2.0
The Ballistic Coefficient
“So,
just what is a ballistic coefficient, and what does it do for a
bullet’s trajectory?” These are questions we have been
asked many, many times, and they are not easy questions to answer.
We will try to answer the first question in this subsection and
then proceed to the second question in the next subsection. In later
subsections we will describe methods used to measure ballistic coefficients,
and give some examples of measured BC values for Sierra’s bullets.
Henceforth, ballistic coefficient will be abbreviated as BC.
2.1
The Ballistic Coefficient Explained
There
are at least three ways to describe the BC. First, it is widely
recognized as a figure of merit for a bullet’s ballistic efficiency.
That is, if a bullet has a high BC, then it will retain its velocity
better as it flies downrange from the muzzle, will resist the wind
better, and will “shoot flatter.” But this description
is qualitative, rather than quantitative. For example, if we compare
two bullets and one has a BC 25% higher than the other, how much
is the improvement in bullet ballistic performance? This question
can be answered only by calculating the trajectories for the two
bullets and then comparing velocity, wind deflection, and drop or
bullet path height versus range from the muzzle. So, the figure
of merit approach really gives only a qualitative insight into bullet
performance, and sometimes this insight is not correct. It often
happens that the bullet with the smaller BC is lighter than the
bullet with the higher BC. The lighter bullet therefore can be fired
at a higher muzzle velocity, and it can then deliver better ballistic
performance just because it leaves the muzzle at a higher velocity.
We will talk more about this later.
The second
way to describe the BC is to use its precise mathematical definition.
Mathematically, the BC defined as is the sectional density of the
bullet divided by the form factor. This definition emerges from
the physics of ballistics and is used in mathematical analysis of
bullet trajectories. But in a practical sense, this definition is
not satisfactory to most people for at least two reasons. The first
is the question of a bullet’s form factor. The form factor
is a property of the shape of the bullet design, but it is no easier
to explain than the BC. The second reason is that this mathematical
definition can lead to an erroneous conclusion. Assume for the moment
that the form factor is just a constant property of the bullet design
(not always true). The sectional density of a bullet is its weight
divided by the square of its diameter. (The square of any number
is the number multiplied by itself). So, to get a large BC we need
a large sectional density. It appears from the mathematics that
a bullet with a very small diameter should have a very large sectional
density because its weight is divided by a very small number, and
this should give it a very high BC. In other words, this line of
reasoning would lead us to expect that small caliber bullets should
have very large BC values. But this is not true because when the
diameter of the bullet is small, the volume also is small. The weight
of the bullet then is small, and the sectional density is necessarily
small also. The net result is that small caliber bullets generally
have lower BC values than larger caliber bullets.
The third
way to describe the ballistic coefficient traces back to the historical
development of the science of ballistics in the latter half of the
19th century. This explanation is lengthier, but it provides a better
understanding of what the BC is and what its role is in trajectory
calculations. The latter half of the 19th century and the early
part of the 20th century was a period of very intensive and fruitful
development in the science of ballistics. The developments in ballistics
were driven by technological advances in guns, projectiles, propellant
ignition, and propellants throughout the 19th century, and by warfare,
particularly in Europe and America. Warfare was almost an international
sport among the kings, emperors, Kaisers and tsars in Europe throughout
the 1800’s. The United States experienced the War of 1812,
the Mexican War, the Civil War, the Indian wars in the West, and
the SpanishAmerican War within that same century. Governments were
eager to fund research, development and manufacturing of improved
guns and gunnery, because battles were generally won by the forces
that had superior arms.
Percussion
ignition was invented in 1807 by the Rev. Alexander Forsythe in
Scotland. In 1814, Joshua Shaw, an artist in Philadelphia, invented
the percussion cap. In 1842, the U.S. Army adopted the percussion
lock for the Model 1842 Springfield Musket, replacing flintlock
ignition in earlier shoulder arms. Rifled muskets and handguns began
to replace smoothbore military weapons in the mid1850’s after
a French Army officer, Capt. Claude Minie, developed a means to
expand a bullet upon firing to cause it to fit the grooves of a
rifled barrel. This advancement combined the rapidity and ease of
loading of round balls—which had been the standard military
projectile for over a century—with the increased range and
deadly accuracy of rifled arms. The range and precision of military
weapons, for both small arms and artillery, was increasing dramatically.
