External Ballistics
(taken from the 5
thEdition of Sierra Manuals)
1.0 Introduction
Since the mid-1980s, shooters have begun to use personal computers with ballistics software
programs to calculate bullet trajectories and to explore variations in trajectories caused by changes in shooting conditions. During this period a number of ballistics software programs have been
developed, especially by suppliers of bullets for reloading rifle and handgun cartridges, and have appeared on the market. Sierra introduced the Infinity exterior ballistics program in 1997. Infinity is the latest version of Sierra’s continuing exterior ballistics program development effort, which began with our first exterior ballistics software in 1967. Today, there are many other exterior ballistics programs, and a few interior ballistics programs, available to shooters.
From our point of view, all these available programs are quite good. Their capabilities vary, and their computational accuracies differ a little. But on the whole they are quite acceptable for almost all shooting purposes. Today’s hand-loaders are fortunate that a wide selection of software programs is available at very attractive prices.
We hope the reader will forgive us when we say that we like Infinity best. These authors take great pride in having had important roles in the development of the Infinity software program. Actually, the development of Infinity was a full team effort. The ballistics experts at Sierra helped greatly to
establish the functional requirements for Infinity, based on their personal expertise and their
interactions with shooters throughout the world. Then, these authors took the major responsibilities for the physics, mathematics, functional design and scientific encoding of the program. Finally, the software professionals at 305 Spin in Sedalia, MO, integrated the scientific program into Microsoft Windows and performed all the functions necessary to make Infinity ―user friendly.‖ The same team has been responsible for implementing continual updates to Infinity, but the principal criticisms and suggestions leading to those updates have been received from shooters using Infinity. We are very grateful to them.
This article on Exterior Ballistics has been written specifically for shooters who use an exterior ballistics software program on a personal computer. This is a departure from the Exterior Ballistics articles that we have contributed to previous editions of the Sierra Reloading Manuals. The historical and mathematical approach to ballistics used in the previous articles has been omitted. This article instead concentrates on using ballistics software to determine ballistic coefficients of bullets, calculate bullet trajectories under a full range of shooting conditions, and answer questions about effects on trajectories caused by shooting conditions. Infinity has been used to support discussions throughout this article, but most of the calculations described should be able to be performed with one or more of the other available programs.
Section 2.0 of this article describes the ballistic coefficient. This section explains what the ballistic coefficient is, how it is related to a drag function, why it must be referenced to sea level altitude and standard atmospheric conditions, how it affects a bullet trajectory, and why in a practical sense a ballistic coefficient changes with bullet velocity. Section 2.0 also describes how ballistic coefficients are measured. It presents lessons we have learned from more than 30 years of practical experience with measurements, and it provides examples of ballistic coefficient measurements we have made.
Section 3.0 describes effects of shooting conditions at the firing point on bullet trajectories. Included are effects of altitude above sea level, atmospheric conditions, winds, and shooting uphill or downhill. Some problems about sighting in a gun are discussed, i.e., using a short target range to sight in at a longer zero range, determining the zero range from where bullets group at a known range, and sighting in at a local target range to be zeroed in at some other shooting location. The concept of point blank range is described, and how to select a zero range in order to maximize the point blank range of any gun for game or silhouette targets. The maximum range of a bullet is discussed, as is the bore elevation angle necessary to achieve that maximum range. Finally in Section 3.0, the maximum height a bullet will reach if fired straight up is described. These last two topics are of great interest in designing and operating outdoor shooting ranges.
Section 4.0 is completely new material not treated anywhere in our previous articles. In the past few years, we have received questions from an increasing number of target shooters about small effects they have observed in bullet trajectories — effects that cannot be explained by available exterior ballistics software programs. All the software programs generally available to shooters use a three degree-of-freedom dynamical model for a flying bullet — that is, a point mass with a ballistic
coefficient. The small, unexplained effects can be attributed to rotational motions of spin-stabilized bullets. Rotational motions of a bullet are modeled only in six degree-of-freedom ballistics programs. Such programs are used in the military.
Accuracy of target rifles has continually improved through the years, particularly in long-range target shooting. The small effects of bullet rotational motions have become observable because rifle
accuracy has improved to a point where these effects can be seen under some conditions. In Section 4.0 we attempt to explain these effects and their causes. These include the yaw of repose of a bullet and an associated cross-range deflection, turning of a bullet to follow a cross-wind and an associated vertical deflection, and turning of a bullet to follow a vertical wind and an associated cross-range deflection. These seem to be the most observable effects of the rotational motions of sporting bullets. Sections 5.0 and 6.0 relate specifically to Sierra’s Infinity program. Section 5.0 describes the content and format of the printout records from Infinity, that is, the trajectory parameters, their physical units, and other information communicated to shooters by the printout records. Section 6.0 is an overview of the capabilities of Infinity, describing its major features, operating modes, and how to use the program for trajectory computations and to answer questions concerning bullet trajectories. It is always a
pleasure to hear from users of Sierra products and a special pleasure to hear from those interested in ballistics. Please do not hesitate to contact us with questions or comments.
