2.5
Examples of Ballistic Coefficient Measurements
This subsection presents a few examples
of ballistic coefficient measurements that we have made by the methods
described in Section 2.3. As you examine the figures presented here,
please bear in mind that they are engineeringtype graphs. Also,
these examples have been selected to illustrate some of the points
made in the preceding discussions of ballistic coefficients.
Figure 2.51 shows BC measurements for
Sierra’s 6.5 mm (.264 inch) 160 grain Semipoint (SMP) bullet
versus velocity. This bullet is very long compared to its caliber,
and it has a flat base, long bearing surface and a rounded point.
Each dot on the figure is a BC measurement made by the initial velocity
and time of flight method for a round fired.
Figure 2.51. BC measurements for the
Sierra 6.5 mm 160 grain Semipoint bullet
The measurements shown in Figure 2.51
were made by reducing the cartridge powder load in each ammunition
round to achieve successively lower velocities. These measurements
were made specifically to illustrate how BC value changes with velocity
for this particular type of bullet. Our usual approach to BC measurements
is to select three or four discrete velocity levels within the appropriate
velocity range for the type of bullet being tested. For example,
for this bullet we would select one level at about 2800 fps, the
next at about 2400 fps, the next at about 2000 fps, and the final
level at about 1600 fps. Ten rounds would be fired at each of these
levels, and the average BC value and statistical variations for
that velocity level would be determined. Then, we would determine
recommended BC values, velocity subranges and subrange velocity
boundaries for the bullet type being tested.
Figure 2.51 shows convincingly that the
BC for this particular bullet type varies continuously with velocity,
increasing in value as the bullet flies downrange and its retained
velocity drops. The three velocity subranges shown in the figure
are recommended for use in computing ballistic trajectories for
this bullet type. Within each subrange a constant BC value is used,
and when the bullet velocity crosses a subrange boundary, the BC
is changed to the new value. This approach permits very accurate
trajectory computations.
The trend in BC values shown in Figure
2.51, to increase in value as velocity decreases in the range above
1600 fps, seems to be common to hollow point and blunt nose bullets.
Spitzer pointed bullets seem to have BC values
that vary little with velocity or have BC values that decrease as
bullet velocity decreases in the velocity range above 1600 fps.
Another observation from the figure is
that the scatter in magnitude of the BC value is quite small at
all velocity levels, indicating that this bullet type is highly
stable at all velocities above 1600 fps. This is generally true
of flat base bullets with long bearing surfaces. There is one point
in Figure 2.51, a low BC value at about 2200 fps, which does not
conform to this observation. Such “wild points” happen
occasionally. When the average characteristics of any bullet are
being measured, it generally is justifiable to ignore such wild
points if there are very few. If there are more than a few such
points, some investigation is necessary to determine the cause.
It is well known that bullet stability
is critical for accuracy, but it is not well understood that there
are different degrees of bullet stability. BC measurements give
us some insights into varying degrees of bullet stability. Figure
2.52 shows BC measurements for Sierra’s 22 caliber (.224 inch
diameter) 69 grain Hollow Point Boat Tail MatchKing bullet as a
function of rifling twist rate. The rifling twist rates in the test
barrels varied from one turn in 7 inches (1 x 7) to one turn in
12 inches (1 x 12), except that we did not have a test barrel with
a 1 x 11 twist rate. All BC measurements were made by the initial
velocity and time of flight method. All rounds were fired at around
2800 fps, which is about a maximum load for this bullet in the 223
Remington cartridge in a bolt action rifle. The figure shows the
number of rounds fired at each rifling twist rate and the individual
BC measurements for each group, together with the average value,
the standard deviation (SD), and the extreme spread (ES) of the
group.
Looking first at the group for the 1x
7 twist rate, the average BC values for this group of 10 rounds
is 0.297 when rounded to three significant figures, sufficient for
trajectory computations. The standard deviation (SD) of the measurements,
0.0022, is less than 1.0 % of the average BC value for the group,
and the extreme spread (ES), 0.0079, is less than 4.0% of the average
BC value. These figures illustrate the criteria that we use (SD
no more than about 1.0% of average value, and ES no more than about
5.0% of average value) to determine whether the measured data are
“good.” If either of these criteria is seriously exceeded,
we look for a reason or repeat the measurements.
The 12 round group for the 1x8 twist rate
also has an average BC value of 0.297. The standard deviation for
the group is 0.0039, which is about 1.3% of the average BC value,
and the ES, 0.0129, is about 4.3% of the average BC value. This
group obviously is not quite as “tight” as the previous
group, but we would not call this “bad” because the SD
does not seriously violate our standard deviation criterion.