The period
between 1855 and about 1870 witnessed much research and development
in breech loading rifles and handguns. The first metallic selfcontained,
internally primed cartridge (the 22 Short rimfire cartridge) was
introduced by Smith & Wesson in 1857 in their Model No. 1 breechloading
revolver. Breechloading rifles firing selfcontained cartridges
appeared in the 1860’s, and some were used during the U.S.
Civil War (e.g., the Spencer carbine and the Henry rifle). In 1866
in the United States, Hiram Berdan obtained a patent on a primer
that was suitable for centerfire cartridges. That same year in England,
Col. Edward Boxer patented a full cartridge for the British Snider
Enfield rifle, which was a centerfire cartridge utilizing the Boxer
primer. (It is interesting to note that later the Berdan primer
was widely adopted on the European continent, while the
Boxer primer became standard
in the United States.) In 1873, the U.S. military adopted the Model
1873 Trapdoor Springfield rifle with the 4570 centerfire cartridge.
In the space of just 31 years the U.S. Army changed from smoothbore
muskets with flintlock ignition to rifles with selfcontained metallic
centerfire cartridges.
Just 11 years later in 1884, a French
physicist named Paul Vielle developed the first smokeless propellant
that was stable and loadable for military purposes. Earlier powder
developments had led up to Vielle’s discovery, but they were
useful only for sporting purposes. The French Army loaded Vielle’s
smokeless propellant in the 8mm Lebel cartridge for the Model 1886
Lebel rifle, the very first military rifle firing a smokeless propellant
cartridge.
Smokeless propellant was quickly adopted
by other nations, including the U.S., and caused significant advancements
in bullet performance and design. Muzzle velocity in military rifles,
which was less than 1400 fps in the 4570 and most other black powder
cartridges, increased to more than 2000 fps in the earliest smokeless
propellant cartridges. This led to the development of jacketed bullets
of smaller caliber and lighter weights, i.e., 7mm, 30, and 8mm calibers,
which could be fired at even higher velocities and not deposit lead
in the barrels at those velocities. Before the end of the 19th century,
pointed bullets and boattail bullets were also developed to significantly
improve bullet ballistic performance.
With all these developments in guns and
ammunition, the need to understand the ballistics of projectiles
became more acute. It was no longer sufficient to target a gun by
hitandmiss methods. Of course, graduated sights had existed on
both smoothbore and rifled muskets for many years, but the elevation
marks on the sights had been determined by firing tests of these
weapons with a specific projectile at a specific muzzle velocity,
at a specific altitude, and with a specific set of weather conditions.
As warfare grew in intensity and mobility, it became vitally necessary
to understand the physics of bullet motion. In other words, it was
necessary to find a way to calculate bullet trajectories as well
as the changes in those trajectories caused by changes in bullets,
muzzle velocities and firing conditions.
An immense
problem thwarted this objective for many years. This problem was
understanding the physics and mathematically describing the aerodynamic
drag force on a projectile. The invention of the ballistic pendulum
by the English ballistician Benjamin Robins in 1740 had led to the
astounding discovery (at that time) that the drag force on a bullet
was many times more powerful than the force due to gravity, and
that it changed markedly with bullet velocity. That event started
a chain of firing tests, instrumentation developments, and theoretical
investigations that lasted at least 200 years. Progress was slow
because aerodynamic drag is a very complex physical process, and
mathematics had to be developed to make accurate computation of
trajectories possible long before the age of computers.
An early observation was that the drag
force was different on every type of projectile, so that measurements
of drag deceleration seemed to be necessary on each type of projectile
over the full velocity range between muzzle velocity and impact
velocity. However, around 1850 Francis Bashforth in England proposed
a practical idea that greatly simplified things and is used in the
present day. He proposed a model bullet, or “standard”
bullet, on which comprehensive measurements of drag deceleration
versus velocity could be made. Then, for other bullets this “standard”
drag deceleration could be scaled by some means, so that exhaustive
drag measurements could be avoided for those bullets.
