Ballistic coefficient

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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.[1] It is inversely proportional to the negative acceleration — 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.

Formulae[edit]

General[edit]

BC_{Physics} = \frac{M}{C_d \cdot A} = \frac{\rho \cdot l}{C_d}

Where as:

  • BCPhysics = ballistic coefficient as used in physics and engineering
  • M = mass
  • A = cross-sectional area
  • Cd = drag coefficient
  • ρ (rho) = average density
  • l = body length

Ballistics[edit]

The formula for calculating the ballistic coefficient for small and large arms projectiles only is as follows:

BC_{Projectile} = \frac{m}{ d^2 \cdot i} [2]

Where as:

  • BCProjectile = ballistic coefficient as used in point mass trajectory from the Siacci method (less than 20 degrees). [3]
  • 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. [4] [5]

Here are several methods to compute i or Cd:

i=\frac{2}{n} \cdot \sqrt{\frac{4n-1}{n}} [6] [7] [8]

Where as:

  • i = Coefficient of form.
  • n = number of calibers of the projectile's ogive.
Where n is unknown:
n = \frac{(4 \cdot l^2 + 1)}{4} [9]
Where as:
  • n = number of calibers of the projectile's ogive.
  • l = length of the head (ogive) in number of calibers.

or

A drag coefficient can also be calculated mathematically:

 C_{d}= \frac{8}{ \rho \cdot v^2 \cdot \pi \cdot d^2} [10]

Where as:

  • Cd = drag coefficient.
  • ρ (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:

i = \frac{C_{p}}{C_{G}} [11]

Where as:

  • i = Coefficient of form.
  • CG = drag coefficient of 1.00 from any “G” model, reference drawing, projectile.[12]
  • Cp = drag coefficient of the actual test projectile at range.

Commercial Use[edit]

This is formula for calculating the ballistic coefficient within smalls arms shooting community, but redundant to BCProjectile:

BC_{Smallarms} = \frac{SD}{i} [13]

Where as:

Bullet performance[edit]

This BC formula gives the ratio of ballistic efficiency compared to the standard G1 model projectile. The standard G1 projectile originates from the somewhat shorter "C" standard reference projectile. The standard C projectile has a 25.4 millimetres (1 in) diameter with a flat base; a 50.8 millimetres (2 in) radius tangential curve for the point; an overall length of 76.2 millimetres (3 in); and an overall weight of 454 grams (1 lb), as defined by the German steel, ammunition and armaments manufacturer Krupp in 1881.[15] By definition, the G1 model standard projectile has a BC of 1 (using Imperial Units). The French Gâvre Commission decided to use this projectile as its first reference projectile, giving the G1 its name.[16][17]

““As a matter of convenience, the ballistic tables are made up for a factitious projectile with a weight of one pound, a diameter of one inch, and a two-caliber radius of ogive. This simplifies the computations of the ballistic coefficient of any projectile. “The ballistic coefficient has thus been defined as the factitious weight of a projectile of one-inch caliber and standard form (i.e., two caliber radius of ogive), which would traverse the standard atmosphere with the same facility as the actual projectile traverses the actual atmosphere.” ”” [18]

A bullet with a high BC will travel farther than one with a low BC because it is affected less by air resistance, and retains more of its initial velocity as it flies downrange from the muzzle (see external ballistics).[19]

When hunting with a rifle, a higher BC is desirable for several reasons. A higher BC results in a flatter trajectory for a given distance, which in turn reduces the effect of errors in estimating the distance to the target. This is particularly important when attempting a clean hit on the vital organs of a game animal. If the target animal is closer than estimated, then the bullet will hit higher than expected. Conversely, if the animal is further than estimated the bullet will hit lower than expected. Such a difference from the point of aim can often make the difference between a clean kill and a wounded animal.

This difference in trajectories becomes more critical at longer ranges. For some cartridges, the difference in two bullet designs fired from the same rifle can result in a difference between the two of over 30 centimetres (12 in) at 500 metres (550 yd). The difference in impact energy can also be great because kinetic energy depends on the square of the velocity. A bullet with a high BC arrives at the target faster and with more energy than one with a low BC.

Since the higher BC bullet gets to the target faster, there is also less time for it to be affected by any crosswind.

Differing mathematical models and bullet ballistic coefficients[edit]

G1 shape standard projectile. All measurements in calibers/diameters.
G7 shape standard projectile. All measurements in calibers/diameters.
Wind drift calculations for rifle bullets of differing G1 BCs fired with a muzzle velocity of 2,950 ft/s (900 m/s) in a 10 mph (16 km/h) crosswind.[20]
Energy calculations for 9.1 grams (140 gr) rifle bullets of differing G1 BCs fired with a muzzle velocity of 2,950 feet per second (900 m/s).[21]

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, boat-tail, 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.
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)[22]
  • G2 (Aberdeen J projectile)
  • G5 (short 7.5° boat-tail, 6.19 calibers long tangent ogive)
  • G6 (flatbase, 6 calibers long secant ogive)
  • G7 (long 7.5° boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets[23])
  • G8 (flatbase, 10 calibers long secant ogive)
  • GL (blunt lead nose)