The groups for the 1x9 and 1x10 rifling
twist rates also satisfy the stan
Figure 2.52. BC measurements versus barrel
twist rates for Sierra’s
dard deviation and extreme spread criteria,
but the average BC values are beginning to decrease. For the 1x9
twist rate, the average BC is 0.295, and for the 1x10 twist rate,
the average BC value is 0.294. The group for the 1x12 rifling twist
rate shows a striking decrease in average BC value and increase
in the scatter in the measurements. We attribute these changes to
a decrease in stability of the bullets fired from the barrels with
the slower rifling twist rates. We emphasize that none of the bullets
tumbled during flight; all bullets printed pointfirst on paper
targets just behind screen 3 in the test setup (see Figure 2.32).
Our interpretation of the data is as follows.
All bullets have some coning motion just after they leave the barrel,
as described in Section 2.4. When the rifling twist rate is fast
(e.g., the 1x7 and 1x8 twist rates in Figure 2.52), the coning
motion is small, and the dominant causes of the scatter in the BC
measurements are random sources of error such as those described
in Section 2.3.1.2. When the rifling twist rate in the barrel is
slower (e.g., the 1x9 and 1x10 twist rates in Figure 2.52), coning
motion increases in magnitude, and it becomes a systematic source
of BC measurement error. This systematic effect causes the average
value of the BC measurements to decrease, while the scatter in the
measurements, caused by random sources of error, does not increase
dramatically. In other words, the increased coning motion causes
all the bullets to experience increased drag, and on average, they
experience the same increase in drag, which causes a reduced average
BC for the group. The random causes of BC error are not overwhelmed
by the coning motion, so that the scatter in the BC measurements
is about the same. When the rifling twist rate is very slow
for the bullet (e.g., the 1x12 twist rate in Figure 2.52), we believe
that the coning motion increases dramatically. It certainly has
a systematic effect on measured BC, and it also has a random roundtoround
variation, which overwhelms the random errors associated with small
variations the bullet shape or construction. In this situation,
fired bullets are only marginally stable, and accuracy is usually
very poor. When long, slender, heavy bullets are used in any caliber,
fast rifling twist rates are necessary for good bullet ballistic
performance and accuracy.
Figure 2.53 shows BC measurements made
for Sierra’s 30 caliber (.308 inch diameter) 190 grain Hollow
Point Boat Tail MatchKing bullet as a function of rifling twist
rate. The twist rates in the test barrels varied from one turn in
8 inches (1x8) to one turn in 14 inches (1x14), except that we did
not have a test barrel with a 1x13 twist rate. All BC measurements
were made by the initial velocity and time of flight method. All
rounds were fired at around 2350 fps using the 308 Winchester cartridge.
Fifteen rounds were fired for each rifling twist rate.
Figure 2.53 shows the same characteristics
for the 190 grain 30 caliber bullet as were observed in Figure 2.52
for the 69 grain 22 caliber bullet. The average BC values for the
groups are relatively consistent for rifling twist rates from 1x8
through 1x11. The criteria for standard deviation and extreme spread
are satisfied very well for these groups, and the scatter patterns
are tight. The group for the 1x12 rifling twist rate has a lower
average BC value, and with the exception of one “wild”
round, the scatter pattern is tight. However, when the rifling twist
rate is 1x14, a dramatic decrease in average BC value occurs, with
a large increase in the scatter of the BC measurements. This bullet
could be used in a barrel with a 1x14 twist rate only if it were
fired at a considerably higher velocity to improve stability, such
as in one of the 300 Magnum cartridge types.
Bullet coning motions usually tend to
damp out as the bullet travels downrange. That is, the coning motion
of a bullet is largest when it leaves the muzzle and grows smaller
as the bullet flies downrange, basically because of air friction.
Some shooters refer to this effect as the bullet “going to
sleep,” and it can be observed in BC measurements. The effective
BC of a bullet is often higher if the measured range between the
initial and final chronographs (for the measurement method of Section
2.3.1) or between the initial chronograph and the time of flight
screen (for the measurement method of Section 2.3.2) is closer to
200 yards rather than 50 or 100 yards.
This effect is illustrated in Figure 2.54
for Sierra’s 30 caliber 190 grain Hollow Point Boat Tail MatchKing
bullet. Two separate sets of BC measurements for this bullet are shown
— one made by the initial velocity and time of flight method
with a 50yard measured distance between the initial chronograph and
the time of flight screen, and the other made by the initial and final
velocity method with a 250 yard measured distance between the two
chronographs.
Figure 2.53. BC measurements versus barrel
twist rates for Sierra’s .308” inch diameter 190 grain
Hollow Point Boat Tail MatchKing bullet
The two groups of measurements were made
at different muzzle velocities (about 135 fps different in average
values), but the velocities were close enough that a valid comparison
between BC values can still be made. It is evident that the average
BC value of 0.532 for the measurements made over the 250 yard distance
is almost 10% higher than the average BC value of 0.485 for the
measurements made over the 50 yard distance. This is attributed
to the coning motion damping out over the longer measurement range.