Bashforth could not have known how successful
his suggestion would be. Ballisticians and physicists were working
intensively to mathematically describe the aerodynamic drag force
and derive the equations of motion of bullet flight. They had recognized
in the equations of motion a theoretical scale factor for aerodynamic
drag that would adjust the standard drag model to fit a nonstandard
bullet. This scale factor turned out to be the form factor of the
nonstandard bullet divided by the sectional density, that is, the
reciprocal of the BC. The form factor was a number that accounted
for the different shape of the nonstandard bullet compared to the
standard bullet.
Bashforth’s suggested standard bullet
had a weight of 1.0 pound, a caliber of 1.0 inch, and a point with
a 1.5 caliber ogive. Firing tests on projectiles of approximately
this shape and weight were conducted in England and Russia between
about 1865 and 1880. However, the definitive drag deceleration tests
were performed by Krupp at their test range in Meppen, Germany,
between 1875 and 1881. In 1883 Col. (later General) Mayevski in
Russia formulated a mathematical representation of the drag force
for the standard bullet. In the 1880’s, an Italian Army team
led by Col. F. Siacci formulated an analytical approach and found
analytical closed form solutions to the equations of motion of bullet
flight for levelfire trajectories. This meant that trajectory calculations
for shoulder arms could be performed algebraically, rather than
by the more tedious methods of calculus. The Siacci team’s
results also showed that not only could the standard drag deceleration
be scaled by using the BC of the nonstandard bullet, but also the
standard trajectory computed for the standard bullet could be scaled
by the same factor to compute an actual trajectory for the nonstandard
bullet. This was a very important breakthrough that greatly reduced
the amount of work in trajectory computations. The Siacci approach
was adopted by Col. James M. Ingalls of the U.S. Army Artillery.
His team produced the Ingalls Tables, first published in 1900, which
in turn became the standard for small arms ballistics used by the
U.S. Army in World War I.
So, the ballistic coefficient actually
is a scale factor. The BC scales the standard drag deceleration
of the standard bullet to fit a nonstandard bullet. However, the
BC works in a reciprocal manner. That is, the higher the BC of a
nonstandard bullet, the lower the drag is compared to the standard
bullet. This is alright, because it means that the higher the BC
of a bullet, the better will be its ballistic performance. The physical
units of the BC are pounds per square inch (lb/in2). The BC value
for the standard bullet then is 1.0 (weight 1.0 lb, diameter 1.0
inch, and form factor 1.0 by definition for the standard bullet).
Ballistic coefficients of most sporting and target bullets have
values less than 1.0, and generally BC values increase as caliber
increases. A bullet can have a BC higher than 1.0. For example,
some heavy 50 caliber bullets have BC values greater than 1.0.
Military agencies in different nations
developed many standard bullets over the years. This was done because
of fundamentally different shapes in military projectiles, such
as sharper points and boat tails. The purpose was to establish better
standards for these classes of bullet shapes. In recent years, however,
this practice has largely been abandoned in the military. With modern
instrumentation and computers, it has become possible to measure
the drag deceleration of every individual projectile type used by
the military. Thus, there is no longer a need for a standard projectile
for military applications. Or, we might say that every type of military
projectile is its own standard.
This is not true for commercial bullets,
however. Ballistic coefficients are used for all commercial bullets
for sporting and targetshooting purposes, mainly because the BC
of each is relatively easy and inexpensive to measure, compared
to measuring the drag deceleration. The standard projectile for
commercial bullets is still nearly identical to Bashforth’s
standard bullet. The standard drag model, also called the standard
drag function, for this projectile is known as G1. The BC values
quoted by all producers of commercial bullets are referenced to
G1. It is important to note that BC values quoted by commercial
producers can not be used with any drag model other than G1. It
is possible to find other standard drag models by looking up historical
military ballistics data. But, if a standard drag model other than
G1 is used, the BC values of bullets must be measured with reference
to that drag model in order to calculate accurate trajectories.
The values are likely to be very different from the values referenced
to G1.
