Ballistic coefficient
Encyclopedia
In ballistics
, the ballistic coefficient (BC) 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. BC is a function of mass, diameter, and drag coefficient
. It is given by the mass
of the object divided by the diameter
squared that it presents to the airflow divided by a dimensionless constant i that relates to the aerodynamics of its shape. Ballistic coefficient has units of lb/in² or kg/m². BCs for bullet
s are normally stated in lb/in² by their manufacturers without referring to this unit.
where:
This is not the same as the BC used by bullet manufacturers. This is the BC as defined by and used in Physics and Engineering. Although it would not be incorrect to use this equation on bullets, the BC obtained from this equation would not give the same value as the BC from a bullet manufacturer because their value is a comparison to the G1 bullet model.
where:
This BC formula gives the ratio of ballistic efficiency compared to the standard G1 model projectile. The standard G1 projectile originates from the "C" standard reference projectile (a 1 pound (454 g), 1 inch (25.4 mm) diameter projectile with a flat base, a length of 3 inches (76.2 mm), and a 2 inch (50.8 mm) radius tangential curve for the point) defined by the German steel, ammunition and armaments manufacturer Krupp
in 1881. The G1 model standard projectile has a BC of 1. The French Gavre Commission decided to use this projectile as their first reference projectile, giving the G1 name.
A bullet with a high BC will travel farther than one with a low BC since it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will “shoot flatter” (see external ballistics
).
When hunting with a rifle, a higher BC is desirable for several reasons. A higher BC results in a flatter trajectory
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 vitals 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 in bullet drop 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 cm (1 foot) at 500 meters (550 yards). 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, it is also less affected by the crosswinds.
), have BC’s in the range 0.12 to slightly over 1.00, with high being the most aerodynamic, and low being 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.
Ammunition makers often offer several bullet weights and types for a given cartridge. Heavy-for-caliber pointed (spitzer) bullets with a boattail design have the high BCs, 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 BC bullets and are often used in long range target matches of 300–1000 meters. The 6 and 6.5 have relatively light recoil compared to larger calibers with high BC bullets and tend to take matches where accuracy is key. Examples include the 6mm PPC, 6mm Norma BR
, 6x47mm SM, 6.5x47mm Lapua
, 6.5 Grendel
and the 6.5-284. The 6.5 mm is also a very 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. The .308 is also a favorite long-range target cartridge.
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. New competitors in the larger caliber category are the .375 and .408 Cheyenne Tactical
and the .416 Barrett
.
article. This article implies that knowing how a BC was established 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 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 .510-773 grain monolithic solid bullet / twist rate 1:15 in) look like this:
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.
, 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 however available.
The resulting drag curve models for several standard projectile shapes or types are referred to as the:
Since these standard projectile shapes differ significantly the Gx BC will also differ significantly from the Gy BC for an identical bullet. To illustrate this a bullet manufacturer has published a G1 BC of 0.659 and a G7 BC of 0.337 for their 7 mm Match Target VLD bullet and has since published the G1 and G7 BCs for most of their target bullets. In general the G1 model yields compariatively high BC values and is often used by the sporting ammunition industry.
(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 hitting 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.
Ballistics
Ballistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...
, the ballistic coefficient (BC) 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. BC is a function of mass, diameter, and drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
. It is given by the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the object divided by the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
squared that it presents to the airflow divided by a dimensionless constant i that relates to the aerodynamics of its shape. Ballistic coefficient has units of lb/in² or kg/m². BCs for bullet
Bullet
A bullet is a projectile propelled by a firearm, sling, or air gun. Bullets do not normally contain explosives, but damage the intended target by impact and penetration...
s are normally stated in lb/in² by their manufacturers without referring to this unit.
Formula
where:
- BC_{Physics} = ballistic coefficient as used in physics and engineering
- M = mass
- A = cross-sectional area
- C_{d} = drag coefficientDrag coefficientIn fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
- ρ (rho) = average densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
- l = body length
This is not the same as the BC used by bullet manufacturers. This is the BC as defined by and used in Physics and Engineering. Although it would not be incorrect to use this equation on bullets, the BC obtained from this equation would not give the same value as the BC from a bullet manufacturer because their value is a comparison to the G1 bullet model.
Bullet performance
The formula for calculating the ballistic coefficient for bullets only is as follows:where:
- BC_{Bullets} = ballistic coefficient
- SD = sectional densitySectional densitySectional density is the ratio of an object's mass to its cross-sectional area. It conveys how well an object's mass is distributed to overcome resistance. For illustration, a needle can penetrate a target medium with less force than a coin of the same mass...