2.0 The Ballistic Coefficient
―So, just what is a ballistic coefficient, and what does it do for a bullet’s trajectory?‖ These are questions we have been asked many, many times, and they are not easy questions to answer. We will try to answer the first question in this subsection and then proceed to the second question in the next subsection. In later subsections we will describe methods used to measure ballistic coefficients, and give some examples of measured BC values for Sierra’s bullets. Henceforth, ballistic coefficient will be abbreviated as BC.
There are at least three ways to describe the BC. First, it is widely recognized as a figure of merit for a bullet’s ballistic efficiency. That is, if a bullet has a high BC, then it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will ―shoot flatter.‖ But this description is qualitative, rather than quantitative. For example, if we compare two bullets and one has a BC 25% higher than the other, how much is the improvement in bullet ballistic performance? This question can be answered only by calculating the trajectories for the two bullets and then comparing velocity, wind deflection, and drop or bullet path height versus range from the muzzle. So, the figure of merit
approach really gives only a qualitative insight into bullet performance, and sometimes this insight is not correct. It often happens that the bullet with the smaller BC is lighter than the bullet with the higher BC. The lighter bullet therefore can be fired at a higher muzzle velocity, and it can then deliver better ballistic performance just because it leaves the muzzle at a higher velocity. We will talk more about this later.
The second way to describe the BC is to use its precise mathematical definition. Mathematically, the BC defined as is the sectional density of the bullet divided by the form factor. This definition emerges from the physics of ballistics and is used in mathematical analysis of bullet trajectories. But in a practical sense, this definition is not satisfactory to most people for at least two reasons. The first is the question of a bullet’s form factor. The form factor is a property of the shape of the bullet design, but it is no easier to explain than the BC. The second reason is that this mathematical definition can lead to an erroneous conclusion. Assume for the moment that the form factor is just a constant
property of the bullet design (not always true). The sectional density of a bullet is its weight divided by the square of its diameter. (The square of any number is the number multiplied by itself). So, to get a large BC we need a large sectional density. It appears from the mathematics that a bullet with a very small diameter should have a very large sectional density because its weight is divided by a very small number, and this should give it a very high BC. In other words, this line of reasoning would lead us to expect that small caliber bullets should have very large BC values. But this is not true because when the diameter of the bullet is small, the volume also is small. The weight of the bullet then is small, and the sectional density is necessarily small also. The net result is that small caliber bullets generally have lower BC values than larger caliber bullets.
The third way to describe the ballistic coefficient traces back to the historical development of the science of ballistics in the latter half of the 19th century. This explanation is lengthier, but it provides a better understanding of what the BC is and what its role is in trajectory calculations. The latter half of the 19th century and the early part of the 20th century was a period of very intensive and fruitful development in the science of ballistics. The developments in ballistics were driven by technological advances in guns, projectiles, propellant ignition, and propellants throughout the 19th century, and by warfare, particularly in Europe and America. Warfare was almost an international sport among the kings, emperors, Kaisers and tsars in Europe throughout the 1800’s. The United States experienced the War of 1812, the Mexican War, the Civil War, the Indian wars in the West, and the
Spanish-American War within that same century. Governments were eager to fund research, development and manufacturing of improved guns and gunnery, because battles were generally won by the forces that had superior arms.
Percussion ignition was invented in 1807 by the Rev. Alexander Forsythe in Scotland. In 1814, Joshua Shaw, an artist in Philadelphia, invented the percussion cap. In 1842, the U.S. Army adopted the percussion lock for the Model 1842 Springfield Musket, replacing flintlock ignition in earlier
shoulder arms. Rifled muskets and handguns began to replace smoothbore military weapons in the mid-1850’s after a French Army officer, Capt. Claude Minie, developed a means to expand a bullet upon firing to cause it to fit the grooves of a rifled barrel. This advancement combined the rapidity and ease of loading of round balls—which had been the standard military projectile for over a century—
with the increased range and deadly accuracy of rifled arms. The range and precision of military weapons, for both small arms and artillery, was increasing dramatically.
The period between 1855 and about 1870 witnessed much research and development in breech loading rifles and handguns. The first metallic self-contained, internally primed cartridge (the 22 Short rimfire cartridge) was introduced by Smith & Wesson in 1857 in their Model No. 1 breech-loading revolver. Breech-loading rifles firing self-contained cartridges appeared in the 1860’s, and some were used during the U.S. Civil War (e.g., the Spencer carbine and the Henry rifle). In 1866 in the United States, Hiram Berdan obtained a patent on a primer that was suitable for centerfire cartridges. That same year in England, Col. Edward Boxer patented a full cartridge for the British Snider Enfield rifle, which was a centerfire cartridge utilizing the Boxer primer. (It is interesting to note that later the Berdan primer was widely adopted on the European continent, while the Boxer primer became standard in the United States.) In 1873, the U.S. military adopted the Model 1873 Trapdoor Springfield rifle with the 45-70 centerfire cartridge. In the space of just 31 years the U.S. Army
changed from smoothbore muskets with flintlock ignition to rifles with self-contained metallic centerfire cartridges.
