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In [[ballistics]], the '''ballistic coefficient''' ('''BC''') of a body is a measure of its ability to overcome air resistance in flight.<ref>[http://arxiv.org/ftp/arxiv/papers/0705/0705.0389.pdf The Truth About Ballistic Coefficients]</ref>  It is [[inversely proportional]] to the negative acceleration &mdash; a high number indicates a low negative acceleration. This is roughly the same as saying that the projectile in question possesses low drag, although some meaning is lost in the generalization. BC is a function of mass, diameter, and [[drag coefficient]].
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==Formulae==
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:<math>BC_{Physics} = \frac{M}{C_d \cdot A} = \frac{\rho \cdot l}{C_d}</math>


Where:
'''MathML'''
*''BC<sub>Physics</sub>'' = ballistic coefficient as used in physics and engineering
:<math forcemathmode="mathml">E=mc^2</math>
*''M'' = mass
*''A'' = cross-sectional area
*''C<sub>d</sub>'' = [[drag coefficient]]
*''<math>\rho</math> (rho)'' = average [[density]]
*''l'' = body length


===Ballistics===
<!--'''PNG''' (currently default in production)
The formula for calculating the ballistic coefficient for small and large arms projectiles ''only'' is as follows:
:<math forcemathmode="png">E=mc^2</math>
:<math>BC_{Projectile} = \frac{m}{ d^2 \cdot i}</math> <ref>{{cite book | last = Moss, Leeming and Farrar| title = Brassey's Land Warfare Series: Military Ballistics | publisher = Royal Military College of Science, Shrivenham, UK | year = 1995 |page=86 | isbn = 978-1857530841 }}</ref>


Whereas:
'''source'''
*''BC<sub>Projectile</sub>'' = ballistic coefficient as used in point mass trajectory from the Siacci method (less than 20 degrees).<ref>{{cite book | last = Cline | first = Donna | title = Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" | publisher = Lattie Stone Ballistics | year = 2002 | page=39 }}</ref>
:<math forcemathmode="source">E=mc^2</math> -->
*''m'' = mass of bullet in '''kg''' or '''lb'''
*''d'' = measured cross section (diameter) of projectile in '''m''' or '''in'''
*''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, Bugless/Coxe; 3 Sky Screen; 4 Sky Screen; Target Zeroing; Doppler radar.<ref>{{cite book | last = Cline | first = Donna | title = Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" | publisher = Lattie Stone Ballistics | year = 2002 | pages=43–48 }}</ref><ref>{{cite book | last = Rinker | first = Robert A. | title = Understanding Firearm Ballistics; 3rd Edition | publisher = Mulberry House Publishing | year = 1999 |page=176 | isbn = 978-0964559844}}</ref>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


Here are several methods to compute '''''i''''' or '''C<sub>d</sub>''':
==Demos==
:<math>i=\frac{2}{n} \cdot \sqrt{\frac{4n-1}{n}}</math> <ref>{{cite book | last = Cline | first = Donna | title = Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" | publisher = Lattie Stone Ballistics | year = 2002 | page=44 }}</ref><ref>{{cite book | last = Rinker | first = Robert A.  | title = Understanding Firearm Ballistics; 3rd Edition | publisher = Mulberry House Publishing | year = 1999 |page=176 | isbn = 978-0964559844}}</ref><ref>''Textbook of Small Arms 1909'' (1909). Great Britain. War Office H.M. Stationery Office. ISBN 978-1847914217</ref>


Whereas:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
*''i'' = Coefficient of form.  
*''n'' = number of calibers of the projectile's ogive.


:Where ''n''  is unknown:


::<math>n = \frac{(4 \cdot l^2 + 1)}{4}</math> <ref>{{cite book | last = Cline | first = Donna | title = Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere" | publisher = Lattie Stone Ballistics | year = 2002 | page=44 }}</ref>
* accessibility:
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


::Whereas:
==Test pages ==
:*''n'' = number of calibers of the projectile's ogive.
:*''l'' = length of the head (ogive) in number of calibers.


