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Physical measure of overcoming air resistance

In ballistics, the ballistic coefficient (BC, C) of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration: a high number indicates a low negative acceleration—the drag on the body is small in proportion to its mass. BC can be expressed with the units kilograms per square meter (kg/m2) or pounds per square inch (lb/in2) (where 1 lb/in2 corresponds to 703.069581).

In 1537, Niccolò Tartaglia performed test firing to determine the maximum angle and range for a shot. His conclusion was near 45 degrees. He noted that the shot trajectory was continuously curved. In 1636, Galileo Galilei published results in "Dialogues Concerning Two New Sciences". He found that a falling body had a constant acceleration. This allowed Galileo to show that a bullet's trajectory was a curve. In 1718, John Keill challenged the Continental Mathematica, "To find the curve that a projectile may describe in the air, on behalf of the simplest assumption of gravity, and the density of the medium uniform, on the other hand, in the duplicate ratio of the velocity of the resistance". This challenge supposes that air resistance increases exponentially to the velocity of a projectile. Keill gave no solution for his challenge. Johann Bernoulli took up this challenge and soon thereafter solved the problem and air resistance varied as "any power" of velocity; known as the Bernoulli equation. This is the precursor to the concept of the "standard projectile". In 1753, Leonhard Euler showed how theoretical trajectories might be calculated using his method as applied to the Bernoulli equation, but only for resistance varying as the square of the velocity. In 1864, the Electro ballistic chronograph was invented, and by 1867 one electro ballistic chronograph was claimed by its inventor to be able to resolve one ten millionth of a second, but the absolute accuracy is unknown.

Many countries and their militaries carried out test firings from the mid eighteenth century on using large ordnance to determine the drag characteristics of each individual projectile. These individual test firings were logged and reported in extensive ballistics tables. Of the test firing, most notably were: Francis Bashforth at Woolwich Marshes & Shoeburyness, England (1864 1889) with velocities to and M. Krupp (1865–1880) of Friedrich Krupp AG at Meppen, Germany. Friedrich Krupp AG continued these test firings to 1930; to a lesser extent General Nikolai V. Mayevski, then a Colonel (1868–1869) at St. Petersburg, Russia; the Commission d'Experience de Gâvre (1873 to 1889) at Le Gâvre, France with velocities to and The British Royal Artillery (1904–1906). The test projectiles (shot) used, vary from spherical, spheroidal, ogival; being hollow, solid and cored in design with the elongated ogival headed projectiles having 1, 1½, 2 and 3 caliber radii. These projectiles varied in size from, at to at

Many militaries up until the 1860s used calculus to compute the projectile trajectory. The numerical computations necessary to calculate just a single trajectory was lengthy, tedious and done by hand. So, investigations to develop a theoretical drag model began. The investigations led to a major simplification in the experimental treatment of drag. This was the concept of a "standard projectile". The ballistic tables are made up for a factitious projectile being defined as: "a factitious weight and with a specific shape and specific dimensions in a ratio of calibers." This simplifies calculation for the ballistic coefficient of a standard model projectile, which could mathematically move through the standard atmosphere with the same ability as any actual projectile could move through the actual atmosphere. Although Bashforth did not conceive the "restricted zone", he showed mathematically there were 5 restricted zones. Bashforth did not propose a standard projectile, but was well aware of the concept.