Just 11 years later in 1884, a French physicist named Paul Vielle developed the first smokeless propellant that was stable and loadable for military purposes. Earlier powder developments had led up to Vielle’s discovery, but they were useful only for sporting purposes. The French Army loaded Vielle’s smokeless propellant in the 8mm Lebel cartridge for the Model 1886 Lebel rifle, the very first military rifle firing a smokeless propellant cartridge.
Smokeless propellant was quickly adopted by other nations, including the U.S., and caused
significant advancements in bullet performance and design. Muzzle velocity in military rifles, which was less than 1400 fps in the 45-70 and most other black powder cartridges, increased to more than 2000 fps in the earliest smokeless propellant cartridges. This led to the development of jacketed bullets of smaller caliber and lighter weights, i.e., 7mm, 30, and 8mm calibers, which could be fired at even higher velocities and not deposit lead in the barrels at those velocities. Before the end of the 19th century, pointed bullets and boat-tail bullets were also developed to significantly improve bullet ballistic performance.
With all these developments in guns and ammunition, the need to understand the ballistics of
projectiles became more acute. It was no longer sufficient to target a gun by hit-and-miss methods. Of course, graduated sights had existed on both smoothbore and rifled muskets for many years, but the elevation marks on the sights had been determined by firing tests of these weapons with a specific projectile at a specific muzzle velocity, at a specific altitude, and with a specific set of weather conditions. As warfare grew in intensity and mobility, it became vitally necessary to understand the physics of bullet motion. In other words, it was necessary to find a way to calculate bullet trajectories as well as the changes in those trajectories caused by changes in bullets, muzzle velocities and firing conditions.
An immense problem thwarted this objective for many years. This problem was understanding the physics and mathematically describing the aerodynamic drag force on a projectile. The invention of the ballistic pendulum by the English ballistician Benjamin Robins in 1740 had led to the astounding discovery (at that time) that the drag force on a bullet was many times more powerful than the force due to gravity, and that it changed markedly with bullet velocity. That event started a chain of firing tests, instrumentation developments, and theoretical investigations that lasted at least 200 years. Progress was slow because aerodynamic drag is a very complex physical process, and mathematics had to be developed to make accurate computation of trajectories possible long before the age of computers.
An early observation was that the drag force was different on every type of projectile, so that
measurements of drag deceleration seemed to be necessary on each type of projectile over the full velocity range between muzzle velocity and impact velocity. However, around 1850 Francis Bashforth in England proposed a practical idea that greatly simplified things and is used in the present day. He proposed a model bullet, or ―standard‖ bullet, on which comprehensive measurements of drag deceleration versus velocity could be made. Then, for other bullets this ―standard‖ drag deceleration could be scaled by some means, so that exhaustive drag measurements could be avoided for those bullets.
Bashforth could not have known how successful his suggestion would be. Ballisticians and physicists were working intensively to mathematically describe the aerodynamic drag force and derive the
equations of motion of bullet flight. They had recognized in the equations of motion a theoretical scale factor for aerodynamic drag that would adjust the standard drag model to fit a nonstandard bullet. This scale factor turned out to be the form factor of the nonstandard bullet divided by the sectional density, that is, the reciprocal of the BC. The form factor was a number that accounted for the different shape of the nonstandard bullet compared to the standard bullet.
Bashforth’s suggested standard bullet had a weight of 1.0 pound, a caliber of 1.0 inch, and a point with a 1.5 caliber ogive. Firing tests on projectiles of approximately this shape and weight were conducted in England and Russia between about 1865 and 1880. However, the definitive drag deceleration tests were performed by Krupp at their test range in Meppen, Germany, between 1875 and 1881. In 1883 Col. (later General) Mayevski in Russia formulated a mathematical representation of the drag force for the standard bullet. In the 1880’s, an Italian Army team led by Col. F. Siacci formulated an analytical approach and found analytical closed form solutions to the equations of motion of bullet flight for level-fire trajectories. This meant that trajectory calculations for shoulder arms could be performed algebraically, rather than by the more tedious methods of calculus. The Siacci team’s results also showed that not only could the standard drag deceleration be scaled by using the BC of the nonstandard bullet, but also the standard trajectory computed for the standard bullet could be scaled by the same factor to compute an actual trajectory for the nonstandard bullet. This was a very important breakthrough that greatly reduced the amount of work in trajectory
computations. The Siacci approach was adopted by Col. James M. Ingalls of the U.S. Army Artillery. His team produced the Ingalls Tables, first published in 1900, which in turn became the standard for small arms ballistics used by the U.S. Army in World War I.
So, the ballistic coefficient actually is a scale factor. The BC scales the standard drag deceleration of the standard bullet to fit a nonstandard bullet. However, the BC works in a reciprocal manner. That is, the higher the BC of a nonstandard bullet, the lower the drag is compared to the standard bullet. This is alright, because it means that the higher the BC of a bullet, the better will be its ballistic
performance. The physical units of the BC are pounds per square inch (lb/in2). The BC value for the standard bullet then is 1.0 (weight 1.0 lb, diameter 1.0 inch, and form factor 1.0 by definition for the standard bullet). Ballistic coefficients of most sporting and target bullets have values less than 1.0, and generally BC values increase as caliber increases. A bullet can have a BC higher than 1.0. For example, some heavy 50 caliber bullets have BC values greater than 1.0.
