Accurate Specifications of Ballistic Coefficients
Bryan Litz
Ballistician, Berger Bullets
If the title sounds strangely familiar to you, it’s because the January issue of Varmint
Hunter had an article titled “Inaccurate Specifications of Ballistic Coefficients” by Michael and
Amy Courtney where measured Ballistic Coefficient (BC) data for several bullets was presented.
If you have it handy, I suggest reviewing that article before reading this one. As the Ballistician
of Berger Bullets, it’s my responsibility to establish the BC’s of our bullets and so Michael and
Amy’s article was very interesting to me, enough so to warrant a response.
I would like to applaud Michael and Amy for their experiments to determine BC. I agree
that the manufacturers claimed values are often in error and can lead to inaccurate trajectory
predictions which ultimately results in missing targets. It was this same situation that drove me
to develop a method for experimentally measuring BC’s for myself several years ago. The
method I developed has matured into the process that is now used to experimentally
determine the official BC’s of Berger Bullets. This article is a brief glimpse into the challenges of
measuring BC, and how the effective BC might be different when fired from different rifles.
I’m not simply responding in defense of my company’s claims. In fact, we know that the
advertised BC of the particular Berger Bullet tested by Michael and Amy Courtney (the .257
caliber 115 grain VLD) was too high, and we’ve taken steps to fix it. As part of a wholesale
reassessment of BC’s for Berger bullets, the BC of this and other bullets have been adjusted
based on experimental testing. Rather than focusing on whose results are ‘more accurate’, the
intent of this article will be to explore some various approaches used to measure the BC’s of
bullets and the unique challenges of each. Unfortunately, due to the broad range of material
covered, I won’t be going into as much detail as some of the subjects’ merit.
My Basic Procedure for Measuring BC
The basic procedure that I use to measure BC’s is shown schematically in Figure 1. The
bullet is fired and its initial velocity is measured with a chronograph. The first acoustic sensor
(microphone) is located at the chronograph, and 3 other sensors are placed in 200 yard
intervals downrange out to 600 yards (4 sensors in all). As the bullet flies past each sensor, the
supersonic ‘crack’ of the bullet registers and is recorded to a single audio file which is
essentially a ‘time stamped’ trajectory for each shot. Typically 5 shots per bullet type are
averaged to arrive at a BC. Together with the muzzle velocity, atmospheric conditions, and
information about the proximity of the sensors to the trajectory, this raw data can be used to
determine the BC of a bullet with great accuracy and precision.
My method differs from the Courtney’s in two fundamental ways. We both measure
the bullets initial velocity, but I’m measuring downrange time of flight whereas they measure
downrange velocity. The ‘two chronograph’ method used by the Courtney’s is inherently more
accurate, all else being equal. However, as is usually the case, all else is not equal. My tof
measurements span 600 yards, whereas the Courtney’s test spans a distance of only 97 yards.
Extending the range between sensors reduces the error caused by uncertainty in the
measurement of distance, time and velocity.
The instrumentation above produces the sound file below for each shot.
The bullets time of flight in each 200 yard segment of it’s trajectory is determined within +/- 0.5 ms and used to
determine the BC, and how the BC changes with velocity.
The other way in which our tests differ is that I’m measuring the bullets flight over
several segments of range, whereas the Courtney test measures the bullets flight over only one
interval. Measuring the bullets flight in several segments allows one to measure how BC
changes with velocity, which will be discussed later.
Now that the basic ideas of the test procedures have been presented, we turn our
attention to the uncertainties involved in such a test, and their impact on the measured BC.
Measurement Uncertainties
The fundamental concept of addressing measurement uncertainty in an experimental
set-up is that you want to make sure that the uncertainty of the measurement is small in
comparison to its magnitude. For example, if one were to attempt to measure the velocity loss
of a bullet over only 10 yards (which is only about 20 fps) with chronographs that have a
claimed error of +/- 3 fps, there would be a very large amount of uncertainty in the
measurement of velocity loss. However, if you place the same two chronographs 100 yards
from each other, the same +/- 3 fps of uncertainty in velocity measurement results in much less
error in BC because the bullet looses about 200 fps in 100 yards. The same principle holds for
range uncertainty. If the chronographs are placed 100 yards apart and there is +/- 1 yard of
uncertainty, there will be more error in the calculated BC than if the chronographs were placed
200 yards apart with the same +/- 1 yard of uncertainty.
