4.4 Lessons Learned From Ballistic Coefficient Measurements
Sierra’s ball by means of firing tests very soon after our research began. Through the years since 1971 Sierra has continued to determine ballistic coefficients for the bullets in Sierra’s line by firing tests. A very large data base of test measurements has been built, and some significant, and often surprising, lessons have been learned from that testing.
Much has been written in previous editions of the Manual about our ballistic coefficient measurements, and this information will continue to be available from Sierra. This section will summarize the principal lessons learned from our testing experience. Any reader interested in the details supporting these observations may refer to the 1984 edition, or contact Sierra for a copy of the Exterior Ballistics section from that edition. The lessons and observations are summarized in the paragraphs below.
1) Ballistic Coefficients Must Be Determined by Firing Tests
In 1971, the ballistic coefficient for every bullet in Sierra’s line had been worked out by the Coxe-Beugless method, whichThe boat tail shape is a type of bullet designed to have low nts of low base drag bullets was available to Sierra at that time. Consequently, a key objective of our early firing tests was to measure the ballistic coefficients of Sierra’s boat tail bullets and compare these to the ballistic coefficients of flat base bullets.
The results from those tests showed very clearly that the boat tail bullets had higher ballistic coefficients than flat base bullets of the same caliber, weight, and point shape. Another very important observation was that the measured ballistic coefficients of the flat base bullets were not equal to values calculated from the Coxe-Beugless method. Consequently, Sierra has chosen to determine ballistic coefficients by means of firing tests.
2) G 1 is the Most Appropriate Drag Function Choice for Sierra’s Bullets
The tests conducted in 1971 also compared four standard drag functions to determine which of these models was best for Sierra’s boat tail bullets. The four drag functions were published by Winchester-Western:
G 1 – for all bullets except those in the categories below
G 5 – for low base drag bullets (i.e., boat tail bullets)
G 6 – for flat base, sharp pointed, full patch bullets
G L – for hollow point, lead nose bullets
Theory says that if the drag model matches the true bullet drag, the ballistic coefficient will be constant at all velocities. If the drag model does not match the true drag, then we will find different values of ballistic coefficient in measurements at different muzzle velocities.
Several bullets were used in the tests, both flat base and boat tail types. For each bullet type, several test rounds (typically 5 to 10) were fired at each of two muzzle velocities (typically 3000 and 2500 fps). For each test round four ballistic coefficients were determined ( C 1 , C 5 , C 6 , and C L ). This procedure gave two sets of four ballistic coefficient values for each bullet type, one set for the higher muzzle velocity and one for the lower. By comparing the values in the two sets, it was easy to see which of the four C’s changed significantly (outside the measurement error band) and which did not. From theory we expected that the best drag model ( G 1 , G 5 , G 6 , or G L ) for each bullet type to be the one for which the corresponding ballistic coefficient ( C 1 , C 5 , C 6 , or C L ) changed the least amount between the two muzzle velocity levels.
Although we expected G 5 to be best for the boat tail bullets, the tests showed that G 1 was better. This was a surprise. The second surprise was that, while G 1 was best for flat base bullets, C 1 was not constant; it varied significantly with muzzle velocity.
With these observations, G 1 was adopted as the standard drag model for all Sierra bullets. Values of ballistic coefficients in all Sierra Manuals are referenced to the G 1 drag function.
Finding that G 1 was best for boat tail bullets was fortunate, because G 1 has been and continues to be used by all commercial manufacturers of bullets. Consequently, ballistic comparison can be made between bullets made by different manufacturers, as well as between bullets made by a single manufacturer. If different drag models were necessary for different bullets or different bullet styles, comparisons based on ballistic coefficients would not be possible. Another advantage of G 1is that it is the drag function for the Ingalls Tables, so that ballistic trajectories can be calculated using those tables.
3) Ballistic Coefficient Changes with Velocity for All Bullets
It was mentioned above that an important result from the drag model comparison tests in 1971 was that the ballistic coefficient C 1 is not constant with velocity even for flat base bullet types. We have found, in fact, that ballistic coefficient changes with velocity for almost every bullet type that we have tested so far.
This discovery presented a serious dilemma. Because drag is far and away the strongest force acting on a bullet in flight, it is necessary to calculate drag very accurately in order to compute bullet trajectories which are accurate. The implication of a ballistic coefficient which varies with velocity is that the corresponding standard drag model does to accurately represent the true bullet drag.
Two approaches are possible. One is to determine a drag model for every bullet in Sierra’s line. This approach is used by the military. It makes sense when there is a small numbach is prohibitively expensive.
