4.3
**
How The Ballistic Coefficient is Measured by Firing Tests
**

**
**
**
Figure 4.3-1
**
is a schematic diagram of the firing test setup that we use. The firing test method requires simultaneous measurement of muzzle velocity and time of flight to a measured range, for each round fired. From these three measured parameters the ballistic coefficient of each individual bullet fired can be calculated. We cannot use the alternative method of first measuring the average muzzle velocity of, say, ten shots, and then measuring the average time of flight of another ten shots, and use these averaged results to calculate an “average” ballis dispersion of the measured ballistic coefficients, and we therefore want to know the standard deviation and extreme spread, as well as the average value of the measurements.

Consequently, the test setup uses three photoelectric screens to trigger two chronographs, one which measures muzzle velocity and the other which measures time of flight of each round fired, as shown in
**
Figure 4.3-1
**
. The signal from Screen 1 starts both chronometers (clocks); the signal from Screen 2 stops the muzzle chronometer; which then reads out the time
**
t
**
**
12
**
that the bullet used to travel the distance
**
d
**
**
12
**
between Screen 1 and Screen 2; and the signal from Screen 3 stops the time of flight chronometer, which then reads out the time
**
t
**
**
13
**
that the bullet used to travel the distance
**
d
**
**
13
**
between Screen 1 and Screen 3. Screen 1 is positioned about ten feet from the muzzle of the test firearm. A sheet of half-inch plywood is used as a blast shield in front of Screen 1. The bullet passes through a small hole in the blast shield. When a round is fired, powder gases exit the muzzle at a velocity approximately 50 percent greater than the bullet velocity. The blast shield is to prevent these gases from triggering Screen 1 before the bullet reaches the screen, which is a circumstance that we have experienced. We have also experienced the situation where the muzzle flash prematurely triggers Screen 1. This occurs quite regularly with blackpowder firearms and occasionally with handguns using smokeless powder loads. Needless to say, premature triggering of Screen 1 gives erroneously low muzzle velocities and erroneously long times of flight.

In Sierra’s test range screen 2 is located ten feet downrange from Screen 1, so

**
d
**
**
12
**
**
= 10.0 (feet)
**

This distance is measured very accurately to assure highly accurate muzzle velocity measurements. Muzzle velocity for each round is calculated by dividing the distance
**
d
**
**
12
**
by the measured time
**
t
**
**
12
**
. Mathematically,

The quantity
**
u
**
**
o
**
is called muzzle velocity. Precisely speaking,
**
u
**
**
o
**
is the measured velocity at a point halfway between Screen 1 and Screen 2, which is also about fifteen feet downrange from the gun muzzle. This is why we say that Sierra’s muzzle velocities are always referred to a point fifteen feet downrange from the muzzle. This same point (halfway between Screen 1 and Screen 2) is also the origin of coefficient calculation, because muzzle velocity is one of the needed parameters and
**
u
**
**
o
**
is valid at this particular point. Therefore, we must correct the other two measured parameters, range and time of flight, to this origin. The corrected range is the distance
**
d
**
**
13
**
less half of
**
d
**
**
12
**
.

and the corrected time of flight is
**
t
**
**
13
**
less half of
**
t
**
**
12
**
:

With these three parameters,
**
u
**
**
o
**
,
**
R
**
, and
**
t
**
**
f
**
, we can calculate the ballistic coefficient of that particular bullet.

Before explaining how that calculation proceeds, let us first comment about measurement accuracy. Screen 3 is movable and we adjust the distance
**
d
**
**
13
**
in order to maintain time
**
t
**
**
13
**
in the neighborhood of 50 milliseconds (a millisecond is one thousandth of a second). For rifle bullet velocities around 3000 fps,
**
d
**
**
13
**
is set at 150 feet; for pistol bullets around 1000 fps,
**
d
**
**
13
**
is set at 50 feet; and so forth. The chronometers read out in units of microseconds (one millionth of a second), with a precision of plus-or-minus a half microsecond. The time of flight chronometer then measures a 50 millisecond time with a precision of one part in 100,000 which contributes an entirely negligible error in measured ballistic coefficient. For the distance
**
d
**
**
13
**
, we maintain a measurement error of one part in 10,000 which converts to 3/16 inch in 150 feet or 1/16 inch in 50 feet. The inaccuracy in the
**
d
**
**
13
**
measurement then contributes a negligible error in measured ballistic coefficient.

