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 engineering-type graphs. Also, these examples have been selected to illustrate some of the points made in the preceding discussions of ballistic coefficients.
Figure 2.5-1 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 2.5-1. BC measurements for the Sierra 6.5 mm 160 grain Semipoint bullet
The measurements shown in Figure 2.5-1 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 2.5-1 shows convincingly that the BC for this particular bullet type varies continuously with velocity, increasing in value as the bullet flies down-range 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 2.5-1, 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 2.5-1, 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 2.5-2 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 1×8 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 1×9 and 1×10 rifling twist rates also satisfy the stan-
Figure 2.5-2. 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 1×9 twist rate, the average BC is 0.295, and for the 1×10 twist rate, the average BC value is 0.294. The group for the 1×12 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 point-first on paper targets just behind screen 3 in the test setup (see Figure 2.3-2).
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 1×7 and 1×8 twist rates in Figure 2.5-2), 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 184.108.40.206. When the rifling twist rate in the barrel is slower (e.g., the 1×9 and 1×10 twist rates in Figure 2.5-2), 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 1×12 twist rate in Figure 2.5-2), we believe that the coning motion increases dramatically. It certainly has a systematic effect on measured BC, and it also has a random round-to-round 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 2.5-3 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 (1×8) to one turn in 14 inches (1×14), except that we did not have a test barrel with a 1×13 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 2.5-3 shows the same characteristics for the 190 grain 30 caliber bullet as were observed in Figure 2.5-2 for the 69 grain 22 caliber bullet. The average BC values for the groups are relatively consistent for rifling twist rates from 1×8 through 1×11. 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 1×12 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 1×14, 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 1×14 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 down-range. 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 2.5-4 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 50-yard 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 2.5-3. 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 round-to-round 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 2.5-4.
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 2.5-5 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 (1×7) 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 2.5-6 and 2.5-7 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 338-378 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 2.5-6. 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 2.5-6. 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 2.5-7, 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 2.5-7, 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 2.5-7. 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 radar-tracking 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 2.5-6 and 2.5-7 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 2.5-8. 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 2.5-8 for Sierra’s 44 caliber (.4295 inch diameter) 240 grain Jacketed Hollow Cavity Sports Master bullet. The BC for this stubby, hollow-point bullet behaves differently compared to the rifle bullets in Figures 2.5-6 and 2.5-7. 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 long-range trajectories for rifle and handgun bullets traveling at low velocities.