There are several important precautions and points to consider when measuring ballistic coefficients by this method.
1. Errors in the measurements. There are many sources of errors that can affect BC measurements. They can be separated into two categories: random errors and systematic errors. A random error is an error that may occur in any test round, but that changes in magnitude or direction (i.e., an erroneous increase or decrease in BC value) from one test round to the next. Typical sources of random errors are round-to-round variations in bullet weight, jacket thickness, core homogeneity, etc. A property of random errors is that they can be effectively removed by averaging the BC measurements of several test rounds. We typically fire at least 10 test rounds for each bullet type at each velocity level that we choose for measuring the BC value.
The other category of errors is systematic errors. A systematic error is a consistent error that occurs in every test round fired and is nearly the same magnitude and always of the same direction (i.e., always an erroneous increase or always an erroneous decrease in BC value) from one round to the next. Systematic errors are very bad, and every effort must be made to eliminate their sources.
The most important sources of systematic errors are errors in the measured distances between chronograph screens and in the measured range distance from the initial velocity chronograph to the final velocity chronograph. For example, suppose the separation distance between screens 1 and 2 in Figure 2.3-1 is supposed to be 10.0 feet. However, when we set up the screens, we make a measurement error of 1/16 inch, so that the true distance is 10.0 feet plus 1/16 inch (10.0052 ft). This error is consistent for every round fired, so that it is a systematic error source for the BC value.
Consider round 1 in Table 2.3-1. The chronograph really measures a travel time between screens 1 and 2, and then divides the erroneous distance 10.0 feet by this travel time to give an initial velocity of 2742 fps, which would contain a systematic velocity error. Then, the travel time between screens must have been 3647 microseconds (that is, 10.0 ft divided by 2742 fps). If the true distance of 10.0052 feet had been divided by this travel time, the true initial velocity should have been 2743.4 fps. Now, most chronographs do not read to tenths of a fps; they round off to the nearest whole fps. So our initial velocity chronograph should have indicated a velocity of 2743 fps, instead of 2742 fps. Using Infinity, it is easy to verify that the BC value for round 1 should then have been 0.4560 for this example, instead of 0.4583. [Note that the roundoff imprecision in the chronographed velocity results in a random error, not a systematic error, in BC value. This error can be effectively removed by averaging over several test rounds.] This example illustrates that a measurement error of about 1 part in 2000 in the separation distance between screens 1 and 2 will cause a systematic error in BC value of about 1 part in 200. That is, the small error in separation distance causes an error in BC value that is ten times larger – a very high sensitivity. A similar analysis will show that a measurement error of 1 part in 2000 in the separation distance between screens 3 and 4 of the final velocity chronograph (see Figure 2.3-1) will cause another systematic error of about 1 part in 200 in the measured BC value. Again, this is a very high error sensitivity. The multiplication factor of 10 in these error sensitivities applies to this particular example. In another situation, the error sensitivities would still be high, but the factor of 10 might change upward or downward.
This high sensitivity of systematic errors in BC to errors in the separation distances between the screens of the chronographs is primarily why we use a separation distance of at least 10 feet between screens in Sierra’s test range. Chronographs are available with screen separation distances of 1 or 2 feet. These chronographs are convenient because of their light weight and portability, and they are adequate for the purposes of load development where velocity measurement errors of 10 or 20 fps are tolerable. However to measure ballistic coefficients, systematic errors must be no more than 1 part in 2000, and hopefully less than that. For a screen separation distance of 1.0 foot, the maximum separation distance error would then need to be no more than 0.006 inch. Mechanical tolerances in mounting the screens on their supporting structure and positioning the active sensor within each screen are likely to be greater than this number. So, these chronographs should never be used for BC measurements, despite their convenience.
The situation is not as critical for the separation distance between the two chronographs. If somehow we made an error of one part in 2000 in measuring the 103 yard separation distance (i.e., an error of about 2 inches), this error source would contribute a systematic error of about 1 part in 2000 in the BC value. This is a one-to-one error sensitivity, but it shows that we must carefully measure the range separation distance between the two chronographs. We cannot afford an error of a yard, or even a foot.