In 1872, Mayevski published his report Traité de Balistique Extérieure, which included the Mayevski model. Using his ballistic tables along with Bashforth's tables from the 1870 report, Mayevski created an analytical math formula that calculated the air resistances of a projectile in terms of log A and the value n. Although Mayevski's math used a differing approach than Bashforth, the resulting calculation of air resistance was the same. Mayevski proposed the restricted zone concept and found there to be six restricted zones for projectiles. Circa 1886, Mayevski published the results from a discussion of experiments made by M. Krupp (1880). Though the ogival headed projectiles used varied greatly in caliber, they had essentially the same proportions as the standard projectile, being mostly 3 caliber in length, with an ogive of 2 calibers radius. Giving the standard projectile dimensionally as and
In 1880, Colonel Francesco Siacci published his work "Balistica". Siacci found as did those who came before him that the resistance and density of the air becomes greater and greater as a projectile displaced the air at higher and higher velocities. Siacci's method was for flat fire trajectories with angles of departure of less than 20 degrees. He found that the angle of departure is sufficiently small to allow for air density to remain the same and was able to reduce the ballistics tables to easily tabulated quadrants giving distance, time, inclination and altitude of the projectile. Using Bashforth's k and Mayevski's tables, Siacci created a four zone model. Siacci used Mayevski's standard projectile. From this method and standard projectile, Siacci formulated a shortcut. Siacci wrote that within any restricted zone, C being the same for two or more projectiles, the trajectories differences will be minor. Therefore, C agrees with an average curve, and this average curve applies for all projectiles. Therefore, a single trajectory can be computed for the standard projectile without having to resort to tedious calculus methods, and then a trajectory for any actual bullet with known C can be computed from the standard trajectory with just simple algebra. The 1870 Bashforth tables were to Mayevski, using his tables, supplemented by the Bashforth tables (to 6 restricted zones) and the Krupp tables. Mayevski conceived a 7th restricted zone and extended the Bashforth tables to Mayevski converted Bashforth's data from Imperial units of measure to metric units of measure (now in SI units of measure). In 1884, James Ingalls published his tables in the U. S. Army Artillery Circular M using the Mayevski tables. Ingalls extended Mayevski's ballistics tables to within an 8th restricted zone, but still with the same n value (1.55) as Mayevski's 7th restricted zone. Ingalls, converted Mayevski's results back to Imperial units. The British Royal Artillery results were very similar to those of Mayevski's and extended their tables to within the 8th restricted zone changing the n value from 1.55 to 1.67. These ballistic tables were published in 1909 and almost identical to those of Ingalls. In 1971 the Sierra Bullet company calculated their ballistic tables to 9 restricted zones but only within
The general form for the calculations of trajectory adopted for the G model is the Siacci method. The standard model projectile is a "fictitious projectile" used as the mathematical basis for the calculation of actual projectile's trajectory when an initial velocity is known. The G1 model projectile adopted is in dimensionless measures of 2 caliber radius ogival head and 3.28 caliber in length. By calculation this leaves the body length 1.96 caliber and head, 1.32 caliber long. For the purposes of mathematical convenience for any standard projectile (G) the C is 1.00. Where as the projectile's sectional density (SD) is dimensionless with a mass of 1 divided by the square of the diameter of 1 caliber equaling an SD of 1. Then the standard projectile is assigned a coefficient of form of 1. Following that C, as a general rule, within flat fire trajectory, is carried out to 2 decimal points. C is commonly found within commercial publications to be carried out to 3 decimal points as few sporting, small arms projectiles rise to the level of 1.00 for a ballistic coefficient.

Sporting bullets, with a calibre d ranging from , have C in the range 0.12 lb/in2 to slightly over 1.00 lb/in2 (84 kg/m2 to 703 kg/m2). Those bullets with the higher BCs are the most aerodynamic, and those with low BCs are the least. Very low drag bullets with C ≥ 1.10 lb/in2 (over 773 kg/m2) can be designed and produced on CNC precision lathes out of mono metal rods, but they often have to be fired from custom made full bore rifles with special barrels. Ammunition makers often offer several bullet weights and types for a given cartridge. Heavy for caliber pointed (spitzer) bullets with a boattail design have BCs at the higher end of the normal range, whereas lighter bullets with square tails and blunt noses have lower BCs. The 6 mm and 6.5 mm cartridges are probably the most well known for having high BCs and are often used in long range target matches of – The 6 and 6.5 have relatively light recoil compared to high BC bullets of greater caliber and tend to be shot by the winner in matches where accuracy is key. Examples include the 6mm PPC, 6mm Norma BR, 6×47mm SM, 6.5×55mm Swedish Mauser, 6.5×47mm Lapua, 6.5 Creedmoor, 6.5 Grendel, .260 Remington, and the 6.5 284. In the United States, hunting cartridges such as the .25 06 Remington (a 6.35 mm caliber), the .270 Winchester (a 6.8 mm caliber), and the .284 Winchester (a 7 mm caliber) are used when high BCs and moderate recoil are desired. The .30 06 Springfield and .308 Winchester cartridges also offer several high BC loads, although the bullet weights are on the heavy side for the available case capacity, and thus are velocity limited by the maximum allowable pressure. In the larger caliber category, the .338 Lapua Magnum and the .50 BMG are popular with very high BC bullets for shooting beyond 1,000 meters. Newer chamberings in the larger caliber category are the .375 and .408 Cheyenne Tactical and the .416 Barrett.

For many years, bullet manufacturers were the main source of ballistic coefficients for use in trajectory calculations. However, in the past decade or so, it has been shown that ballistic coefficient measurements by independent parties can often be more accurate than manufacturer specifications. Since ballistic coefficients depend on the specific firearm and other conditions that vary, it is notable that methods have been developed for individual users to measure their own ballistic coefficients.

Satellites in low Earth orbit (LEO) with high ballistic coefficients experience smaller perturbations to their orbits due to atmospheric drag. The ballistic coefficient of an atmospheric reentry vehicle has a significant effect on its behavior. A very high ballistic coefficient vehicle would lose velocity very slowly and would impact the Earth's surface at higher speeds. In contrast, a low ballistic coefficient vehicle would reach subsonic speeds before reaching the ground. Vehicles that carry nuclear weapons launched by an intercontinental ballistic missile (ICBM), by contrast, have a high ballistic coefficient, ranging between 100 and 5000 lb/ft2, enabling a significantly faster descent from space to the surface. This in turn makes the weapon less affected by crosswinds or other weather phenomena, and harder to track, intercept, or otherwise defend against.