Since these standard projectile shapes differ significantly the Gx BC will also differ significantly from the Gy BC for an identical bullet.[24] To illustrate this the bullet manufacturer Berger has published the G1 and G7 BCs for most of their target, tactical, varmint and hunting bullets.[25] Other manufacturers like Lapua and Nosler also started to publish the G1 and G7 BCs for most of their target bullets.[26][27] 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 lower drag than the applied reference projectile shape. A form factor (i) greater than 1 indicates the particular projectile exhibits more drag than the applied reference projectile shape.[28] 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[edit]

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 coefficients), 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 radars 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 13.0 millimetres (0.510 in), 50.1 grams (773 gr) monolithic solid bullet / twist rate 1:380 millimetres (15 in)) look like this:

Range (m) 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
Ballistic coefficient 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 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.[29]

General trends[edit]

Sporting bullets, with a calibre d ranging from 4.4 to 12.7 millimetres (0.172 to 0.50 in), have BCs in the range 0.12 to slightly over 1.00 lb/in². Those bullets with the higher BCs are the most aerodynamic, and those with low BCs are the least. Very-low-drag bullets 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.[30]

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 300 m (328 yd) – 1,000 m (1,094 yd). 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 mm is also a popular hunting caliber in Europe.

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.

In the larger caliber category, the .338 Lapua Magnum and the .50 BMG are popular with very high BC bullets for shooting beyond 1000 meters. Newer chamberings in the larger caliber category are the .375 and .408 Cheyenne Tactical and the .416 Barrett.

Satellites and reentry vehicles[edit]

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[edit]

  • External ballistics - The behavior of a projectile in flight.
  • Trajectory of a projectile – This page can be used to calculate the trajetory of a projectile excluding air resistance. BC is an easy way to account for air resistance.

Freeware small arms ballistic coefficient calculators[edit]

Ballistic calculators

External links[edit]

References[edit]

  1. ^ The Truth About Ballistic Coefficients
  2. ^ Moss, Leeming and Farrar (1995). Brassey's Land Warfare Seires: Military Ballistics. Royal Military College of Science, Shrivenham, UK. p. 86. ISBN 978-1857530841. 
  3. ^ Cline, Donna (2002). Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere". Lattie Stone Ballistics. p. 39. 
  4. ^ Cline, Donna (2002). Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere". Lattie Stone Ballistics. p. 43-48. 
  5. ^ Rinker, Robert A. (1999). Understanding Firearm Ballistics; 3rd Edition. Mulberry House Publishing. p. 176. ISBN 978-0964559844. 
  6. ^ Cline, Donna (2002). Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere". Lattie Stone Ballistics. p. 44. 
  7. ^ Rinker, Robert A. (1999). Understanding Firearm Ballistics; 3rd Edition. Mulberry House Publishing. p. 176. ISBN 978-0964559844. 
  8. ^ Textbook of Small Arms 1909 (1909). Great Britain. War Office H.M. Stationery Office. ISBN 978-1847914217
  9. ^ Cline, Donna (2002). Exterior Ballistics Explained, Trajectories, Part 3 “Atmosphere". Lattie Stone Ballistics. p. 44. 
  10. ^ Moss, Leeming and Farrar (1995). Brassey's Land Warfare Seires: Military Ballistics. Reading: Royal Military College of Science, Shrivenham, UK. p. 79. ISBN 978-1857530841. 
  11. ^ Historical Summary
  12. ^ Reference Notes for Use in the Course in Gunnery and Ammunition (1917). Coast Artillery School U.S. Army, p12. ASIN B00E0UERI2
  13. ^ Berger Bullets Reloading Manual 1st Edition (2012), Berger Bullets LLC, p814
  14. ^ Hornady Handbook of Cartridge Reloading:Rifle,Pistol Vol. II (1973)| Hornady Manufacturing Company, Fourth Printing July 1978, p505
  15. ^ Exterior Ballistics and Ballistic Coefficients
  16. ^ Weite Schüsse - drei (German)
  17. ^ Historical Summary
  18. ^ Reference Notes for Use in the Course in Gunnery and Ammunition (1917). Coast Artillery School U.S. Army, p12. ASIN B00E0UERI2
  19. ^ The Ballistic Coefficient Explained
  20. ^ JBM Ballistics online trajectory calculator
  21. ^ JBM ballistics online trajectory calculator
  22. ^ Exterior Ballistics and Ballistic Coefficients
  23. ^ A Better Ballistic Coefficient by Bryan Litz, Ballistician Berger Bullets
  24. ^ Ballistic Coefficient Basics
  25. ^ Berger Bullets Technical Specifications
  26. ^ Lapua bullets technical information
  27. ^ Nosler AccuBond Longe Range technical information
  28. ^ Form Factors: A Useful Analysis Tool by Bryan Litz, Chief Ballistician Berger Bullets
  29. ^ .338 Lapua Magnum product brochure
  30. ^ LM Class Bullets, very high BC bullets for windy long ranges.