'''or'''
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
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A drag coefficient can also be calculated mathematically:
*[[Inputtypes|Inputtypes (private Wikis only)]]
:<math> C_{d}= \frac{8}{ \rho \cdot v^2 \cdot \pi \cdot d^2}</math> <ref>{{cite book | last = Moss | first =  Leeming and Farrar| title = Brassey's Land Warfare Series: Military Ballistics | publisher = Royal Military College of Science, Shrivenham, UK | location = Reading | year = 1995 |page=79 | isbn = 978-1857530841 }}</ref>
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
Whereas:
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*''C<sub>d</sub>'' = drag coefficient.
*''<math>\rho</math> (rho)'' = density of the projectile.
*''v'' = projectile velocity at range.
* ''[[π]] (pi) ≈ 3.14159
*''d'' = measured cross section (diameter) of projectile in '''m''' or '''in'''
 
'''or'''
 
From standard physics as applied to “G” models:
:<math>i = \frac{C_{G}}{C_{p}}</math> <ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
Whereas:
*''i'' = Coefficient of form.
*''C<sub>G</sub>'' =  drag coefficient of 1.00 from any “G” model, reference drawing, projectile.<ref>''Reference Notes for Use in the Course in Gunnery and Ammunition'' (1917). Coast Artillery School U.S. Army, p12. ASIN B00E0UERI2</ref>
*''C<sub>p</sub>'' = drag coefficient of the actual test projectile at range.
 
===Commercial Use===
This is formula for calculating the ballistic coefficient within smalls arms shooting community, but redundant to '''''BC<sub>Projectile</sub>''''':
:<math>BC_{Smallarms} = \frac{SD}{i}</math> <ref>''Berger Bullets Reloading Manual 1st Edition'' (2012), Berger Bullets LLC, p814</ref>
 
Whereas:
*''BC<sub>Smallarms</sub>'' = Ballistic coefficient 
*''SD'' = [[Sectional density]]
*''i'' = Coefficient of form (form factor <ref>''Hornady Handbook of Cartridge Reloading:Rifle,Pistol Vol. II'' (1973)| Hornady Manufacturing Company, Fourth Printing July 1978, p505</ref>)
 
==Historical Precedents of Ballistic Coefficient, Methods and the Standard Model Projectile “G”==
*Within this section, some of the mathematical terms and denotations are from historical references and not within Wikipedia standards. When needed, these terms and denotations shall be followed in bold font  to denote the accepted Wikipedia standard. All Ordnance size and velocities are nominal or averaged as is historically and mathematically customary.
 
===Background===
In 1537,  [[Niccolò Tartaglia]]  did some test firing to determine the maximum angle and range for a [[Projectile|shot]]. His conclusion was near 45 degrees. He noted that the shot [[trajectory]] was continuously [[Parabola|curved]].<ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
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.<ref>Galileo, Galilei, ''Dialogues Concerning Two New Science'' , 2010; pg Fourth Day , THE MOTION OF PROJECTILES, Digireads.com ISBN 978-1420938159</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
Circa 1665, [[Sir Isaac Newton]] derived the [[Drag|law of air resistance]] and stated it was [[Inverse-square law|inversely]] proportional to the 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 weight of the shot. Newton’s experiments were only at low velocities to about {{convert|260|m/s|ft/s|0|abbr=on}}.<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 1, Cambridge at the University Press</ref><ref>William and Robert Cambers; ''CHAMBERS'S ENCYCLOPAEDIA Vol. VIII'', 1891; pg 438, J.B. Lippincott, USA</ref><ref>Pejsa, Arthur, ''New Exact Small Arms Ballistics: The Source Book for Riflemen'', 2008; pg29,  Kenwood Publishing ISBN 978-0974990262</ref>
 
In 1718, [[John Keill]] challenged the Continental Mathematica, ''“To find the curve that a projectile may describes in the air, on behalf of the  simplest assumption of [[Gravity of Earth|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 is 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 differential equation|Bernoulli equation]].  This is the precursor to the concept of the “standard projectile”.<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 1, Cambridge at the University Press</ref>
 
In 1742, [[Benjamin Robins]] invented the [[ballistic pendulum]]. This was  a simple mechanical device that could measure a projectiles velocity. Robins reported muzzle velocities ranging from {{convert|1400|ft/s|m/s|0|abbr=on}} to {{convert|1700|ft/s|m/s|0|abbr=on}}. In his book published that same year “New Principles of Gunnery”, he uses numerical integration from [[Euler method|Euler's method]] and found that air resistance “varies as the square of the velocity, but insists it changes at the [[speed of sound]].” <ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 4, 6, Cambridge at the University Press</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Pejsa, Arthur, ''Modern Practical Ballistics 2nd Edition'', 1991; pg6,  Kenwood Publishing ISBN 978-0961277635</ref>
 