The second approach is the one we adopted. We determined by computational experiments that if we define several velocity ranges for each bullet (we use three to five) and use a constant ballistic coefficient within each range (the average value for that velocity range), we can calculate highly accurate bullet trajectories by allowing the ballistic coefficient to change value when the bullet velocity crosses a boundary between ranges. This strategem allows all the classical methods of trajectory calculation to be used, and an overall bullet trajectory is made up of a series of segments. Within any one segment the ballistic coefficient is constant, but it changes value from one segment to another. The velocity boundaries between segments are chosen so that the ballistic coefficient variation within a segment is at worst no more than plus-or-minus 5 percent of the average value within the segment.
4) The Firing Test Method Measures
a Ballistic Coefficient for the Shooting System
The firing test method measures an effective ballistic coefficient for the shooting system (rifle, cartridge, and bullet), and not just the bullet alone. Theoretically, ballistic coefficient is a fundamental property of bullet shape, and an absolutely perfect measurement of ballistic coefficient would require that the bullet be perfectly stabilized in flight. The firing test method, on the other hand; yields an effective ballistic coefficient for the bullet even if it is not perfectly stabilized (which is usually the case).
A bullet is gyroscopically stabilized by its spin, which is imparted by the rifling in the barrel. If a bullet is perfectly stabilized, then its longitudinal axis (which is also its spin axis) is exactly aligned with its direction of flight, that is, with its velocity vector. A bullet which is imperfectly stabilized is not tumbling, but is undergoing more complex motions. One possible motion is a precession of the bullet axis around the direction of flight in a coning motion, and the other is a nutation or nodding of the bullet axis. Imperfect stabilization can be caused by a variety of physical factors; examples are a small center of gravity offset from the bullet longitudinal axis, a small aberration in point shape or tail shape, and tipoff moments applied to the bullet when it exits the gun muzzle by barrel whip or uneven escape of powder gases around the bullet base. A bullet undergoing such motions has a larger area for air drag to act upon than it would if perfectly stabilized, and therefore, since drag is greater the effective ballistic coefficient is smaller. The physical effects which cause imperfect stability are random from round to round. Thus, they cause a dispersion (or scatter) in measured ballistic coefficient from round to round. We have learned that this scatter is an excellent indicator of how well stabilized the bullets are. If the scatter is small, the bullets are well stabilized.
There is an argument that the very largest measured value is the best value to accept as the ballistic coefficient of a test bullet at each test velocity, because it corresponds to the most well stabilized flight condition. From a purely scientific viewpoint this is probably correct. However, we believe the majority of shooters are most interested in the nominal performance that they can expect from their loads rather than “perfect” performance. Therefore, Sierra has always usedaverage values of measured ballistic coefficient to calculate trajectories and characterize bullet ballistic performance, with one exception explained below.
The scatter in ballistic coefficient values measured at very low muzzle velocities occurs because the rifling twist rate does not stabilize the bullets well. This is especially true for long heavy bullets fired at velocities below 2000 fps. It is fair in this special case to accept only the highest values as valid measurements of ballistic coefficient for the following reason. It is very unlikely that these bullets would be fired at muzzle velocities below 2000 fps, especially not from guns with twists too slow to stabilize the bullets well. We need to know the ballistic coefficients at low velocities in order to calculate long range trajectories for these bullets. It is safe to assume that when the bullets have been fired at higher muzzle velocities and have traveled far enough for remaining velocity to fall below 2000 fps, they are well stabilized in the low velocity region of flight, and consequently the highest measured values of ballistic coefficient are appropriate at low velocities.
5) Ballistic Coefficients Change
Markedly at Velocities Near the Speed of Sound
We have found that very large changes in ballistic coefficient can be expected in the range of 900 to 1200 fps. These large changes reflect the fact that the analytical drag model does not match the true drag behavior in this velocity range near the speed of sound. These ballistic coefficient variations can have important effects on bullet trajectories for certain rifle cartridges, such as the .45-70, .444 Marlin, .44 Magnum (rifle), and other low velocity types. Handgun bullet trajectories are also seriously affected by ballistic coefficient variations, and this topic is discussed in more detail inSection 4.5.
6) Ballistic Coefficients Are Not Predictable
Finally, and perhaps most important, our experience has shown that the ballistic coefficient of a bullet is not really a predictable parameter. We have already remarked that when a bullet is not well stabilized in flight the apparent ballistic coefficient can be much smaller than the value determined from the shape of the bullet only. But we have found that even for well stabilized bullets the sectional density/form factor relationship described in Section 4.1 agrees only roughly with measured ballistic coefficients. Stated another way, two bullets with the same point shape and tail shape, but of different weights and/or diameters, usually do not have the same form factor according to our experience. This is yet another argument for measuring ballistic coefficients. We know, of course, that for many years simple theoretical calculations of ballistic coefficients provided adequate ballistics for most shooting situations. But we now know we need measured ballistic coefficients to calculate highly accurate long range trajectories to match the exterior ballistics potential of modern firearms and high performance cartridges.
4.4 Lessons Learned From Ballistic Coefficient Measurements