, SD = mass of bullet in pounds or kilograms divided by its caliber squared in inches or meters; units are lb/in^{2} or kg/m^{2}. - i = form factor, i = ; (C_{G} ~ 0.5191)
- C_{B} = Drag coefficient of the bullet
- C_{G} = Drag coefficient of the G1 model bullet
- M = Mass of object, lb or kg
- d = diameter of the object, in or m
This BC formula gives the ratio of ballistic efficiency compared to the standard G1 model projectile. The standard G1 projectile originates from the "C" standard reference projectile (a 1 pound (454 g), 1 inch (25.4 mm) diameter projectile with a flat base, a length of 3 inches (76.2 mm), and a 2 inch (50.8 mm) radius tangential curve for the point) defined by the German steel, ammunition and armaments manufacturer Krupp
Krupp
The Krupp family , a prominent 400-year-old German dynasty from Essen, have become famous for their steel production and for their manufacture of ammunition and armaments. The family business, known as Friedrich Krupp AG Hoesch-Krupp, was the largest company in Europe at the beginning of the 20th...
in 1881. The G1 model standard projectile has a BC of 1. The French Gavre Commission decided to use this projectile as their first reference projectile, giving the G1 name.
A bullet with a high BC will travel farther than one with a low BC since it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will “shoot flatter” (see external ballistics
External ballistics
External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target.-Forces...
).
When hunting with a rifle, a higher BC is desirable for several reasons. A higher BC results in a flatter trajectory
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
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 vitals 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 in bullet drop 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 cm (1 foot) at 500 meters (550 yards). The difference in impact energy can also be great because kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
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, it is also less affected by the crosswinds.
General trends
Sporting bullets, with a calibre d ranging from 0.172 to 0.50 inches (4.50 to 12.7 mm.50 BMG
The .50 Browning Machine Gun or 12.7×99mm NATO is a cartridge developed for the Browning .50 caliber machine gun in the late 1910s. Entering service officially in 1921, the round is based on a greatly scaled-up .30-06 cartridge...
), have BC’s in the range 0.12 to slightly over 1.00, with high being the most aerodynamic, and low being the least. Very-low-drag bullet
Very-low-drag bullet
Very-low-drag bullets are primarily a small arms ballistics development of the 1980s–1990s, driven by shooters' desire for bullets that will give a higher degree of accuracy and kinetic efficiency, especially at extended ranges. To achieve this the projectile must minimize air resistance in flight...
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.
Ammunition makers often offer several bullet weights and types for a given cartridge. Heavy-for-caliber pointed (spitzer) bullets with a boattail design have the high BCs, 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 BC bullets and are often used in long range target matches of 300–1000 meters. The 6 and 6.5 have relatively light recoil compared to larger calibers with high BC bullets and tend to take matches where accuracy is key. Examples include the 6mm PPC, 6mm Norma BR
6mm Norma BR
The 6mm BR is a centerfire cartridge created for benchrest shooting. The cartridge is also known as the 6mm Bench Rest or simply 6 BR, and has also developed a following among varmint hunters because of its efficiency...
, 6x47mm SM, 6.5x47mm Lapua
6.5x47mm Lapua
6.5x47mm Lapua is a rifle cartridge that was developed specifically for 300m competition by ammunition maker Nammo Lapua and the Swiss rifle manufacturer Grünig & Elmiger AG in 2005.-Features:The cartridge has many special features including: :...
, 6.5 Grendel
6.5 Grendel
The 6.5 Grendel is a proprietary intermediate cartridge developed in 2002 by Arne Brennan . This cartridge is an improved version of the PPC cartridge family designed by Dr. Lou Palmisano and Ferris Pindell in 1974. The first 6.5mm member of the PPC cartridge family was created in 1984 by Dr...
and the 6.5-284. The 6.5 mm is also a very popular hunting caliber in Europe.
In the United States, hunting cartridges such as the .25-06 Remington
.25-06 Remington
The .25-06 Remington had been a wildcat cartridge for half a century before being standardized by Remington in 1969. It is based on the .30-06 Springfield cartridge necked-down to .257 inch caliber with no other changes...
(a 6.35 mm caliber), the .270 Winchester
.270 Winchester
The .270 Winchester was developed by Winchester Repeating Arms Company in 1923 and unveiled in 1925 as a chambering for their bolt-action Model 54. The cartridge is based upon the .30-06 Springfield...
(a 6.8 mm caliber), and the .284 Winchester
.284 Winchester
The .284 Winchester is an example of a commercially rather unsuccessful cartridge that has enjoyed a resurgence in interest due to interest from long-range competitive shooters...
(a 7 mm caliber) are used when high BCs and moderate recoil are desired. The .30-06 Springfield
.30-06 Springfield
The .30-06 Springfield cartridge or 7.62×63mm in metric notation, was introduced to the United States Army in 1906 and standardized, and was in use until the 1960s and early 1970s. It replaced the .30-03, 6 mm Lee Navy, and .30 US Army...
and .308 Winchester
.308 Winchester
The .308 Winchester is a rifle cartridge and is the commercial cartridge upon which the military 7.62x51mm NATO centerfire cartridge is based. The .308 Winchester was introduced in 1952, two years prior to the NATO adoption of the 7.62x51mm NATO T65...
cartridges also offer several high-BC loads, although the bullet weights are on the heavy side. The .308 is also a favorite long-range target cartridge.