In 1753, [[Leonhard Euler]]  showed how a 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.<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 6, Cambridge at the University Press</ref>
 
In 1844, the [[Gun chronograph|Electro-ballistic chronograph]] was invented and by 1867 the electro-ballistic chronograph was accurate to with in one ten millionth of a second.<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 13, Cambridge at the University Press</ref>
 
===Test Firing===
Many countries and their militaries carried out test firings from the mid eighteenth century on using large [[ordnance ammunition|ordnance]] to determine the [[drag]] characteristics of each individual projectile. These individual test firings were logged and reported in extensive ballistics tables.<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 5-11, Cambridge at the University Press</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 39, Lattie Stone Ballistics</ref>
 
Of the test firing, most notably were: Francis Bashforth at Woolwich Marshes & Shoeburyness, England  (1864-1889) with velocities to {{convert|2800|ft/s|m/s|0|abbr=on}} and M. Krupp (1865-1880) of [[Krupp|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 {{convert|1830|m/s|ft/s|0|abbr=on}} and the The [[Royal Artillery|British Royal Artillery]] (1904-1906).<ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; pg Protest, 1, H.M Stationary Office, Harrison & Sons, London</ref><ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 13, 135, Cambridge at the University Press</ref><ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg18,19, D. Van Nostrand Publisher</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 39-40, Lattie Stone Ballistics</ref>
 
The test projectiles (shot) used, vary from [[spherical]], [[spheroidal]], [[Ogive|ogival]]; being hollow, solid and cored in design with the elongated ogival-headed projectiles having 1, 1½, 2 and 3 [[caliber]] [[Radius|radii]]. These projectiles varied in size from, {{convert|75|mm|in|1|abbr=on}} at {{convert|3|kg|lbs|1|abbr=on}} to {{convert|254|mm|in|1|abbr=on}} at {{convert|187|kg|lbs|1|abbr=on}} <ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; pg 3-4, H.M Stationary Office, Harrison & Sons, London</ref><ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 36, 135, Cambridge at the University Press</ref><ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg19, D. Van Nostrand Publisher</ref>
 
===Methods and the Standard Projectile===
Many military's up until the 1860s had to used [[calculus]] to compute a bullet trajectory. The numerical computations necessary to calculate just a single trajectory was lengthy, tedious and done by hand. So, an investigation  to develop a theoretical drag model began. This competition did lead 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 [[Artificiality|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 [[Atmosphere|standard atmosphere]] with the same ability as any actual projectile could move through the actual atmosphere.<ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg8, D. Van Nostrand Publisher</ref><ref>Coast Artillery School Press, ''Reference Notes for Use in the Course in Gunnery and Ammunition'', 1917; pg 12, Coast Artillery School, ASIN:B00E0UERI2</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
====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 (v<sup>2</sup>) from {{convert|830|ft/s|m/s|0|abbr=on}} to {{convert|430|ft/s|m/s|0|abbr=on}} and with the cube of velocity (v<sup>3</sup>) from {{convert|1000|ft/s|m/s|0|abbr=on}} to {{convert|830|ft/s|m/s|0|abbr=on}}. As of his 1880 report, he found that drag varied by v<sup>6</sup>  from {{convert|1100|ft/s|m/s|0|abbr=on}} to {{convert|1040|ft/s|m/s|0|abbr=on}}. Bashforth used [[Rifling|rifled]] guns of {{convert|3|in|mm|0|abbr=on}}, {{convert|5|in|mm|0|abbr=on}}, {{convert|7|in|mm|0|abbr=on}} and {{convert|9|in|mm|0|abbr=on}};  smooth-bore guns of similar caliber for firing spherical shot and howitzers propelled elongated projectiles having an ogival-head of 1½ caliber radius.<ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; pg 4, H.M Stationary Office, Harrison & Sons, London</ref><ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 135-136, Cambridge at the University Press</ref><ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg19, D. Van Nostrand Publisher</ref>
 