2. The measured BC value must be calculated for each round individually, and then statistical analysis can be applied to the results. We need to obtain an average BC value from our measurements, to reduce or eliminate random errors, and to obtain a proper value for trajectory computations. To do this, we must calculate the BC of each test round individually, and then average the resulting values to obtain the average BC for the bullet type at each velocity level chosen for the measurements. We cannot first average all the initial velocity values, then average all the final velocity values, and then calculate a BC value from these average velocity values. This approach would lead to an erroneous average BC value, because of the laws of mathematical statistics. Also, there is useful information in the standard deviation and extreme spread of the individual BC values. These statistical parameters yield some knowledge of the stability of the bullets as they fly, as well as the quality provided by the manufacturing process.
3. The BC value for a bullet is likely to vary with bullet velocity as the bullet flies. We have made this point in all of the previous editions of Sierra’s Reloading Manuals. The reason is that the standard drag model (called the G1 drag model) is not a perfect representation of the aerodynamic drag on sporting bullets over the full range of bullet velocities. Therefore, Sierra follows a policy of measuring the BC of each bullet at several different velocity levels and then publishing BC values for each bullet within up to five velocity ranges that together span the total velocity range for the bullet. A glance at the table of BC values for Sierra bullets elsewhere in this manual will illustrate how these values are published. Sierra’s Infinity exterior ballistics program uses all five BC values for each bullet to compute trajectories for any Sierra bullet.
4. The measured BC value is valid for a certain range of bullet velocity. When the range distance between the two chronographs (see Figure 2.3-1) is relatively short — like 100 yds for rifle bullets or 50 yds for handgun bullets — the difference between the initial velocity and final velocity of each round should be no more than 10 percent of the initial velocity. For example, for round 1 in Table 2.3-1, the difference between the initial velocity (2742 fps) and the final velocity (2549 fps) is 193 fps, which is about 7 percent of the initial velocity. In this situation the BC value derived for each round characterizes the bullet performance in the range between the initial and final velocities. An alternative point of view is that the BC value is valid for a velocity that is midway between the initial and final velocity values. Thus, for round 1 in Table 2.3-1 the BC value 0.4583 is considered valid for a velocity of 2646 fps. This point of view is justified because the difference between the initial and final velocities is a small fraction of the initial velocity. We use this approach when we measure BC values.
Another situation that sometimes arises is that a single BC value is needed for a certain hunting or target shooting situation. For example, if you are a hunter and use a particular cartridge load for certain game, and you use a bullet for which the BC is not known, you may need a single value of BC valid for range distances out to a maximum of 400 yds. You can use the procedure explained above to measure an effective BC by placing the initial velocity chronograph near the muzzle of your rifle and the final velocity chronograph 400 yds downrange (carefully measured). This BC value will serve for all ballistic calculations, such as finding effects of changing altitude, changing weather, winds, uphill/downhill shooting, etc. However, you must think of this BC as valid for (a) the muzzle velocity of your cartridge and (b) a range distance of no more than 400 yards. If you change either of these parameters, the BC value may not be valid for accurate trajectory calculations.
5. BC values determined by using Sierra’s Infinity software using the process described are referenced to sea level standard atmospheric conditions. We have made the point in previous editions of the Sierra Reloading Manuals that measured BC values must be reduced to sea level altitude and standard atmospheric conditions at sea level, and we explained how to perform the necessary calculations. This is effectively done in the Sierra Infinity program (and we believe it is done also in other ballistic software programs). When using Infinity, simply enter the altitude, temperature, barometric pressure, and relative humidity at the firing point when beginning the computations. The BC used by Infinity for each round fired then is assumed to be the value for sea level standard conditions. Thus when the calculated velocity or time of flight values equal the measured values (after being corrected for the defined altitude, pressure, etc. during computations) the input BC is referenced to sea-level standard atmospheric conditions. Note that the barometric pressure entered into Infinity is from a barometer at the range or a local weather report. It is NOT the absolute pressure for the range altitude. The absolute pressure necessary for trajectory computations is calculated within Infinity from the altitude and atmospheric data for the firing point.
6. Protect the chronographs from stray bullets. This is a practical consideration when measuring ballistic coefficients. It is extremely embarrassing (especially when the equipment does not belong to you) and very expensive when a stray bullet destroys a screen or an electronics enclosure. The final velocity chronograph is especially vulnerable because it is far from the muzzle. We use a paper target at the downrange location before the final velocity chronograph is moved into position, and fire several shots to make sure the rifle is properly sighted to put bullets through the “window” in the screen. Then, the final velocity chronograph is moved into position for the firing tests. At Sierra’s test range, armor plates also are used to protect the structure and electronics of both chronographs. Even though all firing is done from machine rests, these plates bear the scars of some accidental stray bullets.