The error in ballistic coefficient is then caused almost entirely by the muzzle chronograph, and there are four contributing sources. First is the imprecision of the muzzle velocity chronometer, and this source is no worse than one part in 5000. Then there is the error in measuring distance
**
d
**
**
12
**
, and this is held to 1/16 inch, or one part in 2000. The other two error sources are the uncertainties in position of the bullet in its passage through Screen 1 and Screen 2 when each screen senses the bullet presence and triggers. Although we have tried many experiments, there is no simple way of determining how large these uncertainties are. However, from a combination of analysis and experimentation, we believe that the total error in the measured ballistic coefficient is no larger than two percent.

Now, turning to the calculation of ballistic coefficient from measured values of muzzle velocity
**
u
**
**
o
**
, range
**
R
**
, and time of flight
**
t
**
**
f
**
, we will explain this calculation using the
**
S
**
and
**
T
**
functions associated with the Ingalls Tables. Our computer program for this calculation uses computational algorithms which we consider to be more exact that the
**
S
**
and
**
T
**
functions, but which are difficult to explain without calculus. The method we use, however, is exactly the same as the method presented in the following explanation. Using the
**
S
**
and
**
T
**
(Space and Time) functions just simplifies the explanation. The Ingalls Tables and the
**
S
**
and
**
T
**
functions are well presented in Hatcher’s Notebook.

The range
**
R
**
can be expressed using the
**
S
**
function by the following equation (for level fire):

**
R = C [S(u
**
**
o
**
**
) - S(u
**
**
f
**
**
)]
**

where:

**
R
**
is the measured range in feet.

**
C
**
is the bullet ballistic coefficient.

**
u
**
**
o
**
is the measured muzzle velocity in fps.

**
u
**
**
f
**
is the final velocity of the bullet when it passes through Screen 3 (which is not known at this time).

**
S(u
**
**
o
**
**
)
**
is read as “the
**
S
**
function value at the muzzle velocity
**
u
**
**
o
**
,” and this value would be found in the Ingalls Tables beside the numerical value of
**
u
**
**
o
**
in the velocity column.
**
S
**
values are in units of feet.

**
S(u
**
**
f
**
**
)
**
is read as “the
**
S
**
function value at the final velocity
**
u
**
**
f
**
,” and it also would be found in the Ingalls Tables if
**
u
**
**
f
**
were known.

The time of flight
**
t
**
**
f
**
can be expressed using the
**
T
**
function by a similar equation:

**
t
**
**
f
**
**
= C[T(u
**
**
o
**
**
) - T(u
**
**
f
**
**
)]
**

where
**
t
**
**
f
**
is the measured time of flight

**
T(u
**
**
o
**
**
)
**
is “the
**
T
**
function value at the muzzle velocity
**
u
**
**
o
**
,” in seconds.

**
T(u
**
**
f
**
**
)
**
is “the
**
T
**
function value at the final velocity
**
u
**
**
f
**
” in seconds.

and
**
C
**
,
**
u
**
**
o
**
, and
**
u
**
**
f
**
are defined above.

At this point we have two equations (which express two physical conditions, one for range and the other for time of flight), but each equation contains two unknown parameters,
**
C
**
and
**
u
**
**
f
**
. In principle, there is enough known information to solve these equations simultaneously for both unknown quantities. As a first step, we form the ratio.

This new parameter
**
V
**
, which has units of velocity, is known because
**
R
**
and
**
t
**
**
f
**
are known. Notice, however, that forming this ratio has eliminated the ballistic coefficient
**
C
**
from the equation, so that this equation contains only one unknown parameter
**
u
**
**
f
**
. This equation, though is nonlinear. It is quite easy, however, to solve the equation graphically, and
**
Figure 4.3-2
**
illustrates this technique.