Bashforth uses ''b'' as the variable for ballistic coefficient.  When ''b'' is equal to or less than v<sup>2</sup>, then ''b'' is equal to ''P'' for the drag of a projectile. It would be found that air does not deflect off the front of a projectile in the same direction, when there are of differing shapes. This prompted the introduction of a second factor to ''b'', the coefficient of form (''i''). This is particularly true at high velocities, greater than {{convert|830|ft/s|m/s|0|abbr=on}}. Hence, Bashforth introrduced the “undetermined multiplier” of any power called the <math>k</math> [[Factor graph|factor]] that compensate for this unknown effects of drag above {{convert|830|ft/s|m/s|0|abbr=on}}; <math>k > i </math>.  Bashforth then integrated <math>k</math> and <math>i</math> as <math>K_v</math>.<ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; pg 3,5-6, H.M Stationary Office, Harrison & Sons, London</ref><ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg 1, Cambridge at the University Press</ref><ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg7-8, 31, 136, D. Van Nostrand Publisher</ref><ref>[http://mrpogson.com/homepage/ballistics.html]</ref>
 
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.<ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; pg 5-6, H.M Stationary Office, Harrison & Sons, London</ref>
 
====The Mayevski/Siacci Method====
In 1872, General Mayevski published his report ''Trité 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 [[Logarithm|log A]] and the [[Variable (mathematics)|value]] ''n''. Although Mayevski's math used a differing approach than Bashforth, the resulting calculations of air resistance was the same. Mayevski proposed the restricted zone concept and found there to be 6 restricted zones for projectiles .<ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; pg pgvii, Cambridge at the University Press</ref><ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg14,18-31, D. Van Nostrand Publisher</ref><ref>Mayevski, Nikolai V, ''Trité Balistique Extérieure'' , 1872; pg58, 133 of the Schedules, Gauthier-Villars, Printer Bookseller office longitudes of the Polytechnic School, (Translated from French )</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 40, Lattie Stone Ballistics</ref>
 
Circa 1886, General 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 were essential of the same as the standard projectile, being mostly 3 caliber in length, with an ogive of 2 calibers radius. Giving the standard projectile dimensionally as {{convert|10|cm|in|1|abbr=on}} and {{convert|1|kg|lbs|1|abbr=on}}.<ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg19, D. Van Nostrand Publisher</ref><ref>Mayevski, Nikolai V, ''Trité Balistique Extérieure'' , 1872; pg58, Gauthier-Villars, Printer Bookseller office longitudes of the Polytechnic School, (Translated from French )</ref><ref>Siacci, Francesco,''Balistica; Second edition entirely recast'' , 1888; pg7, Torino F.Casanova, Publisher bookseller Via Academy of Sciences, (Translated from Italian and Latin)</ref>
 
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.<ref>Siacci, Francesco,''Balistica; Second edition entirely recast'' , 1888; pg3, Torino F.Casanova, Publisher bookseller Via Academy of Sciences, (Translated from Italian and Latin)</ref>
 
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 4 zone model. Siacci used Mayevski's standard projectile. From this method and standard projectile, Siacci formulated a short cut.<ref>Siacci, Francesco, ''Balistica; Second edition entirely recast'' , 1888; pg5,8, Torino F.Casanova, Publisher bookseller Via Academy of Sciences, (Translated from Italian and Latin)</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 39, Lattie Stone Ballistics</ref>
 
Siacci found that within a low velocity restricted zone, projectiles of similar shape, and velocity in the same air density behave similar; <math>\tfrac{\delta w}{d^2}</math> or <math>\tfrac{\delta}{C}</math>. Siacci used the variable <math>C</math> for ballistic coefficient.  Meaning, air density is the generally the same for flat-fire trajectories, thus sectional density is equal to the ballistic coefficient and air density can be dropped. Then as the velocity rises to Bashforth's <math>k</math> for high velocity when <math>C</math> requires the introduction of <math>i</math>. Following within today's currently used ballistic trajectory tables for an average ballistic coeficient: <math>\tfrac{m}{d^2} \cdot \tfrac{p_0}{p}</math> would equal <math>\tfrac{m}{d^2 i}</math> equals <math>\tfrac{SD}{i}</math> as <math>BC</math>.<ref>Siacci, Francesco,''Balistica; Second edition entirely recast'' , 1888; pg4, Torino F.Casanova, Publisher bookseller Via Academy of Sciences, (Translated from Italian and Latin)</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 42, Lattie Stone Ballistics</ref>
 