I have spoken with the Courtneys about their procedures, and it was no surprise to me
that they carefully addressed the measurement uncertainties involved in their experiments to
measure BC (they both hold PhD’s, Michaels is in experimental physics from MIT!). Although
they spared the readers the scientific details in their article, we can rest assured that the
experiment was well executed. For example, they did not overlook the step of calibrating their
chronographs by shooting thru them in a close tandem arrangement to insure they registered
the same velocity (minus ~2-4 fps for the downrange chrono).
My test procedure uses long ranges between sensors in an attempt to minimize the
effects of measurement uncertainty, and to capture BC in multiple segments. In fact, the
increased range between sensors was the driving force that pushed me to use acoustic sensors
rather than chronographs downrange. In order to measure downrange velocity, you have to
‘thread the needle’ with each shot by putting it straight thru the skyscreens of the near and far
chronograph. This practice would inevitably lead to destruction of one or more of the
downrange chronographs if they’re placed much past 200-300 yards. I chose to use acoustic
sensors and measure time of flight because the microphones are able to detect the passage of
the bullet from 10’s of feet away. This adds the unique challenge of accounting for the time lag
between when the bullet passed by the mic until the shock wave reaches it, but it also allows
testing at extended ranges because the worry of destroying equipment is removed.
The estimated uncertainties of the key variables involved in my test procedure are:
distance between mics (+/- 1.5 ft), initial velocity measurement (+/- 3 fps), bullet’s time of flight
measurement (+/- 0.0005 sec), distance from bullet to mic as it passes (+/- 1 foot). The
cumulative effects of the uncertainties in my method result in average BC measurements that
are repeatable within +/- 1%. In other words, if I test a particular bullet type at one date and
location, then re-test that same bullet type at another date and location; having instrumented
the range from scratch on both occasions, my repeat measurement of that same bullet type will
result in a BC that is within +/- 1% of the previously measured value. There are exceptions to
this statement, but very few.
I’ve briefly described two approaches to experimentally measuring a bullets BC. How do
the results compare? Well it just so happens that the Courtney’s and I have measured some of
the same bullets for BC, and I will now present a comparison of our results.
Comparing results
The first bullet that I’ll present results for is the .224 caliber 40 grain Hornady Vmax.
This is an interesting case because it represents the ‘bookend’ as the lowest BC bullet tested by
the Courtneys. You can see in
.224 caliber 40 grain Hornady Vmax
Figure 2 that the advertised BC
for this bullet (0.200), the BC
measured by the Courtneys for
an average velocity of 2905 fps
(0.199), and the BC measured
by myself for this bullet (0.196
@ 2905 fps) are all within 2% of
each other, which is fantastic
agreement. Since we don’t
know what velocity the BC
given by Hornady is valid for,
I’ve shown it as being a
constant 0.200 for all speeds,
Figure 2. There is very good agreement on the BC of the 40
even though we know that BC
grain Hornady Vmax. All sources are within 2%.
changes with speed as indicated
by my measurements.
Another interesting thing to point out about this bullet is that since it has such a low BC,
it suffers a large amount of velocity decay compared to higher BC bullets. Because of the
principle of measurement uncertainties discussed above, the BC of this bullet is easier to
measure accurately than higher BC bullets because the amount of velocity lost is large
compared to the uncertainty of the velocity and time of flight measurement. Higher BC bullets
are more challenging to measure accurately because by definition, they lose less velocity over a
given range.
The next bullet that I’ll present is the .30 caliber 125 grain Nosler Ballistic Tip.
This bullet is interesting because the Courtney’s tested it from 3 different rifles and measured a
different BC from each rifle.
.308 caliber 125 grain Nosler Ballistic Tip
Figure 3 shows how our results
compare to each other, and to
Nosler’s advertised BC for this
bullet.
In this case, the ‘Litz’
measured BC was higher than
the ‘Courtney’ measured BC by
an average of 8%. In Figure 3, I
Ballistic Coefficients From Various Sources
show the error bars associated
with the BC’s based on the
repeatability of each tests
measurements.
The interesting thing to
note about this bullet is how the
measured BC changes with
velocity. Even though the
Courtney’s fired this bullet from
3 different rifles, the BC they
measured for this bullet was
lower than the Litz measured BC
by roughly the same amount
each time.
