2.3.1
Initial Velocity and Final Velocity Method
2.3.1.1 Measurement Procedure
This first method is illustrated in Figure
2.31. This method uses two chronographs for each bullet fired to
measure an initial velocity and a final velocity at a measured range
distance between the chronographs. The initial velocity chronograph
is usually placed near the muzzle of the gun, as shown in Figure
2.31. A blast shield with a small hole for bullet passage usually
is used to keep the muzzle flash or blast from disturbing the screens
of the initial velocity chronograph. If the screens are photoelectric
types, as is the usual case, the muzzle flash may trigger screen
1 before the bullet arrives. Also, because the powder gases exit
the muzzle at about 1.5 times the bullet velocity, the gases can
trigger screen 1. Or, the muzzle blast can cause screen 1 or 2 to
bend or vibrate. Any of these effects will cause an erroneous measurement
of the initial velocity, which will, in turn, cause an error in
the measured BC value.
The final velocity chronograph is placed
downrange at a carefully measured distance from the initial velocity
chronograph. This range distance is measured from the center point
between screens 1 and 2 to the center point between screens 3 and
4. This is because each chronograph really measures the bullet travel
time between the two screens to which it is connected (i.e., between
screens 1 and 2 or between screens 3 and 4). Then, the velocity
is obtained by dividing the precisely measured distance between
the pair of screens by the measured bullet travel time. This calculation
is performed within the electronics of the chronograph. Because
the bullet slows down a tiny bit as it travels between the two screens,
the measurement of velocity is considered to be valid at the center
point between the pair of screens. Of course, if the separation
distance between screens 1 and 2 is the same as between screens
3 and 4, the range distance between the two chronographs may be
measured from screen 1 to screen 3.
The measurement procedure for each bullet
fired is to record the initial and final velocities as well as the
range distance between the two chronographs. The altitude, temperature,
barometric pressure, and relative humidity at the firing point must
also be recorded. If the shooting range is not level, the elevation
angle must be recorded, especially if it exceeds about 3 degrees
either upward or downward. If there is any appreciable wind at the
firing point, BC measurements should not be attempted.
With these data for each bullet fired,
an exterior ballistics software program for a personal computer
can be used to calculate the ballistic coefficient. Some exterior
ballistics programs contain an optional routine for computation
of the BC value, but that is not necessary. The normal trajectory
computation routine can be used in an iterative fashion for each
bullet fired. That is, first initialize the program by entering
the altitude, temperature, barometric pressure, humidity and range
elevation angle at the firing point. Then, for each round fired
enter the measured initial velocity as the “muzzle velocity”
for the trajectory calculation. Then, guess a BC value, calculate
a trajectory over the measured range distance, and examine the calculated
final velocity. If the calculated final velocity is higher
than the measured number, the BC
value is too high.
(Conversely, if the calculated final velocity is lower
than the measured value, the BC value
is too low.)
Then, reduce (or increase) the BC value a little, calculate another
trajectory over the measured range distance, and examine the calculated
final velocity. The calculated final velocity should be nearer the
measured value. If the calculated value from this second iteration
is higher (or lower) than the measured value, reduce (or increase)
the BC value in the program and perform another iteration of the
calculations. After a few iterations, this method will “home
in” on a correct value for the measured BC for the first round
fired. Of course, this is a BC value for which the calculated final
velocity matches the measured final velocity as closely as possible.
This resulting BC value is considered valid for a bullet velocity
midway between the initial and final velocities for that round.
For the other rounds fired, the resulting BC for the
first round can be used as the initial guess for the BC value, and
fewer iterations will be required to reach a correct BC value for
each of those rounds.
Figure 2.31 Test Range Setup for Initial
and Final Velocity Method for BC Measurement
An example
has been prepared to illustrate this method of determining the BC.
Suppose we have developed a load for the 308 Winchester (7.62 x
51 mm NATO) cartridge in a boltaction rifle that pushes a certain
.308 diameter 160 grain bullet at
a muzzle velocity of about 2750 fps. We do not know the BC of this
bullet type and want to measure it on our local shooting range.
(Actually, we do not need to know the weight of the bullet or even
the caliber to determine the BC. We need only the firing test data
as described below.) Suppose that our shooting range is located
at an altitude of 790 feet above sea level, and we perform the shooting
tests on a day when the temperature is 78° F, the barometric
pressure is 30.15 inches of Mercury, and the relative humidity is
80%. Note that the barometric pressure is obtained from a barometer
at the range or from a local weather report for the time of day
when the firing tests take place. The test range is level, and the
range distance between chronographs is 103 yards, which is measured
precisely and accurately when we set up the range for the tests.
We fire, say, ten rounds to obtain an average BC value for this
bullet type at velocities in the vicinity of 2750 fps. We record
the altitude, atmospheric conditions, range distance between the
chronographs, and the initial and final velocities for each round
fired. Then, we retire to our computer at home. We start up the Sierra Infinity
program, and it comes up automatically
in the “Trajectory” mode of operation. Suppose that for
the first round fired at the range, the initial velocity was 2742
fps and the final velocity was 2549 fps, as read from the initial
and final velocity chronographs. In the Infinity
program we select any 30 caliber
bullet in the “Load Bullet” library, and transfer it to
the “Active Bullets” list in the upper right corner of
