There are two definitions of ballistic coefficient in common
use, one produces ballistic coefficients with units and the
other produces ballistic coefficients without units. That would
seem to create a contradiction until you realize that the units
can be mathematically removed from ballistic coefficients, thus
standardizing them across all systems of measurement. The need
to standardize ballistic coefficients is selfevident given
that, except for the United States, the industrialized world
uses the Metric system.
Researchers use a definition that separates the shape of a
bullet from its mass and diameter. The shape of the bullet is
represented by the term "form factor" while the mass
and diameter are represented by the term "sectional density."
Ballistic coefficients calculated by this method have the same
units as sectional density.
Industry, for the most part, defines ballistic coefficient
as a ratio of the drag deceleration of an actual bullet as it
relates to the drag deceleration of a standard bullet without
treating shape and sectional density separately. This method,
while once very difficult to do, is now readily done with modern
instruments. For example, the Oehler Model
43 from 1992 calculated the ballistic coefficient for every
shot fired in real time using the drag deceleration method without
knowing anything about the bullet's shape. Ballistic coefficients
calculated by this method have no units.
It's important to note that ballistic coefficients with units
are equivalent to ballistic coefficients produced without units
only when ballistic coefficients have the units of lb/inē (pounds
per square inch). Converting such ballistic coefficients to
g/cmē or kg/mē results in values that can't be used with most
software, charts or graphs.
Drag Deceleration Method:
The drag deceleration definition of ballistic
coefficient is the drag deceleration of the standard bullet
divided by the drag deceleration of the actual bullet as shown
in the following equation.
The units of deceleration, such as feet/secē or
meters/secē, cancel out and BC becomes a ratio that has no units,
which is to say it's dimensionless. You can find this method
defined in industry documentation such as in Sierra reloading
manuals (2nd, 3rd, and 4th editions). Also, in his book
Modern Practical Ballistics, Arthur Pejsa defines Ballistic
coefficient as "Ratio of the retardation A of a projectile
to that of a reference projectile." Retardation is just
another term for drag deceleration.
Each standard projectile is designated by a "G"
number, so if the G7 projectile is used then the ballistic coefficient
is a G7 BC. It's easy to see from the equation that every standard
projectile has a ballistic coefficient of 1.
Form Factor Method:
Form factor is a representation of a bullet's
shape or form and is the ratio of an actual bullet's drag coefficient
divided by a standard bullet's drag coefficient as shown in
the following equation..
Drag coefficient is a dimensionless quantity as
is form factor. Each standard projectile is designated by a
"G" number, so if the drag coefficient is for the
G7 projectile then the form factor is related to that same projectile.
It's easy to see from the equation that every standard projectile
has a form factor value of 1.
Ballistic coefficient is then defined by the following
equation.
Which can be simplified into the following equation.
The resulting ballistic coefficient has the same
units as sectional density, which are lb/inē for the English
measurement system. For a standard bullet to have a BC of 1
it's sectional density must also be 1, and that only works out
when using pounds for mass and inches for bullet diameter.
It might seem odd that the scientific definition
depends on standard projectiles having a sectional density of
1 pound/inchē, but this is the legacy of Francis Bashforth of
England who was the first to propose using standard projectiles
(circa 1870). Bashforth's proposed standard bullet had a weight
of 1 pound and a diameter of 1 inch, which gives it a sectional
density of 1.0 lb/inē. Contrary to what many have been taught,
the pound is the unit of mass in the Imperial Absolute system
and the Poundal is the unit of force. That's also why lbf/s
(poundfeet/second) is the correct unit for bullet momentum.
As an example, given a bullet with a diameter
of 0.338 inches and a mass of 300 grains the sectional density
would be 0.375 lb/inē in the English system. If that bullet
has a form factor of 1.169 relative to the G7 standard projectile
then the example bullet would have a G7 BC of 0.321 lb/inē.
If using grams and centimeters in the metric system that same
bullet's sectional density would be 26.375 g/cmē and it would
have a G7 BC of 22.562 g/cmē. Using kiligrams and meters would
result in a sectional density of 263.747 kg/mē with a G7 BC
of 225.618 kg/mē.
No software designed for dimensionless ballistic
coefficients or for ballistic coefficients in units of lb/inē
would work with ballistic coefficients of 22.562 g/cmē or 225.618
kg/mē. From a software developer's standpoint there needs to
be a method of producing useful ballistic coefficients for users
of the metric system. The following equation solves that problem.
Using values from the example bullet and plugging
them into the equation for units in g/cmē gives the following.
The resulting ballistic coefficient is the same
as it was when in units of lb/inē, and because the units cancel
out, the value is now dimensionless just like the ballistic
coefficients produced by the drag deceleration method. This
simple step standardizes ballistic coefficient values across
all systems of measurement, including the English system.
Being the ballistic coefficient value in lb/inē
doesn't change when the units are mathematically removed, the
obvious shortcut is to just drop the lb/inē units, which is
not the same as ignoring them.
These dimensionless ballistic coefficient can't
be plugged back into equations and used for research purposes
without adding the units back, but other than that there seems
to be no reason for retaining units for published ballistic
coefficients. In fact, publishing ballistic coefficients with
units of lb/inē creates confusion for users of the metric system,
which is virtually everyone outside the United States. The burden
of standardizing ballistic coefficients belongs to bullet and
ammunition manufactures as well as software developers.