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]].<ref>Siacci, Francesco,''Balistica; Second edition entirely recast'' , 1888; pg5, Torino F.Casanova, Publisher bookseller Via Academy of Sciences, (Translated from Italian and Latin)</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
==== The Ballistic Tables====
The aforementioned ballistics tables are generally: functions, air density, projectile time at range, range, degree of projectile departure,  weight  and diameter to facilitate the calculation of ballistic [[formula]]e. These formulae produce the projectile velocity at range, drag and trajectories. The modern day commercially published ballistic tables or software computed ballistics tables for small arms, sporting ammunition are [[External ballistics|exterior ballistic]], trajectory tables.<ref>Bashforth, Francis; ''Reports on experiments made with the Bashforth chronograph...'', 1878-1879; H.M Stationary Office, Harrison & Sons, London</ref><ref>Bashforth, Francis, ''A revised account of the experiments made with the Bashforth chronograph...'', 1890; Cambridge at the University Press</ref><ref>Mayevski, Nikolai V, ''Trité Balistique Extérieure'' , 1872; pg i of the Schedules, Gauthier-Villars, Printer Bookseller office longitudes of the Polytechnic School, (Translated from French )</ref>
 
The 1870 Bashforth tables were to {{convert|2800|ft/s|m/s|0|abbr=on}}. 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 {{convert|1100|m/s|ft/s|0|abbr=on}}.  Mayevski converted Bashforth's data from [[Imperial units|Imperial units of measure]] to [[Metric system|metric units of measure]] (now in [[International System of Units|SI units of measure]]). In 1884, [[James Monroe Ingalls|James Ingalls]] published his tables in the U.S. Army Artillery Circular M using the Mayevski tables. Ingalls extended Mayevski's ballistics tables to {{convert|5000|ft/s|m/s|0|abbr=on}} 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 {{convert|5000|ft/s|m/s|0|abbr=on}} 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 those of Ingalls. In 1971 the Sierra Bullet company calculated their ballistic tables to 9 restricted zones but only within {{convert|4400|ft/s|m/s|0|abbr=on}}.<ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg19, D. Van Nostrand Publisher</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 40, Lattie Stone Ballistics</ref>
 
====The G Model====
In 1881, the Commission d'Experience de Gâvre did a comprehensive survey of data available from their tests as well as other countries. After adopting a standard atmospheric condition for the drag data the Gavre drag function was adopted. This drag function was known as the Gavre function and the standard projectile adopted was the Type 1 projectile. Thereafter, the Type 1 standard projectile was renamed by Ballistics Section of [[Aberdeen Proving Grounds]] in Maryland, USA as G<sub>1</sub> after the Commission d'Experience de Gâvre. For practical purposes the subscript 1 in G<sub>1</sub> is generally written in normal font size as G1.<ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 40, Lattie Stone Ballistics</ref>
 
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 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.<ref>Coast Artillery School Press, ''Reference Notes for Use in the Course in Gunnery and Ammunition'', 1917; pg 12, Coast Artillery School, ASIN:B00E0UERI2</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
Over the years there has been some confusion{{cn|date=November 2014}} as to adopted size, weight and radius ogival-head of the G1 standard projectile. This misconception may be explained by Colonel Ingalls in the 1886 publication,  Exterior Ballistics in the Plan Fire; page 15, ''In the following tables the first and second columns give the velocities and corresponding resistance, in [[Pound (mass)|pounds]], to an elongated one inch in diameter and having an ogival head of one and a half calibers. They were deduced from Bashforth's experiments by Professor  A. G. Greenhill, and are taken from his papers published in the Proceedings of the Royal Artillery Institution, No 2, Vol. XIII.'' Further it is discussed that said projectile's [[weight]] was one pound.<ref>Ingalls, James M., ''Exterior Ballistics in the Plan Fire'', 1886; pg15, D. Van Nostrand Publisher</ref>
 
For the purposes of mathematical convenience for any standard projectile (G) the ''BC'' is 1.00. Where as the projectiles [[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 <math>BC = \tfrac{SD}{i} = \tfrac{1}{1} = 1.00</math>.  ''BC'', as a general rule, within flat-fire trajectory, is carried out to 2 decimal points. ''BC'' 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.<ref>Coast Artillery School Press, ''Reference Notes for Use in the Course in Gunnery and Ammunition'', 1917; pg 12, Coast Artillery School, ASIN:B00E0UERI2</ref>
 
When using the Siacci method for diffing G models, the formula used to compute the trajectories is the same. What differs is retardation factors found through testing of actual projectiles that are similar in shape to the standard project reference. This creates slightly different set of retardation factors between differing G models. When the correct G model retardation factors are applied within the Siacci mathematical formula for the same G model ''BC'', a corrected trajectory can be calculated for any G model.
 