Note that the scatter pattern for the 250 yard measurements is slightly
worse than the scatter pattern for the 50 yard measurements. However,
recall that we believe the difference in the average BC values is
caused by systematic coning motions, while the scatter pattern in
each case is caused by random roundtoround variations in bullet
characteristics.
It is often neither possible nor practical
to have large measurement range distances, such as 200 yards, and
this can be a disadvantage in both of these methods of measuring
ballistic coefficients. To begin with, the downrange screens are
at greater risk of being struck by stray bullets because of aiming
errors or cartridge loading errors. If a stray bullet strikes a
screen or an electronics box, the result is both embarrassing and
expensive. The test sequence is interrupted, and a new screen must
be purchased. If a final velocity chronograph is located downrange,
it must be read for each round fired. Often this necessitates walking
the 200 yards or so downrange to read
the chronograph. Figure
2.54.
Of course, the final velocity instrument
can be placed at the firing point, but then two coaxial cables must
be routed from the final velocity screens back to the firing point
to conduct the start and stop signals for the chronograph. Electrical
pulses travel on coaxial cables at speeds of 60 to 80 percent of
the speed of light, or 0.6 to 0.8 foot per nanosecond. This may
cause a significant time delay for pulses traveling over those cables,
and the rise and fall times of the electrical pulse signals are
also lengthened. These effects can cause systematic errors in both
methods of measuring BC values when long runs of electrical cables
are necessary. When setting up to measure ballistic coefficients,
great care must be taken to minimize these effects.
Figure 2.55 shows BC measurements for
Sierra’s 22 caliber 80 grain Hollow Point Boat Tail MatchKing
bullet. Five groups of rounds were fired at muzzle velocities ranging
from about 2800 fps to about 1600 fps. Note first that this bullet
has a surprisingly high BC for a 22 caliber bullet. In fact, it
is higher than the BC values of some 30 caliber bullets with weights
up to 150 grains.
The two groups of measurements at about
2000 fps and 1600 fps have average BC values that are lower than
the measurements at the higher velocity levels. Also, the scatter
pattern of the group fired at about 1600 fps is somewhat larger
than the scatter patterns of the groups fired at higher velocities.
We believe that these effects are caused by coning motions of the
bullets. The rifling twist rate in the barrel (1x7) was just not
fast enough to well stabilize the bullets fired with muzzle velocities
near 2000 or 1600 fps. The increased coning motion at 2000 fps causes
a systematic decrease in the BC, while the coning motion at 1600
fps is severe enough to cause both a systematic decrease and random
variations in the BC values. This illustrates a disadvantage of
measuring ballistic coefficients using the initial and final velocity
method or the initial velocity and time of flight method. The only
way to get BC measurements at low bullet velocities with either
of these methods
is to fire the bullets at low muzzle velocities
where rifling twist rates are not fast enough to stabilize the bullets
sufficiently well to get highly accurate measurements.
When ballistic coefficients can be measured
by the method described in Section 2.3.3 using a Doppler radar system,
the disadvantages of the other two methods are completely avoided.
The measurements are more accurate and complete, and important characteristics
of ballistic coefficients are fully revealed. Figures 2.56 and
2.57 are BC measurements made using the Doppler radar at the Yuma
Proving Ground for two Sierra bullets, the .338 inch diameter 300
grain MatchKing and the .224 inch diameter 77 grain MatchKing. [Note
in both these figures that the velocity axis has been reversed from
the previous graphs. Bullet velocity starts at a high value at the
left end of the axis and decreases toward the right end of the axis.]
The 338 MatchKing rounds were fired at about 2950 fps in 338378
Weatherby cartridges at an elevation angle of 20 degrees at the
firing point. Each round was tracked downrange until each bullet
was “lost” by the radar as it sank into ground clutter
(low brush and other objects interfering with radar signal trans
Figure 2.56. BC measurements
by the Doppler radar method for
mission/reception). This occurred when
the velocity of each bullet was about 700 fps, well below the speed
of sound. The BC values were calculated for three rounds and are
plotted in Figure 2.56. The dots in the graph generally indicate
very close BC measurements for all three bullets, except where the
dots are separated a small distance, where they indicate values
for individual bullets. The vertical bars indicate scatter in the
BC values for the three rounds where these BC values were calculated
at the same velocity. The BC values shown in the figure are typical
for all the test rounds fired with this bullet.