Velocity (fps) Litz BC
Courtney BC
% difference
Remember that the ‘Litz’
2875
0.345
0.319 (.30/30)
7.5%
2665
0.340 0.308 (.308 Win)
9.4%
BC was established by measuring
2132
0.329
0.306 (.30-06)
7.0%
the bullets time of flight in
Average: 0.338
0.311
8.0%
multiple segments for several
Figure 3. On average, the Litz BC was 8% higher than the
trajectories, all fired from the
Courtney BC for the 125 grain Ballistic Tip.
same rifle. The fact that the
dependence of BC on velocity is roughly the same for both the Litz and Courtney tests shows
that there is a repeatable dependency of BC on velocity. The fact that the Courtney BC is on
average 8% lower than the Litz BC for this will be discussed after all the comparison data is
presented.
The final bullet that was
tested by both parties is the .308
caliber 150 grain Nosler Ballistic
Tip. The Courtney’s only
measured this bullet from one
speed and from one rifle. Their
measured BC was 7.3% lower
than the ‘Litz’ BC, which is
consistent with the results of the
125 grain Ballistic Tip.
.308 caliber Nosler 150 grain Ballistic Tip
Ballistic Coefficients From Various Sources
Why Were Different BC’s
Measured?
Two independent and
supposedly ‘authoritative’
sources have measured BC’s for
the same bullets and came up
with different results in 2 out of 3
cases. What gives? This question
was the source of many long and
educational emails between
myself and the Courtneys earlier
Figure 4. Comparison of BC’s for the 150 grain Ballistic Tip.
this year. Our current belief is
that both the Courtney and Litz measurements of BC are accurate. Our working hypothesis for
how this is possible is that there are differences in the test rifles that might be causing the
bullets to actually fly with more drag in the Courtney test over the 97 yards they used to test.
The rifles used by the Courtneys to conduct their testing are for the most part typical
light weight factory hunting rifles. Most of the rifles I use in my testing are custom heavy
barreled rifles with match chambers. Our hypothesis is that the light weight barrels used by the
Courtney’s generate a significant amount of ‘barrel whip’, which induces a relatively large
amount of initial ‘tip off’ rate for the bullet. The large initial yaw rate damps out quickly as the
bullet ‘goes to sleep’ over a distance of about 50 to 100 yards. Since the Courtney’s test only
spans 97 yards, the bullets are flying with significant yaw levels for most of that distance. A 6
Degree Of Freedom (6-DOF) computer simulation was run to illustrate the reduction in BC and
shift in point of impact for a bullet fired from a barrel with various initial yaw rates. The results
showed that if a bullet were fired with enough of a ‘tip off’ rate from the muzzle to cause a
maximum of 10 degrees of yaw, that yaw would dampen to 5 degrees by 50 yards, and 2
degrees by 100 yards. A bullet fired like this would experience a 7.5% reduction in effective BC,
and strike about 4” from center at 100 yards compared to a bullet fired with no ‘tip off’ rate.
The Courtneys described to me that they’ve observed up to 6” difference in point of impact at
100 yards when working up loads for different weight bullets in their light barreled .30-06 rifle.
This point of impact shift is too much to be simply a difference in gravity drop, but is more likely
to be evidence that barrel whip is large enough to displace the shot by that much.
Damping of a bullets oscillations over the first 100 yards
Figure 5. Given an initial ‘tip-off’ rate of 100 radians per second, the bullet reaches a maximum
yaw angle of about 11 degrees which damps to about 2 degrees at 100 yards as the bullet ‘goes to
sleep’. The pitching and yawing motion of the bullet can cause over 7% reduction in BC. After
100 yards, the oscillations continue to damp below 2 degrees, and the BC is essentially up to its
max value.
That my test procedure results in higher measured BC’s for the same bullets is
consistent with the fact that I’m using heavier barrels which have less ‘whip’, and impart less of
an initial ‘tip off’ rate for the bullets. It also makes sense that our measurements for the .224
caliber 40 grain Hornady Amax agree so well. A bullet that small and light would not cause
enough recoil and barrel whip to create much of a ‘tip-off’ rate even from a skinny barrel.
All 3 of the .30 caliber rifles used by the Courtneys to test the 125 and 150 grain Ballistic
Tips wore ‘sporter’ weight barrels, and all 3 produced lower measured BC’s than my medium
Palma contour match barrel.
The details given above support our working hypothesis that barrel whip might be the
reason we measured different BCs for the same bullets. Over the coming months, the
Courtneys and I plan to put this hypothesis to the test. One way this can be done is to conduct
the Courtneys two chronograph test with the second chronograph placed farther downrange.