Military agencies in different nations developed many standard bullets over the years. This was done because of fundamentally different shapes in military projectiles, such as sharper points and boat tails. The purpose was to establish better standards for these classes of bullet shapes. In recent years, however, this practice has largely been abandoned in the military. With modern
instrumentation and computers, it has become possible to measure the drag deceleration of every individual projectile type used by the military. Thus, there is no longer a need for a standard projectile for military applications. Or, we might say that every type of military projectile is its own standard.
This is not true for commercial bullets, however. Ballistic coefficients are used for all commercial bullets for sporting and target-shooting purposes, mainly because the BC of each is relatively easy and inexpensive to measure, compared to measuring the drag deceleration. The standard projectile for commercial bullets is still nearly identical to Bashforth’s standard bullet. The standard drag model, also called the standard drag function, for this projectile is known as G1. The BC values quoted by all producers of commercial bullets are referenced to G1. It is important to note that BC values quoted by commercial producers cannot be used with any drag model other than G1. It is possible to find other standard drag models by looking up historical military ballistics data. But, if a standard drag model other than G1 is used, the BC values of bullets must be measured with reference to that drag model in order to calculate accurate trajectories. The values are likely to be very different from the values referenced to G1.
2.2 Bigger Is Not Always Better
It is absolutely true that, if two bullets are fired with the same muzzle velocity at the same firing point and with the same weather conditions, the bullet with the higher BC will arrive first at a specific target, will have higher velocity and energy when it arrives, will suffer less wind deflection, and will have less drop than the bullet with the lower BC. This will happen regardless of caliber, bullet weight, or bullet type. But notice the big IF condition in this statement. If the bullets are fired with different muzzle velocities, at different altitudes, or under different weather conditions, any conclusions from the comparison may not be entirely correct.
What this means is that when comparing the ballistic performance of different bullets, all the firing conditions must be taken into account. Another very important consideration is that when comparing bullet ballistic performance, the performance must be evaluated for the purpose and objectives to be accomplished. For example, the usual purpose may be target shooting or hunting. If we wish to choose a bullet for the purpose of target shooting, the main objectives are maximizing accuracy and minimizing wind deflection. A very important advantage of handloading is that the muzzle velocity produced by the cartridge can be adjusted so that the gun delivers its best accuracy. Such a muzzle velocity is usually a little less than the maximum safe velocity of the cartridge in that gun, and this velocity usually must be discovered by trial at the shooting range. Target shooters with modern high-power target rifles generally are achieving accuracies on the order of 0.2 minutes of angle (MOA) or less. On the other hand, ―shooting flat‖ is not very important, because the ranges to the targets are fixed distances, sighting shots are usually allowed, and sights on guns are allowed to be adjusted between stages of the matches. However, sensitivity to crosswind is very important, because matches take place under variable wind conditions. Even if a shooter is highly experienced in estimating windage corrections, it is best that the bullet selected for the match have the smallest practical sensitivity to crosswind.
For hunting purposes ―shooting flat‖ is a major objective, as are adequate retained energy and momentum over the effective range of the gun for the intended game, adequate accuracy, and low sensitivity to crosswinds and vertical winds. A gun that ―shoots flat‖ produces small bullet drop within the effective range for the intended game. This is very important, because it is difficult for a hunter to estimate range to a game animal under practical conditions in the field. Of course, there are several optical and electro-optical range finders available, but most hunters cannot be assured that their game will be patient and stand perfectly still while they attempt to use a range finder under non-hunter-friendly field conditions. The concept of point blank range is one very practical way to ease
the range-estimating problem for hunters in the field (more about this in a later section). A ―flat shooting‖ gun inherently has more point blank range than a gun with a ―rainbow‖ trajectory. ―Flat shooting‖ requires high muzzle velocities and bullets with high ballistic efficiency.
Adequate accuracy for hunting usually means groups of 1.0 to 1.5 MOA for medium and large game, and 0.5 to 0.75 MOA for varmints. Wind sensitivity is very important, because field environments typically have windy conditions. Crosswinds may happen anywhere, and vertical winds are
experienced in hilly or mountainous regions. The wind deflection sensitivity of a bullet to a vertical wind is exactly the same as its sensitivity to a crosswind. In other words, suppose that a 1.0 mph crosswind from the shooter’s right to left will cause a deflection of the bullet of, say, 6 inches to the left at the target. Then a vertical wind of 1.0 mph upward will also cause a 6 inch deflection of the bullet in the upward direction. This is very important when shooting across a canyon or along a hillside when the wind is blowing.
