectory calculations for shoulder arms could be performed algebraically, rather than by the more tedious methods of calculus. The Siacci team’s results also showed that not only could the standard drag deceleration be scaled by using the BC of the nonstandard bullet, but also the standard trajectory computed for the standard bullet could be scaled by the same factor to compute an actual trajectory for the nonstandard bullet. This was a very important breakthrough that greatly reduced the amount of work in trajectory computations. The Siacci approach was adopted by Col. James M. Ingalls of the U.S. Army Artillery. His team produced the Ingalls Tables, first published in 1900, which in turn became the standard for small arms ballistics used by the U.S. Army in World War I.

So, the ballistic coefficient actually is a scale factor. The BC scales the standard drag deceleration of the standard bullet to fit a nonstandard bullet. However, the BC works in a reciprocal manner. That is, the higher the BC of a nonstandard bullet, the lower the drag is compared to the standard bullet. This is alright, because it means that the higher the BC of a bullet, the better will be its ballistic performance. The physical units of the BC are pounds per square inch (lb/in2). The BC value for the standard bullet then is 1.0 (weight 1.0 lb, diameter 1.0 inch, and form factor 1.0 by definition for the standard bullet). Ballistic coefficients of most sporting and target bullets have values less than 1.0, and generally BC values increase as caliber increases. A bullet can have a BC higher than 1.0. For example, some heavy 50 caliber bullets have BC values greater than 1.0.

Military agencies in different nations developed many standard bullets over the years. This was done because of fundamentally different shapes in military projectiles, such as sharper points and boat tails. The purpose was to establish better standards for these classes of bullet shapes. In recent years, however, this practice has largely been abandoned in the military. With modern instrumentation and computers, it has become possible to measure the drag deceleration of every individual projectile type used by the military. Thus, there is no longer a need for a standard projectile for military applications. Or, we might say that every type of military projectile is its own standard.

This is not true for commercial bullets, however. Ballistic coefficients are used for all commercial bullets for sporting and target-shooting purposes, mainly because the BC of each is relatively easy and inexpensive to measure, compared to measuring the drag deceleration. The standard projectile for commercial bullets is still nearly identical to Bashforth’s standard bullet. The standard drag model, also called the standard drag function, for this projectile is known as G1. The BC values quoted by all producers of commercial bullets are referenced to G1. It is important to note that BC values quoted by commercial producers can not be used with any drag model other than G1. It is possible to find other standard drag models by looking up historical military ballistics data. But, if a standard drag model other than G1 is used, the BC values of bullets must be measured with reference to that drag model in order to calculate accurate trajectories. The values are likely to be very different from the values referenced to G1.