The increased distance between chronographs should yield higher average BC measurements
because the bullet flies with less drag after it ‘goes to sleep’. Another way to test the
hypothesis with the two chronograph method is to place the near chronograph 100 yards
downrange, and the far chronograph 200 yards downrange. In this arrangement, most of the
bullets oscillations should be damped before it starts being measured. We will keep you posted
on the results of our continued experiments in BC measurement.
In Closing
So it turns out that it might be possible to measure different BC’s for the same bullet,
and have them both be right! Great, where does that leave the shooter who needs an accurate
BC to calculate drop? Which ‘experimentally measured’ BC should you use?
If your long range rifle wears a heavy barrel as most do, it’s clear that the BC of bullets
fired from your barrel will be closer to the BC’s I measured with my heavy test barrels. What if
you have a rifle with a thin lightweight barrel that has more severe ‘whip’? In that case, I still
believe that the BC’s measured by my method over 600 yards will be better to use. The reason
is because if your rifle imparts the same initial yaw as the Courtney test rifles, and the bullet
flies with some initial yaw, most of that yaw is damped out by the time the bullet gets to 100
yards. After that, the bullet is flying with near zero yaw, and has a BC closer to the one
measured by the Litz method over long range. In other words, using the Courtney BC assumes
that the reduced BC that was caused by barrel whip at close range applies to the entire
trajectory. A trajectory predicted with this BC would be over predicting the actual average drag
for the bullet on a long range trajectory.
In conclusion, it is our belief that the Courtneys accurately measured the effective BC’s
over the first 97 yards of flight. However, we think that those BC’s were probably influenced by
a significant amount of bullet yawing (from excessive barrel whip) that was going on early in the
bullets flight. As a result, the BC’s measured by the Courtney’s test would not be valid for long
range trajectories after the bullet has completely damped the initial yaw rate induced by barrel
whip.
As for the .257 caliber 115 grain Berger VLD, it turns out that the Courtney’s were
testing a very old version of that bullet that was made when Berger was still operating out of
Arizona and under different management. We discovered that the .257 bullets tested by the
Courtney’s had some dimensional differences from the current version of that bullet, which are
consistent with their low measured BC. The Courtney’s are being supplied with more current
samples of that bullet to test, which we expect to result in a higher measured BC.
Berger has refined our assessment of that bullets BC. The old advertised BC was 0.523
(based on a computer prediction program), and the new BC for that bullet is assessed at 0.479.
This bullet has yet to be actually ‘shot’ for BC. It’s currently assessed value of 0.479 is based on
the experimental measurements of similar bullets that have been tested, and is likely to be
accurate within +/- 3%. The newly assessed BC of 0.479 is the average BC for this bullet
between 3000 and 1500 fps. When this bullet is actually fired in a BC test, it’s advertised BC will
be adjusted (if necessary) to match the experimental result.
Our goal at Berger is
.257 Caliber 115 grain Berger VLD
not to lure shooters to our
bullets by advertising
exaggerated BC’s. Our goal is
to enrich the shooting
experience by making good
bullets and representing
Figure 6. Dimensions of Bergers .257 caliber 115 grain VLD.
them with accurate BC’s so
shooters can enjoy the success of hitting their targets.
It is very encouraging to see shooters putting the advertised BC’s to the test by doing
their own experiments. I would encourage everyone who’s interested to put the numbers to
the test for yourself. Remember to give careful consideration to the potential error sources
involved in your test set-up, in particular the effect that barrel whip might have on your
measured BC over short range.
The Courtney’s and I will continue to work together on the details of our BC
measurements, in particular the current .257 caliber 115 grain VLD, and report again whenever
we have something to share.
About the Author
Bryan Litz earned his BS in Aerospace Engineering from The Pennsylvania State
University in 2002. For the next 6 years, he worked as a civilian for the US Air Force on the
design, modeling and simulation of air-to-air missiles. During that time, Bryan established a
part time consulting company called Applied Ballistics, LLC where he provided bullet designs,
ballistics software development support, and ballistic testing services for many customers in
the industry including Berger Bullets. In the fall of 2008, Bryan took a full time position with
Berger as their Chief Ballastician.
As well as being a student of ballistics, Bryan is also very familiar with the practical
application of long range shooting. In 2008 he won the National Individual Palma
Championship at Camp Perry Ohio, The Midwest Palma Championship, 3 long range NRA
Regional Tournaments, and was part of the winning US team in the international Spirit Of
America Fullbore rifle match in Raton, NM.