the blank part of the screen. Then, we initialize the trajectory
computation as follows:
Trajectory Parameters
Units: Full English (since we are working
in the English system of units) Muzzle Velocity: 2742 fps (for the
first round fired) Maximum Range: 103 yds (this is as far as we
need the trajectory to be computed) Range Increment: 1 yd (because
the distance between chronographs is 103 yds, which is not divisible
by any number other than 1) Zero Range: 103 yds (the distance between
chronographs) Elevation Angle: 0 (because the test range is level)
Sight Height: 1.5 inches (choice for telescope sight on the rifle)
Environment Parameters
Barometric Pressure: 30.15 in Mercury
(from a barometer at the range or a local weather report) Temperature:
78° F (from a thermometer at the range or local weather report)
Altitude: 790 ft (can be obtained from a topographical map or other
source) Humidity: 80 % (relative humidity from a weather station
at the range or from the local weather report) Wind Direction: Any
number between 0 and 12 o’clock is OK
Horizontal Wind Velocity: 0 mph (no wind
is very important) Vertical Wind Velocity: 0 mph (no wind is very
important) At this point we have initialized a trajectory computation
for some 30 caliber bullet (we don’t care which one) with the
correct muzzle velocity, range distance between the chronographs,
trajectory calculation parameters for our purposes, and environmental
conditions at the firing point. But, we haven’t performed any
trajectory computation yet, so there is nothing on the monitor screen
yet. Now, although we will not use it explicitly, we must “Calculate”
a trajectory so that the “Trajectory Variations” menu
item will be available to us. We then go to the Infinity
toolbar at the top of the monitor
screen and select “Trajectory Variations.” From the dropdown
menu that appears, we select “Ballistic Coefficients.”
In the sidebar at the right side of the monitor screen, we then
see five values of ballistic coefficient listed. These values mean
nothing to us since they are for the bullet that we chose to load
into the “Active Bullet” list, not for the bullet that
we are testing.
It is necessary now to make an initial
guess for the BC value of our test bullet. If we guess well, we
will not have to make many computation iterations to find the correct
BC value. For a 30 caliber bullet that weighs 160 grains and has
a Spitzer (sharp pointed) shape, the BC value at around 2600 fps
should be somewhere near 0.5. So, let us choose this value as the
initial guess. We then change the five numbers in the righthand
sidebar on the monitor screen to the value 0.5.
We change all the BC numbers for a particular
reason. As we will explain later, Infinity
allows the ballistic coefficient
of each bullet type to change with bullet velocity as it flies downrange
and slows down. This is because the measured BC of a bullet does
change with velocity, and accounting for such changes can increase
the accuracy of trajectory computations within Infinity.
We use five velocity regions for
this purpose. Within each velocity region there is a single value
of BC valid for that region, and there is a value for each of the
five regions. There are then four velocity boundaries separating
these regions. When the velocity of a bullet falls through one of
these boundaries, Infinity automatically
changes the BC to the value for the new region. In our current case,
we do not know whether our test bullet starting at 2742 fps and
ending up at 2549 fps crosses a velocity boundary for the bullet
we are using.Yes, we could look to see and make a more educated
selection of the one or two BC values that we would need to change,
but if we change all five of the values in the sidebar we will be
sure to be safe.
After we change the BC numbers in the
sidebar to 0.5, we are ready to begin the iterative search procedure
for the correct BC value for the first round fired. Table 2.31
summarizes the computations in the search procedure. To begin the
procedure, click the “Calculate” button on the bottom
of the monitor screen. Infinity performs
the first trajectory computation, the trajectory parameters appear
on the screen, and we immediately scroll down to the final parameter
values at 103 yds range. We find that the computed final velocity
at 103 yds is 2564.8 fps for this first iteration (see Table 2.31).
This is higher than the measured final velocity for this round 2549
fps, so the next guess for BC needs to be lower than 0.5.
At this point we have no idea how much
to lower the next BC guess, but let’s try 0.4. We set the five
BC numbers in the sidebar to 0.4 and click the “Calculate”
button for the second trajectory computation. The computed final
velocity at 103 yds for the BC equal to 0.4 is 2521.5 fps, which
is too low compared to the measured 2549 fps. So for the third iteration,
the guess for BC needs to be raised. Let’s try something halfway
between 0.4 and 0.5; that is 0.45. We change all five BC numbers in the sidebar
to 0.45 and click the “Calculate” button for the third
trajectory computation. For this third iteration, we find that the
computed final velocity at 103 yds is 2545.5 fps, which is closer
but still lower than the measured 2549 fps. So, for the next iteration
let’s raise the BC guess to 0.46.
Again, we change all five BC numbers in
the sidebar to 0.46 and click the “Calculate” button for
the fourth trajectory computation. For this fourth iteration, we
find that the computed final velocity at 103 yds is 2549.7 fps (as
shown in Table 2.31). This is just a little higher than the measured
2549 fps. So, for the next iteration we must lower the BC guess
just a little.
Table 2.31 shows the final three iterations,
each of which follows the same procedure. In each iteration we change
the BC guess by a smaller amount so that the computed final velocity
approaches the measured final velocity. The seventh iteration, which
has a BC of 0.4583, produces a final computed velocity equal to
the measured velocity, and this BC therefore is the correct value
for this first fired round.
Table 2.31 Example of Ballistic Coefficient
Iterative Search Procedure
Test Range Parameters:
Distance between chronographs: 103 yds
Range altitude: 790 ft above sea level Temperature: 78º F
Barometric pressure: 30.15 in Mercury
Relative humidity: 80% Exterior Ballistics Program: Sierra infinity
Test Round 1: Initial velocity 2742 fps;
final velocity 2549 fps
Iteration 

