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 kilogram-force per square meter (kgf/m2) or pounds per square inch (lb/in2) (where 1 lb/in2 corresponds to ).
Formulas
General The ballistic coefficient provides a value representing how an object's velocity and acceleration through a fluid of a given density will relate to each other; for a given velocity through a given fluid, a higher coefficient means the projectile will be affected less by drag, i.e. it will be accelerated less by the drag force.
where: b,physics is the ballistic coefficient as used in physics and engineering is the mass of the object is the cross sectional area d is the drag coefficient is the density of the object is the characteristic body length such that the volume of the object equals D is the drag force, is the acceleration experienced by the object, fluid is the density of the fluid - e.g. the density of dry air at and is , is the speed of the object relative to the fluid.
Examples
For objects whose volume can be compared based on a known comparison of their projected frontal area, we can compare ballistic coefficients; for example, for the shapes on the right, the two cylinders are of unknown length and the cone is of unknown angle, making it impossible to compare their ballistic coefficients. This comparison table assumes all of the objects are of the same characteristic density, so they will have the same mass at the same volume, i.e. the volume divided by the projected frontal area, , multiplied by some shared density , equals the term in the Ballistic Coefficient formula. Note that "angled cube", in this case, means the flow is perpendicular to an edge, not a corner.
+ Ballistic Coefficient Comparison of Shapes, Reynolds number approximately 104 Shape CD V/A Cb/ Relative A=1 V=1 A=1 V=1 A V Sphere 0.47 Half-Sphere 0.42 Cube 1.05 1 1 Angled Cube 0.80
Ballistics The formula for calculating the ballistic coefficient for small and large arms projectiles only is as follows:
where: Cb,projectile, ballistic coefficient as used in point mass trajectory from the Siacci method (less than 20 degrees). m, mass of bullet , measured cross-sectional diameter of projectile Note that the cross-sectional area from the original formula, expressed in , is i, coefficient of form
The coefficient of form, i, can be derived by 6 methods and applied differently depending on the trajectory models used: G model, Beugless/Coxe, 3 Sky Screen, 4 Sky Screen, target zeroing, and Doppler radar.
Here are several methods to compute i or Cd:
You can assume the projectile is a "standard" ogive, ignoring the various ways an ogive can be specified as well as non-ogive projectiles.Textbook of Small Arms 1909 (1909). Great Britain. War Office H.M. Stationery Office.
where:
or
A drag coefficient can also be calculated mathematically:
where: Cd, drag coefficient. , density of the projectile. v, projectile velocity at range. (pi) = 3.14159... d, measured cross-sectional diameter of projectile
or
From standard physics as applied to "G" models:
where: i, coefficient of form. CG, drag coefficient of 1.00 from any "G" model, reference drawing, projectile.Reference Notes for Use in the Course in Gunnery and Ammunition (1917). Coast Artillery School U.S. Army, p12. ASIN B00E0UERI2 Cp, drag coefficient of the actual test projectile at range.
Commercial use This formula is for calculating the ballistic coefficient within the small arms shooting community, but is redundant with Cb,projectile: Berger Bullets Reloading Manual 1st Edition (2012), Berger Bullets LLC, p814
where: Cb,small-arms, ballistic coefficient SD, sectional density i, coefficient of form (form factorHornady Handbook of Cartridge Reloading: Rifle, Pistol Vol. II (1973); Hornady Manufacturing Company, Fourth Printing July 1978, p505)
History Background 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.Historical Summary
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.Galileo, Galilei, Dialogues Concerning Two New Science, 2010; p. Fourth Day, The Motion of Projectiles, Digireads.com
Circa 1665, Sir Isaac Newton derived the law of air resistance. Newton's experiments on drag were through air and fluids. He showed that drag on shot increases proportionately with the density of the air (or the fluid), cross sectional area, and the square of the speed. Newton's experiments were only at low velocities to about .Bashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; page 1, Cambridge at the University PressWilliam and Robert Cambers; Chamber's Encyclopaedia Volume VIII, 1891; page 438, J.B. Lippincott, USAPejsa, Arthur, New Exact Small Arms Ballistics: The Source Book for Riflemen, 2008; page 29, Kenwood Publishing
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 1742, Benjamin Robins invented the ballistic pendulum. This was a simple mechanical device that could measure a projectile's velocity. Robins reported muzzle velocities ranging from to . In his book published that same year "New Principles of Gunnery", he uses numerical integration from Euler's method and found that air resistance varies as the square of the velocity, but insisted that it changes at the speed of sound.Bashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; pages 4, 6, Cambridge at the University PressPejsa, Arthur, Modern Practical Ballistics 2nd Edition, 1991; page 6, Kenwood Publishing
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.Bashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; page 6, Cambridge at the University Press
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.Bashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; page 13, Cambridge at the University Press
Test firing 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.Bashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; pp. 5–11, Cambridge at the University PressCline, Donna, Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" The Point-Mass Trajectory: The Siacci Method Ballistic Coefficient, 2002; page 39, Lattie Stone Ballistics
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).Bashforth, Francis; Reports on experiments made with the Bashforth chronograph..., 1878-1879; p. Protest, 1, H.M Stationery Office, Harrison & Sons, LondonBashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; pages 13, 135, Cambridge at the University PressIngalls, James M., Exterior Ballistics in the Plan Fire, 1886; pages 18,19, D. Van Nostrand PublisherCline, Donna, Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" The Point-Mass Trajectory: The Siacci Method Ballistic Coefficient, 2002; page 39–40, Lattie Stone Ballistics
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, , 2 and 3 caliber radii. These projectiles varied in size from, at to at Bashforth, Francis; Reports on experiments made with the Bashforth chronograph..., 1878-1879; pages 3–4, H.M Majesty's Stationery Office, Harrison & Sons, LondonBashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; pages 36, 135, Cambridge at the University PressIngalls, James M., Exterior Ballistics in the Plan Fire, 1886; page 19, D. Van Nostrand Publisher
Methods and the standard projectile 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.Ingalls, James M., Exterior Ballistics in the Plan Fire, 1886; page 8, D. Van Nostrand PublisherCoast Artillery School Press, Reference Notes for Use in the Course in Gunnery and Ammunition, 1917; page 12, Coast Artillery School, ASIN:B00E0UERI2
The Bashforth method In 1870, Bashforth publishes a report containing his ballistic tables. Bashforth found that the drag of his test projectiles varied with the square of velocity (v2) from to and with the cube of velocity (v3) from to . As of his 1880 report, he found that drag varied by v6 from to . Bashforth used rifled guns of , , and ; smooth-bore guns of similar caliber for firing spherical shot and howitzers propelled elongated projectiles having an ogival-head of caliber radius.Bashforth, Francis; Reports on experiments made with the Bashforth chronograph..., 1878-1879; page 4, H.M Majesty's Stationery Office, Harrison & Sons, LondonBashforth, Francis, A revised account of the experiments made with the Bashforth chronograph..., 1890; pp. 135–136, Cambridge at the University Press