Another method of determining trajectory and ballistic coefficient was developed and published by Wallace H. Coxe and Edgar Beugless of DuPont in 1936. This method is by shape comparison an logarithmic scale as drawn on 10 charts. The method estimates the ballistic coefficient related to the drag model of the Ingalls tables. When matching an actual projectile against the drawn caliber radii of Chart No. 1, it will provide ''i'' and by using Chart No. 2, ''C ''can be quickly calculated. Coxe and Beugless used the variabale ''C'' for ballistic coefficient.<ref>Coxe, Wallace H. and Bugless, Edgar, ''Exterior Balistics Charts'', 1936; pg Charts 1, 2 , Ballistic Engineers of the Burnside Laboratory, of E.I. Du Pont De Nemours & Company Inc., Wilmington, Delaware</ref><ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref>
 
The Siacci method was abandoned by the end of the [[World War I]] for artillery fire. But the [[Ordnance Corps (United States Army)|U.S. Army Ordnance Corps]] continued using the Siacci method into the middle of the 20th century for [[Direct fire|direct (flat-fire)]] tank gunnery. The development of the [[Analog computer|electromechanical analog computer]] contributed to the calculation of aerial bombing trajectories during [[World War II]]. After World War II the advent of the [[Computer|silicon semiconductor based digital computer]] made it possible to create trajectories for the guided missiles/bombs, intercontinental ballistic missiles and space vehicles.<ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 39, Lattie Stone Ballistics</ref>
 
Between World War I and II the U.S. Army Ballistics research laboratories at Aberdeen Proving Grounds, Maryland, USA developed the standard models for G2, G5, G6. In 1965, [[Olin Corporation|Winchester Western]] published a set of ballistics tables for G1, G5, G6 and GL. In 1971 [[Sierra Bullets|Sierra Bullet Company]] retested all their bullets and concluded that the G5 model was not the best model for their boat tail bullets and started using the G1 model. This was fortunate, as the entire commercial sporting and firearms industries had based their calculations on the G1 model. The G1 model and Mayevski/Siacci Method continue to be the industry standard today. This benefit allows for comparison of all ballistic tables for trajectory within the commercial sporting and firearms industry.<ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Cline, Donna, ''Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere" The “Point-Mass Trajectory: The Siacci Method Ballistic Coefficient'', 2002; pg 42, Lattie Stone Ballistics</ref>
 
In recent years there have been vast advancements in the calculation of flat-fire trajectories with the advent of [[Doppler radar]] and the personal computer and handheld comuting devices.  Also, the newer [[External_ballistics#Pejsa_model|methodology]] proposed by Dr. Arthur Pejsa and the use of the G7 model used by Mr. Brian Litz, ballistic engineer for Berger Bullets, LLC for calculating boat tailed spitzer rifle bullet trajectories have improved the prediction of flat-fire trajectories.<ref>[http://www.exteriorballistics.com/ebexplained/4th/30.cfm Historical Summary]</ref><ref>Pejsa, Arthur, ''Modern Practical Ballistics 2nd Edition'', 1991;  Kenwood Publishing ISBN 978-0961277635</ref><ref>''Berger Bullets Reloading Manual 1st Edition'' 2012; Berger Bullets LLC; p161, ISBN 978-0-615-63762-4</ref>
 
==Differing mathematical models and bullet ballistic coefficients==
[[File:G1 Shape Standard Projectile Measurements in Calibers.png|thumb| G1 shape standard projectile. All measurements in calibers/diameters.]]
[[File:G7 Shape Standard Projectile Measurements in Calibers.png|thumb| G7 shape standard projectile. All measurements in calibers/diameters.]]
 
[[File:Effect of BC on Wind Drift.jpg|thumb| Wind drift calculations for rifle bullets of differing G1 BCs fired with a muzzle velocity of {{convert|2950|ft/s|abbr=on}} in a {{convert|10|mph|abbr=on}} crosswind.<ref>[http://www.jbmballistics.com/cgi-bin/jbmtraj-5.1.cgi JBM Ballistics online trajectory calculator]</ref>]]
[[File:Effect of BC on Energy Retained.jpg|thumb|Energy calculations for {{convert|140|gr|disp=flip}} rifle bullets of differing G1 BCs fired with a muzzle velocity of {{convert|2950|ft/s}}.<ref>[http://www.jbmballistics.com/cgi-bin/jbmtraj-5.1.cgi JBM ballistics online trajectory calculator]</ref>]]
 