To illustrate the points made above in this subsection, we will consider two simplified examples, shown in Tables 2.2-1 and 2.2-2. Two popular cartridges are used for these examples. The first (see Table 2.2-1) is the 308 Winchester, also known as the 7.62 x 51 mm NATO cartridge. This cartridge is very popular for hunting medium game, such as deer and antelope. It is also used for target
shooting. For example, in Service Rifle competitions, they are fired in the M14 and M1A rifles, and in Match Rifle competitions they are fired in bolt action rifles. The second cartridge (see Table 2-2-2) is the 300 Winchester Magnum. This cartridge is very popular for both hunting large game, such as elk, moose and bear, and for target shooting, particularly the Long Range target competitions. These cartridges have been selected as examples because there is such a wide variety of 30 caliber
bullets available for handloading.
The examples are simplified in that the numbers in Tables 2.2-1 and 2.2-2 are for sea level altitude and standard atmospheric conditions. So, the performance of each bullet is not calculated for realistic field conditions, but these examples validly illustrate our key points. The numbers in the tables have been calculated using Sierra’s Infinity Exterior Ballistics Program.
Each table is for one of the two example cartridges. Furthermore, each table is separated into two sections: one for hunting purposes and the other for target purposes. In the hunting purposes section, the first column in the table lists each bullet selected for comparison. The next column contains the muzzle velocity of each bullet. Each listed velocity is at or near the top end of the velocity range for that bullet recommended in the Reloading Data Section of this Manual for the cartridge in each table. The third column lists the ballistic coefficient of each bullet at the muzzle velocity. Later in this section we will describe how BC varies with bullet velocity. The number in the table compares the bullets at the muzzle velocity level for each. The BC variations as the bullet flies are taken into account in the trajectory calculations. The fourth column lists the energy of each bullet at the muzzle.
For hunting purposes it has been assumed that the effective range of fire is 400 yards, that the rifle is zeroed in at 250 yards, and that a telescope sight is used, with the centerline of the telescope 1.5 inches above the cen-terline of the bore. Then for these assumptions, the fifth column shows the maximum bullet path height (sometimes called the maximum ordinate) above the hunter’s line of sight through the telescope, together with the downrange position from the muzzle at which this maximum bullet path height occurs.
The remaining four columns in this section of the table show ballistics properties at the 400 yard maximum effective range point. Column 6 lists the remaining velocity; column 7 lists the bullet energy; column 8 lists the distance the bullet passes below the hunter’s line of sight at 400 yards;
and the last column lists the wind deflection sensitivity — that is, the inches of deflection per mile per hour of either crosswind or vertical wind.
In the second section of Tables 2.2-1 and 2.2-2 for target shooting purposes, the first column lists the target bullets selected for comparison; the second column shows the muzzle velocity for each bullet; and the third column lists the BC value at the muzzle velocity. Again, each listed velocity is at or near the top end of the velocity range for that bullet recommended in the Reloading Data Section of this Manual for the cartridge in each table. For purposes of comparison, it is assumed that a near-maximum load delivers the best accuracy, which is not always true.
Two range distances are considered for target shooting — 600 yards and 1000 yards. Column 4 shows the remaining velocity of each bullet at 600 yards, and column 5 lists the wind deflection sensitivity at 600 yards. Columns 6 and 7 show these same two parameters at 1000 yards from the muzzle. Note that the wind deflection sensitivity values listed in the tables are per mile per hour of crosswind or vertical wind. In other words, if the wind speed is 10 mph, the bullets will deflect 10 times the amount shown in the tables.
Consider first the 308 Winchester cartridge in Table 2.2-1. For hunting purposes, the table shows that the bullet that shoots flattest is the 150 grain SBT (Spitzer Boat Tail) at 2800 fps muzzle velocity. However, the bullet with minimum wind deflection sensitivity is the 200 grain SBT at 2400 fps muzzle velocity. So, the lighter 150 grain bullet with smaller BC passes about 4 inches closer to the line of sight at 400 yards than does the 200 grain bullet with a significantly higher BC. On the other hand, the heavier bullet deflects considerably less in a crosswind or vertical wind. If a
crosswind or vertical wind speed were 10 mph, the 150 grain bullet would be deflected 16.2 inches, while the 200 grain bullet would be deflected just 12.8 inches. So, the choice of hunting bullets depends on which is more important to the shooter: a flatter trajectory or sensitivity to wind conditions.
For target shooting purposes, the bullet with the minimum wind deflection sensitivity in the 308 Winchester cartridge is the 200 grain MatchKing fired at 2450 fps muzzle velocity. In this case, the 200 grain MatchKing is better than the 220 grain MatchKing just because the heavier bullet cannot be fired at a high enough muzzle velocity. If the muzzle velocity of the 220 grain bullet could be increased to around 2300 fps, then its higher BC would give it less wind sensitivity than the 200 grain bullet. But the cartridge does not have enough powder capacity to safely allow the velocity increase.