In the larger caliber category, the .338 Lapua Magnum and the .50 BMG
.50 BMG
The .50 Browning Machine Gun or 12.7×99mm NATO is a cartridge developed for the Browning .50 caliber machine gun in the late 1910s. Entering service officially in 1921, the round is based on a greatly scaled-up .30-06 cartridge...
are popular with very high BC bullets for shooting beyond 1000 meters. New competitors in the larger caliber category are the .375 and .408 Cheyenne Tactical
.408 Chey Tac
The .408 Cheyenne Tactical is a specialized rimless, bottlenecked, centerfire cartridge for military long-range sniper rifles that was developed by Dr. John D. Taylor and machinist William O. Wordman...
and the .416 Barrett
.416 Barrett
The .416 Barrett or 10.6x83mm centerfire rifle cartridge is a proprietary bottlenecked centrefire rifle cartridge designed in 2005. It is an alternative to the .50 BMG in long-range high-caliber rifles. It was designed in response to a request for a medium/heavy rifle/cartridge combination that was...
.
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 for these BC statements or differing range-speed measurements on which the stated G1 BC averages are based. The BC changes during a projectile's flight and stated BCs are always averages for particular range-speed regimes. Some more explanation about the transient nature of a projectile's G1 BC (it rises above or gets under a stated average value for a certain speed-range regime) during flight can be found at the external ballisticsExternal ballistics
External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target.-Forces...
article. This article implies that knowing how a BC was established 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
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
s), Doppler radar
Doppler radar
A Doppler radar is a specialized radar that makes use of the Doppler effect to produce velocity data about objects at a distance. It does this by beaming a microwave signal towards a desired target and listening for its reflection, then analyzing how the frequency of the returned signal has been...
-measurements are required. The normal shooting or aerodynamics enthusiast, however, has no access to such expensive professional measurement devices. Weibel 1000e Doppler radar
Doppler radar
A Doppler radar is a specialized radar that makes use of the Doppler effect to produce velocity data about objects at a distance. It does this by beaming a microwave signal towards a desired target and listening for its reflection, then analyzing how the frequency of the returned signal has been...
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
Very-low-drag bullet
Very-low-drag bullets are primarily a small arms ballistics development of the 1980s–1990s, driven by shooters' desire for bullets that will give a higher degree of accuracy and kinetic efficiency, especially at extended ranges. To achieve this the projectile must minimize air resistance in flight...
(Lost River J40 .510-773 grain monolithic solid bullet / twist rate 1:15 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.
Differing mathematical models and bullet ballistic coefficients
Most ballistic mathematical models and hence tables or software takes for granted that one specific drag function correctly describes the drag and hence the flight characteristics of a bullet related to its ballistics coefficient. Those models do not differentiate between wadcutterWadcutter
A wadcutter is a special-purpose bullet specially designed for shooting paper targets, usually at close range and at subsonic velocities typically under 800 ft/s . They are often used in handgun and airgun competitions...
, flat-based, spitzer, boat-tail, very-low-drag
Very-low-drag bullet
Very-low-drag bullets are primarily a small arms ballistics development of the 1980s–1990s, driven by shooters' desire for bullets that will give a higher degree of accuracy and kinetic efficiency, especially at extended ranges. To achieve this the projectile must minimize air resistance in flight...
, 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 however available.
The resulting drag curve models for several standard projectile shapes or types are referred to as the:
- G1 or Ingalls (by far the most popular)
- 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)
- 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. To illustrate this a bullet manufacturer has published a G1 BC of 0.659 and a G7 BC of 0.337 for their 7 mm Match Target VLD bullet and has since published the G1 and G7 BCs for most of their target bullets. In general the G1 model yields compariatively high BC values and is often used by the sporting ammunition industry.
Satellites and reentry vehicles
Satellites in Low Earth OrbitLow Earth orbit
A low Earth orbit is generally defined as an orbit within the locus extending from the Earth’s surface up to an altitude of 2,000 km...
(LEO) with high ballistic coefficients experience smaller perturbations to their orbits due to atmospheric drag.
The ballistic coefficient of an atmospheric reentry
Atmospheric reentry
Atmospheric entry is the movement of human-made or natural objects as they enter the atmosphere of a celestial body from outer space—in the case of Earth from an altitude above the Kármán Line,...
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 hitting 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 ballisticsExternal ballisticsExternal ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target.-Forces...
- The behavior of a projectile in flight. - Trajectory of a projectileTrajectory of a projectileIn physics, the ballistic trajectory of a projectile is the path that a thrown or launched projectile will take under the action of gravity, neglecting all other forces, such as friction from air resistance, without propulsion....
Ballistic coefficient can be use to calculated the trajectory of a projectile. Although the above page currently lacks any practical application of BC. For information on calculating the trajectory of a projectile excluding air resistance see Trajectory of a projectile. BC is an easy way to account for air resistance.