Most ballistic mathematical models and hence tables or software take for granted that one specific drag function correctly describes the drag and hence the flight characteristics of a bullet related to its ballistic coefficient. Those models do not differentiate between [[wadcutter]], flat-based, [[spitzer bullet|spitzer]], boat-tail, [[very-low-drag bullet|very-low-drag]], etc. bullet types or shapes. They assume one invariable drag function as indicated by the published BC. Several different drag curve models optimized for several standard projectile shapes are available, however.<br>
The resulting drag curve models for several standard projectile shapes or types are referred to as:
* G1 or Ingalls (flatbase with 2 caliber (blunt) nose ogive - by far the most popular)<ref>[http://www.shootingsoftware.com/coefficients.htm Exterior Ballistics and Ballistic Coefficients]</ref>
* G2 (Aberdeen J projectile)
* G5 (short 7.5° boat-tail, 6.19 calibers long [[Nose cone design#Tangent ogive|tangent ogive]])
* G6 (flatbase, 6 calibers long [[Nose cone design#Secant ogive|secant ogive]])
* G7 (long 7.5° boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets<ref>[http://02b0516.netsolhost.com/blog1/?p=62 A Better Ballistic Coefficient by Bryan Litz, Ballistician Berger Bullets]</ref>)
* G8 (flatbase, 10 calibers long secant ogive)
* GL (blunt lead nose)
 
Since these standard projectile shapes differ significantly the G''x'' BC will also differ significantly from the G''y'' BC for an identical bullet.<ref>[http://www.shootersforum.com/ballistics-internal-external/61432-ballistic-coefficient-basics.html Ballistic Coefficient Basics]</ref>  To illustrate this the bullet manufacturer [[berger bullets|Berger]] has published the G1 and G7 BCs for most of their target, tactical, varmint and hunting bullets.<ref>[http://www.bergerbullets.com/products/all-bullets/ Berger Bullets Technical Specifications]</ref> Other manufacturers like Lapua and Nosler also started to publish the G1 and G7 BCs for most of their target bullets.<ref>[http://www.lapua.com/en/products/reloading Lapua bullets technical information]</ref><ref>[http://www.nosler.com/Bullets/AccuBond_LongRange.aspx Nosler AccuBond Longe Range technical information]</ref> How much a projectile deviates from the applied reference projectile is mathematically expressed by the form factor (''i''). The applied reference projectile shape always has a form factor (''i'') of exactly 1. When a particular projectile has a sub 1 form factor (''i'') this indicates that the particular projectile exhibits higher drag than the applied reference projectile shape. A form factor (''i'') greater than 1 indicates the particular projectile exhibits lower drag than the applied reference projectile shape.<ref>[http://www.bergerbullets.com/form-factors-a-useful-analysis-tool/ Form Factors: A Useful Analysis Tool by Bryan Litz, Chief Ballistician Berger Bullets]</ref> In general the G1 model yields comparatively high BC values and is often used by the sporting ammunition industry.
 
===The transient nature of bullet ballistic coefficients===
Variations in BC claims for exactly the same projectiles can be explained by differences in the ambient [[air density]] used to compute specific values or differing range-speed measurements on which the stated G1 BC averages are based. Also, the BC changes during a projectile's flight, and stated BCs are always averages for particular range-speed regimes. Further explanation about the variable nature of a projectile's G1 BC during flight can be found at the [[external ballistics]] article. The external ballistics article implies that knowing how a BC was determined is almost as important as knowing the stated BC value itself.
 
For the precise establishment of BCs (or perhaps the scientifically better expressed [[drag coefficient]]s), [[Doppler radar]]-measurements are required. The normal shooting or aerodynamics enthusiast, however, has no access to such expensive professional measurement devices. Weibel 1000e or Infinition BR-1001 [[Doppler radar]]s are used by governments, professional ballisticians, defense forces, and a few ammunition manufacturers to obtain exact real world data on the flight behavior of projectiles of interest.
 