For the 300 Winchester Magnum cartridge, Table 2.2-2 shows that for hunting purposes the 180 grain SBT GameKing bullet loaded to 3100 fps muzzle velocity shoots the flattest of all five bullets listed. The 200 grain SBT GameKing bullet loaded to 2900 fps has the least wind deflection
sensitivity, but it is just a tiny bit better than the 180 grain bullet. So, in this example the numbers in the table indicate that the 180 grain SBT GameKing bullet is probably the best choice for hunting. However, some hunters
Table 2.2-1 Ballistic Coefficient Effects for the 308 Winchester Cartridge
HUNTING
PURPOSES
At 400 yds Range
(Zero at 250 yds)
Selected
Mzzl.
Vel.
BC at
Mzl
Energy
Max
Bullet
Velocity Energy Bullet
Wind
Drift
Bullet
(fps)
Mzzl.
Vel.
(ft-lbs)
Path (in)
(fps)
(ft-lbs) Path (in)
(in/mph)
125 gr SPT 3000 0.279 2498 3.43 @ 1781 880 -19.33 2.09 Pro-Hunter 143 yds 150 gr SBT 2800 0.380 2611 3.68 @ 1899 1201 -19.29 1.62 GameKing 143 yds 165 gr SBT 2600 0.404 2476 4.32 @ 1810 1201 -21.93 1.62 GameKing 139 yds 180 gr SBT 2500 0.505 2498 4.49 @ 1866 1391 -21.99 1.34 GameKing 138 yds 200 gr SBT 2400 0.552 2558 4.66 @ 1832 1490 -23.36 1.28 GameKing 136 yds
TARGET SHOOTING PURPOSES
At 600 yds Range
At 1000 yds
Range
Selected
Mzzl.
Vel.
BC at
Velocity
Wind Drift
Velocity
Wind
Drift
Bullet
(fps)
Mzzl.
Vel.
(fps)
(in/mph)
(fps)
(in/mph)
155 gr
HPBT 2800 0.450 1699 3.21 1170 10.81
175 gr HPBT 2600 0.496 1648 3.13 1194 10.20 MatchKing 180 gr HPBT 2600 0.496 1650 3.14 1201 10.14 MatchKing 190 gr HPBT 2550 0.533 1664 2.97 1225 9.58 MatchKing 200 gr HPBT 2450 0.565 1632 2.93 1230 9.31 MatchKing 220 gr HPBT 2200 0.629 1504 3.04 1171 9.47 MatchKing
Table 2.2-2 Ballistic Coefficient Effects for the 300 Win Magnum Cartridge
HUNTING PURPOSES
At 400 yards Range
(Zero at 250 yards)
Selected
Mzzl.
Vel.
BC at
Mzzl.
Energy
Max Bullet Velocity Energy Bullet
Wind
Drift
Bullet
(fps)
Mzzl.
Vel.
(ft-lbs)
Path (in)
(fps)
(ft-lbs) Path (in)
(in/mph)
165 gr SBT 3200 0.404 3751 2.57 @ 2294 1927 -13.72 1.21 GameKing 144 yds 180 gr SBT 3100 0.501 3840 2.65@ 2373 2250 -13.61 0.98 GameKing 142 yds 180 gr SPT 3100 0.407 3840 2.79 @ 2224 1977 -14.68 1.24 Pro-Hunter 142 yds 200 gr SBT 2900 0.560 3734 3.06 @ 2264 2276 -15.29 0.96 GameKing 141 yds
220 gr RN 2750 0.310 3694 4.05 @ 1809 1598 -21.28 1.89 Pro-Hunter 140 yds
TARGET SHOOTING
PURPOSES
At 600 yds Range At 1000 yds Range
Selected
Mzzl.
Vel.
BC at
Velocity
Wind
Drift
Velocity
Wind
Drift
Bullet
(fps)
Mzzl. Vel.
(fps)
(in/mph)
(fps)
(in/mph)
168 gr HPBT 3200 0.462 2024 2.54 1379 8.61 MatchKing 180 gr HPBT 3100 0.475 2026 2.46 1467 7.93 MatchKing 190 gr HPBT 3000 0.533 2022 2.31 1491 7.46 MatchKing 200 gr HPBT 2900 0.565 1992 2.27 1497 7.24 MatchKing 220 gr HPBT 2750 0.629 1952 2.16 1505 6.82 MatchKing 240 gr HPBT 2800 0.711 2078 1.82 1658 5.66 MatchKing
prefer a flat base rather than a boat tail bullet shape, and others prefer a round nose bullet for some game in some terrain. So, Table 2.2-2 includes the 180 grain SPT (Spitzer) Pro-Hunter bullet for direct comparison with the 180 grain SBT GameKing, as well as the 220 grain RN (Round Nose) Pro-Hunter bullet. One can see the 180 grain Spitzer flat base bullet has about 20% lower BC than the 180 grain Spitzer boat tail bullet, and it loses velocity and energy faster, and has about a 25% increase in wind sensitivity. The 220 grain round nose bullet is the heaviest in the table, but it has the lowest BC, loses velocity and energy rapidly as it flies, has the worst trajectory curvature, and
has the worst wind sensitivity. This is a fine bullet for heavy game, but it rapidly loses its advantages at longer ranges.