BC value 

Computed final velocity 

1 

0.5 

2564.8 

2 

0.4 

2521.5 

3 

0.45 

2545.5 

4 

0.46 

2549.7 

5 

0.458 

2548.9 

6 

0.459 

2549.3 

7 

0.4583 

2549.0 

Test Round 2: Initial velocity 2751 fps;
final velocity 2556 fps
1 

0.4583 

2557.6 

2 

0.457 

2557.1 

3 

0.455 

2556.2 

4 

0.454 

2555.8 

5 

0.4545 

2556.0 

Table 2.31 also shows the iterations
for the second fired round. In this example we suppose that the
second round has a measured velocity of 2751 fps and a measured
final velocity of 2556 fps. To initialize for the second round,
we momentarily return to the “Operations” selection on
the Infinity toolbar
at the top of the monitor screen, select the “Trajectory”
mode of operation, and change the “Muzzle Velocity” entry
in the “Trajectory Parameters” sidebar to 2751 fps. Again,
we must “Calculate” so that the “Trajectory Variations”
menu item is available. Then, we return to the “Trajectory
Variations” selection on the Infinity
toolbar, again select “Ballistic
Coefficients” on the dropdown menu. We verify that all five
entries in the BC sidebar on the monitor have the value 0.4583 from
the first round. Note that we have not changed any of the other
Trajectory Parameters or Environment Parameters, since they all
have the same values for our firing tests.
Table 2.31 shows the sequence of iterations
for the second round. The first iteration with a BC guess of 0.4583
produces a computed final velocity of 2557.6 fps, higher than the
measured final velocity of 2556 fps. The second iteration with a
BC guess of 0.457 produces a computed final velocity of 2557.1,
closer but still higher than the measured 2556 fps. The third iteration
with a BC guess of 0.455 produces a computed final velocity of 2556.2
fps, even closer but still a little higher than the measured 2556
fps. The fourth iteration with a BC guess of 0.454 produces a computed
final velocity of 2555.8 fps, which is lower than the measured 2556
fps. Since the results of the third and fourth iterations equally
straddle the measured 2556 fps, the fifth BC guess is chosen as
0.4545, halfway between 0.455 and 0.454. This final iteration produces
a computed final velocity equal to the measured 2556 fps, so this
value is the correct value for the second bullet fired.
The same procedure should be followed for each of
the remaining eight bullets in the test series that we fired at
the test range. In this way, we will derive the measured BC values
for all ten bullets and can apply statistical analysis to this limited
sample of test bullets for this type of bullet at velocities between
the initial and final values.
The computations in this example have
been explained in some detail, so that the reader can repeat these
calculations step by step if he or she uses Sierra’s Infinity
program. If a different exterior
ballistics program is used, the detailed steps of the procedure
should be changed because any other program will function a little
differently, but the basic method will not change. The idea is to
find a BC value that makes the computed final velocity equal to
the measured final velocity for each bullet tested. This will be
an iterative, trial and error procedure. This example will still
be useful as a guide even if a different exterior ballistics program
is used, because BC values very close to the ones produced by using
Infinity should
result. This can serve as a check on the procedure developed for
any other program.