Doppler radar measurement results for a lathe turned monolithic solid .50 BMG [[very-low-drag bullet]] (Lost River J40 {{convert|0.510|in|disp=flip}}, {{convert|773|gr|disp=flip}} monolithic solid bullet / twist rate 1:{{convert|15|in|disp=flip}}) look like this:
 
{| class="wikitable"
! Range (m)
| 500
| 600
| 700
| 800
| 900
| 1000
| 1100
| 1200
| 1300
| 1400
| 1500
| 1600
| 1700
| 1800
| 1900
| 2000
|-
! Ballistic coefficient (G1)
| 1.040
| 1.051
| 1.057
| 1.063
| 1.064
| 1.067
| 1.068
| 1.068
| 1.068
| 1.066
| 1.064
| 1.060
| 1.056
| 1.050
| 1.042
| 1.032
|}
 
The initial rise in the BC value is attributed to a projectile's always present yaw and [[precession]] out of the bore. The test results were obtained from many shots, not just a single shot. The bullet was assigned 1.062 for its BC number by the bullet's manufacturer, Lost River Ballistic Technologies.
 
Measurements on other bullets can give totally different results. How different speed regimes affect several 8.6&nbsp;mm (.338 in calibre) rifle bullets made by the Finnish ammunition manufacturer Lapua can be seen in the .338 Lapua Magnum product brochure which states Doppler radar established BC data.<ref>[http://www.lapua.com/upload/downloads/brochures/2011/lapuaspecialpurpose2011eng.pdf .338 Lapua Magnum product brochure]</ref>
 
===General trends===
Sporting bullets, with a [[caliber|calibre]] ''d'' ranging from {{convert|0.172|to|0.50|in|1|disp=flip}}, have BCs in the range 0.12 to slightly over 1.00&nbsp;lb/in². Those bullets with the higher BCs are  the most aerodynamic, and those with low BCs are the least. [[Very-low-drag bullet]]s with BCs ≥ 1.10 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.<ref>[http://www.lima-wiederladetechnik.de/Englisch/LM-Class-Bullets.htm LM Class Bullets, very high BC bullets for windy long ranges.]</ref>
 
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&nbsp;mm and 6.5&nbsp;mm cartridges are probably the most well known for having high BCs and are often used in long range target matches of {{convert|300|m|yd|0|abbr=on}} – {{convert|1000|m|yd|0|abbr=on}}.  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]], 6x47mm SM, [[6.5×55mm]] Swedish Mauser, [[6.5x47mm Lapua]], [[6.5 Creedmoor]], [[6.5 Grendel]], [[.260 Remington]], and the [[6.5-284]]. The 6.5&nbsp;mm is also a  popular hunting caliber in Europe.
 
In the United States, hunting cartridges such as the [[.25-06 Remington]] (a 6.35&nbsp;mm caliber), the [[.270 Winchester]] (a 6.8&nbsp;mm caliber), and the [[.284 Winchester]] (a 7&nbsp;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.
 
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 [[.408 Chey Tac|.375 and .408 Cheyenne Tactical]] and the [[.416 Barrett]].
 
==Satellites and reentry vehicles==
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 would reach subsonic speeds before reaching the ground.
 
In general, reentry vehicles that carry human beings back to Earth from space have high drag and a correspondingly low ballistic coefficient. Vehicles that carry nuclear weapons launched by an [[intercontinental ballistic missile]] (ICBM), by contrast, have a high ballistic coefficient, which enables them to travel rapidly from space to a target on land. That makes the weapon less affected by crosswinds or other weather phenomena, and harder to track, intercept, or otherwise defend against.
 
==See also==
*[[External ballistics]] - The behavior of a projectile in flight.
*[[Trajectory of a projectile]]
 
==References==
{{reflist|30em}}
 
==External links==
*[http://www.reentrynews.com/Defs.html Aerospace Corporation Definition]
*[http://www.chuckhawks.com/bc.htm Chuck Hawks Article on Ballistic Coefficient]
*[http://www.frfrogspad.com/bcdata.htm Ballistic Coefficient Tables]
*[http://www.exteriorballistics.com/ebexplained/index.cfm Exterior Ballistics.com]
*[http://www.nennstiel-ruprecht.de/bullfly/ballcoeff.htm#header How do bullets fly? The ballistic coefficient (bc) by Ruprecht Nennstiel, Wiesbaden, Germany]
*[http://www.precisionshooting.com.au/downloads/ballisticcoefficients-explained(4).pdf Ballistic Coefficients - Explained]
[http://www.jbmballistics.com/ballistics/calculators/calculators.shtml Ballistic calculators]
 
 
{{DEFAULTSORT:Ballistic Coefficient}}
[[Category:Projectiles]]
[[Category:Aerodynamics]]
[[Category:Ballistics]]

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