For target shooting with the 300 Winchester Magnum, Table 2.2-2 shows that the best bullet is the 240 grain MatchKing at 2800 fps muzzle velocity. This is the heaviest bullet and has the largest BC. This illustrates a principle that many target shooters have found generally true with magnum
cartridges for target shooting. That is, select the bullet with highest BC value and load it as fast as it will go and still deliver maximum accuracy considering recoil sensitivity of the shooter as well as accuracy capability of the rifle.
So, considering BC, bigger is sometimes better, but not always. The purposes of the shooter, the shooting situation, and the limitations of the gun and cartridge must be taken into account in choosing a bullet. The best tool to use to examine all the possibilities is one of the ballistics computation software programs for the personal computer. All types of ―what if‖ questions can be explored at the keyboard.
2.3 How the Ballistic Coefficient is Measured
We first began to investigate just how to determine the BC values of various bullets in 1969. We learned very quickly that BC values for all bullets need to be measured by firing tests; there is no other way to make an accurate determination. It is true that in 1936, E. I. Du Pont de Nemours & Company, Inc., published a brochure prepared by two ballistics engineers on their staff, Wallace H. Coxe and Edgar Beugless. This brochure, titled Exterior Ballistics Charts, described a method of finding the form factor of a bullet by matching the point shape against a set of ogive contours, and then looking up the form factor value in a table of values. With the form factor known, the BC could then be obtained from a chart in the brochure. The brochure also contained several pages of nomographs and simple computational techniques to determine trajectory variables, such as
remaining velocity, maximum trajectory height, wind deflection, etc., versus range from the muzzle. The work of Coxe and Beugless was a great step forward at the time. They presented the first method of BC estimation available to the general shooting community, and their nomographs presented a useful method of calculating ballistics parameters long before the age of computers. Their methods were used by handload developers and wildcatters until several years after the end of World War II. Today, however, the work of Coxe and Beugless is mainly of historical interest. We found in 1969 and 1970 that BC values determined by their method simply are not accurate enough by modern standards. BC values really must be measured by firing tests. We have used three methods of measuring BC values from firing tests. The first two of these methods can be used by shooters equipped with a pair of chronographs, a computer, and exterior ballistics software such as the Sierra Infinity program.
2.3.1 Initial Velocity and Final Velocity Method
2.3.1.1 Measurement Procedure
This first method is illustrated in Figure 2.3-1. This method uses two chronographs for each bullet fired to measure an initial velocity and a final velocity at a measured range distance between the chronographs. The initial velocity chronograph is usually placed near the muzzle of the gun, as shown in Figure 2.3-1. A blast shield with a small hole for bullet passage usually is used to keep the muzzle flash or blast from disturbing the screens of the initial velocity chronograph. If the screens are
photoelectric types, as is the usual case, the muzzle flash may trigger screen 1 before the bullet arrives. Also, because the powder gases exit the muzzle at about 1.5 times the bullet velocity, the gases can trigger screen 1. Or, the muzzle blast can cause screen 1 or 2 to bend or vibrate. Any of these effects will cause an erroneous measurement of the initial velocity, which will, in turn, cause an error in the measured BC value.
The final velocity chronograph is placed downrange at a carefully measured distance from the initial velocity chronograph. This range distance is measured from the center point between screens 1 and 2 to the center point between screens 3 and 4. This is because each chronograph really measures the bullet travel time between the two screens to which it is connected (i.e., between screens 1 and 2 or between screens 3 and 4). Then, the velocity is obtained by dividing the precisely measured
distance between the pair of screens by the measured bullet travel time. This calculation is performed within the electronics of the chronograph. Because the bullet slows down a tiny bit as it travels
between the two screens, the measurement of velocity is considered to be valid at the center point between the pair of screens. Of course, if the separation distance between screens 1 and 2 is the same as between screens 3 and 4, the range distance between the two chronographs may be measured from screen 1 to screen 3.
The measurement procedure for each bullet fired is to record the initial and final velocities as well as the range distance between the two chronographs. The altitude, temperature, barometric pressure, and relative humidity at the firing point must also be recorded. If the shooting range is not level, the elevation angle must be recorded, especially if it exceeds about 3 degrees either upward or
downward. If there is any appreciable wind at the firing point, BC measurements should not be attempted.
With these data for each bullet fired, an exterior ballistics software program for a personal computer can be used to calculate the ballistic coefficient. Some exterior ballistics programs contain an optional routine for computation of the BC value, but that is not necessary. The normal trajectory computation routine can be used in an iterative fashion for each bullet fired. That is, first initialize the program by entering the altitude, temperature, barometric pressure, humidity and range elevation angle at the firing point. Then, for each round fired enter the measured initial velocity as the ―muzzle velocity‖ for the trajectory calculation. Then, guess a BC value, calculate a trajectory over the measured range distance, and examine the calculated final velocity. If the calculated final velocity is higher than the measured number, the BC value is too high. (Conversely, if the calculated final velocity is lower than the measured value, the BC value is too low.) Then, reduce (or increase) the BC value a little,
calculate another trajectory over the measured range distance, and examine the calculated final velocity. The calculated final velocity should be nearer the measured value. If the calculated value from this second iteration is higher (or lower) than the measured value, reduce (or increase) the BC value in the program and perform another iteration of the calculations. After a few iterations, this method will ―home in‖ on a correct value for the measured BC for the first round fired. Of course, this is a BC value for which the calculated final velocity matches the measured final velocity as closely as possible. This resulting BC value is considered valid for a bullet velocity midway between the initial and final velocities for that round.