**
Figure 4.3-2
**
is a cartesian graph with velocity
**
u
**
plotted along the horizontal axis and the ratio

plotted along the vertical axis. The horizontal axis begins at the value
**
u
**
**
o
**
(the muzzle velocity), and
**
u
**
decreases in magnitude as we move to the right. If
**
u
**
**
o
**
is 3000 fps, for example, the value 2500 fps might be near the arrowhead at the right end of the horizontal axis. The value
**
u
**
**
o
**
also appears on the vertical axis near the top, and the value 2500 fps might be down near the intersection with the horizontal axis. The value
**
V = R/t
**
**
f
**
is located on the vertical axis, and a straight horizontal line is drawn through this point as shown in
**
Figure 4.3-2
**
. Next, a velocity value
**
u
**
**
1
**
, is chosen and the ratio

is calculated. Remember that, knowing
**
u
**
**
o
**
and
**
u
**
**
1
**
, we can simply look up values of
**
S(u
**
**
o
**
**
)
**
,
**
S(u
**
**
1
**
**
)
**
, and
**
T(u
**
**
o
**
**
)
**
, and
**
T(u
**
**
1
**
**
)
**
in the Ingalls Tables. The resulting ratio number (together with
**
u
**
**
1
**
) locates a point on the graph. This is the x-point in
**
Figure 4.3-2
**
directly above the velocity value
**
u
**
**
1
**
.

We can then choose another value
**
u
**
**
2
**
, calculate the corresponding ratio value, and plot that value as the x-point above
**
u
**
**
2
**
in the figure. Clearly, we can continue this process for any number of such points, as the figure shows. We can draw a smooth curve joining the x-points as shown. The desired solution
**
u
**
**
f
**
is the velocity coordinate where the ratio curve intersects the straight line, because this is the value of velocity which makes the ratio function equal to the
**
V
**
parameter. Mathematically, the ratio curve in
**
Figure 4.3-2
**
is a monotonically decreasing function. This assures us that there is a single unique solution for
**
u
**
**
f
**
, so when we find this intersection point in
**
Figure 4.3-2
**
, it is the correct solution.

Once the final velocity
**
u
**
**
f
**
is known {and therefore
**
S(u
**
**
f
**
**
)
**
and
**
T(u
**
**
f
**
**
)
**
} then the ballistic coefficient can be calculated from either of the following equations:

Let us point out also that the parameter
**
V
**
is not the average of the muzzle velocity
**
u
**
**
o
**
and the final velocity
**
u
**
**
f
**
, but it is very close to that value, particularly for times of flight
**
t
**
**
f
**
near 50 milliseconds. Consequently, a good first guess in the iterative calculation procedure described above is

**
u
**
**
1
**
**
= 2V - u
**
**
o
**

This greatly reduces the number of x-point calculations needed, since we are only interested in that part of the ratio curve near the intersection point in
**
Figure 4.3-2
**
.

The reader is probably wondering at this point why we do not measure the final velocity rather than time of flight, because if we measured muzzle velocity
**
u
**
**
o
**
, final velocity
**
u
**
**
f
**
, and range
**
d
**
**
13
**
, we could calculate measured ballistic coefficient without resorting to all the complex manipulations described above to find
**
u
**
**
f
**
, and it would entail using just one more screen, Screen 4, located ten feet downrange from Screen 3. A little earlier we made the point that the measurement error in the ballistic coefficient is driven mainly by error sources associated with the muzzle velocity chronograph, and that the errors associated with measuring distance
**
d
**
**
13
**
and time
**
t
**
**
13
**
have negligible effects on ballistic coefficient accuracy in our method. If instead we were to use a final velocity chronograph, then error sources associated with that chronograph would also drive ballistic coefficient accuracy, and the total error in ballistic coefficient could be significantly larger. For this reason we chose not to use the alternative method involving the final velocity measurement.

Of course, our ballistic coefficient measurements never take place at sea level standard atmospheric conditions, so we must convert the measured results to those conditions. Sierra’s test range is almost exactly 150 feet above sea level, and temperature, barometric pressure, and relative humidity are always recorded for each test series. Then, we use the method explained in
**
Section 5.1
**
of this Manual to make the conversion. Our measured ballistic coefficient value is the “equivalent” ballistic coefficient of that section; we calculate the altitude, temperature, pressure, and relative humidity correction factors
**
F
**
**
A
**
,
**
F
**
**
T
**
,
**
F
**
**
P
**
, and
**
F
**
**
RH
**
; and then the sea level standard ballistic coefficient
**
C
**
o is calculated from