For the 22 caliber 77 grain MatchKing
in Figure 2.57, all rounds were fired at about 2600 fps in 223
Remington cartridges at about 20 degrees elevation angle. Each round
was tracked until velocity fell to about 600 fps, where the radar
signal was lost in ground clutter. Again, BC values for three rounds
were calculated and plotted in Figure 2.57, and the vertical bars
indicate the scatter in BC values for the three bullets. The BC
values shown in the figure are typical for all the test rounds fired.
The Doppler radar method of BC measurement
is clearly the best for several reasons. First, each bullet is fired
at the maximum muzzle velocity obtainable from the gun and cartridge,
and then is observed by the radar almost throughout its entire flight.
There is no need to download cartridges to measure BC values at low
velocity and suffer the errors caused by reduced bullet stability,
in turn caused by the reduced spin rate. A second significant reason
is that each bullet can be allowed to travel downrange from the
Figure 2.57. BC measurements by the Doppler
radar method for Sierra’s .224 inch diameter 77 grain Hollow
Point Boat Tail MatchKing bullet
muzzle for 150 or so yards before BC measurements
begin, so that the initial coning motion of the bullet at the muzzle
can damp out or at least damp to its minimum value. BC measurements
can then be computed for each round from the radartracking data
as frequently as desired along the bullet trajectory. The third
major reason is that each bullet can be observed throughout its
range of velocities, as it slows from supersonic velocities through
transonic velocities, through the speed of sound (about 1120 fps),
and then on down to low subsonic velocities.
Figures 2.56 and 2.57 show some remarkable
BC characteristics for these rifle bullets. At supersonic velocities,
the BC of each type of bullet is nearly constant, showing that the
G1 drag model is appropriate for these sporting bullets in this
velocity range. When bullet velocity falls below about 1600 fps
in the transonic velocity range, the BC of each bullet type decreases
dramatically. A minimum BC value is reached just above the speed
of sound. A dramatic increase in BC value occurs just below the
speed of sound. A maximum BC value is reached when the bullet velocity
is about 1000 fps, and then the BC value decreases as bullet velocity
falls to lower subsonic levels.
A similar type of BC variation has been
observed for handgun bullets. An
Figure 2.58. BC measurements by the initial
velocity and time of flight method for Sierra’s .44 caliber
240 grain Jacketed Hollow Cavity Sports Master handgun bullet
example is shown in Figure 2.58 for Sierra’s
44 caliber (.4295 inch diameter) 240 grain Jacketed Hollow Cavity
Sports Master bullet. The BC for this stubby, hollowpoint bullet
behaves differently compared to the rifle bullets in Figures 2.56
and 2.57. It rises dramatically just above the speed of sound,
falls dramatically just below the speed of sound, and then rises
to a peak value at about 1050 fps, decreasing from this peak at
lower subsonic velocities. These measurements were made by the initial
velocity and time of flight method. This same ballistic coefficient
behavior has been observed for Sierra’s 9mm 115 grain Full
Metal Jacket Tournament Master bullet, and for a 41 caliber 220
grain Full Patch Jacketed bullet (no longer in production). This
behavior appears to be characteristic of many, if not all, handgun
bullets.
As mentioned earlier, this ballistic coefficient
behavior implies that the G1 drag model does not characterize sporting
bullets for rifles or handguns very well at velocities lower than
about 1600 fps. This in turn means that we cannot calculate highly
accurate trajectories for bullets at low velocities. This situation
is somewhat mitigated by the fact that the total aerodynamic drag
on a bullet decreases dramatically as bullet velocity falls through
the speed of sound to subsonic velocity levels.
This is an area of intensive research
by these authors. We are privileged to have access to Doppler radar
tracking data for a large number of bullets tested at the Yuma Proving
Ground, through the courtesy of the YPG and the Association of Firearm
and Toolmark Examiners. Our research has two prime objectives. The
first is to better understand BC measurement techniques. We must
find a way to measure bullet ballistic coefficients in Sierra’s
test range with the limited capabilities of that approach, and then
to correct those measurements to true bullet BC values based on
what we learn from the Doppler radar method. The Doppler radar method,
while clearly the best method, has overwhelming practical disadvantages
that prevent its use by commercial bullet manufacturers such as
Sierra. The cost of the instrumentation is several hundred thousand
dollars, a trained crew is necessary to operate and maintain the
radar, and a test range several miles in length is necessary for
testing bullets. Our approach is to use the other two methods of
BC measurement in Sierra’s test range and to use the Doppler
radar data for selected bullets to learn how to interpret the test
data from Sierra’s range to obtain true values of bullet BC.
The second prime objective of our research
is to determine a modification of the G1 drag function for velocities
below 1600 fps. This change must make the modified G1 function better
characterize rifle and handgun bullets at low velocities, so that
BC values referenced to this modified function do not change so
radically with bullet velocity. Then, we can compute accurate longrange
trajectories for rifle and handgun bullets traveling at low velocities.