For the other rounds fired, the resulting BC for the first round can be used as the initial guess for the BC value, and fewer iterations will be required to reach a correct BC value for each of those rounds.
Figure 2.3-1 Test Range Setup for Initial and Final Velocity Method for BC Measurement
An example has been prepared to illustrate this method of determining the BC. Suppose we have developed a load for the 308 Winchester (7.62 x 51 mm NATO) cartridge in a bolt-action rifle that pushes a certain .308 diameter 160 grain bullet at a muzzle velocity of about 2750 fps. We do not know the BC of this bullet type and want to measure it on our local shooting range. (Actually, we do not need to know the weight of the bullet or even the caliber to determine the BC. We need only the firing test data as described below.) Suppose that our shooting range is located at an altitude of 790 feet above sea level, and we perform the shooting tests on a day when the temperature is 78° F, the barometric pressure is 30.15 inches of Mercury, and the relative humidity is 80%. Note that the barometric pressure is obtained from a barometer at the range or from a local weather report for the time of day when the firing tests take place. The test range is level, and the range distance between chronographs is 103 yards, which is measured precisely and accurately when we set up the range for the tests. We fire, say, ten rounds to obtain an average BC value for this bullet type at velocities in the vicinity of 2750 fps. We record the altitude, atmospheric conditions, range distance between the chronographs, and the initial and final velocities for each round fired. Then, we retire to our computer at home.
We start up the Sierra Infinity program, and it comes up automatically in the ―Trajectory‖ mode of operation. Suppose that for the first round fired at the range, the initial velocity was 2742 fps and the final velocity was 2549 fps, as read from the initial and final velocity chronographs. In the Infinity program we select any 30 caliber bullet in the ―Load Bullet‖ library, and transfer it to the ―Active Bullets‖ list in the upper right corner of the blank part of the screen. Then, we initialize the trajectory computation as follows:
Trajectory Parameters
Units: Full English (since we are working in the English system of units) Muzzle Velocity: 2742 fps (for the first round fired) Maximum Range: 103 yds (this is as far as we need the trajectory to be
computed) Range Increment: 1 yd (because the distance between chronographs is 103 yds, which is not divisible by any number other than 1) Zero Range: 103 yds (the distance between chronographs) Elevation Angle: 0 (because the test range is level) Sight Height: 1.5 inches (choice for telescope sight on the rifle)
Environment Parameters
Barometric Pressure: 30.15 in Mercury (from a barometer at the range or a local weather report) Temperature: 78° F (from a thermometer at the range or local weather report) Altitude: 790 ft (can be obtained from a topographical map or other source) Humidity: 80 % (relative humidity from a weather station at the range or from the local weather report) Wind Direction: Any number between 0 and 12 o’clock is OK
Horizontal Wind Velocity: 0 mph (no wind is very important) Vertical Wind Velocity: 0 mph (no wind is very important) At this point we have initialized a trajectory computation for some 30 caliber bullet (we don’t care which one) with the correct muzzle velocity, range distance between the chronographs, trajectory calculation parameters for our purposes, and environmental conditions at the firing point. But, we haven’t performed any trajectory computation yet, so there is nothing on the monitor screen yet. Now, although we will not use it explicitly, we must ―Calculate‖ a trajectory so that the ―Trajectory Variations‖ menu item will be available to us.
We then go to the Infinity toolbar at the top of the monitor screen and select ―Trajectory Variations.‖ From the dropdown menu that appears, we select ―Ballistic Coefficients.‖ In the sidebar at the right side of the monitor screen, we then see five values of ballistic coefficient listed. These values mean nothing to us since they are for the bullet that we chose to load into the ―Active Bullet‖ list, not for the bullet that we are testing.
It is necessary now to make an initial guess for the BC value of our test bullet. If we guess well, we will not have to make many computation iterations to find the correct BC value. For a 30 caliber bullet that weighs 160 grains and has a Spitzer (sharp pointed) shape, the BC value at around 2600 fps should be somewhere near 0.5. So, let us choose this value as the initial guess. We then change the five numbers in the right-hand sidebar on the monitor screen to the value 0.5.
We change all the BC numbers for a particular reason. As we will explain later, Infinity allows the ballistic coefficient of each bullet type to change with bullet velocity as it flies downrange and slows down. This is because the measured BC of a bullet does change with velocity, and accounting for such changes can increase the accuracy of trajectory computations within Infinity. We use five
velocity regions for this purpose. Within each velocity region there is a single value of BC valid for that region, and there is a value for each of the five regions. There are then four velocity boundaries
separating these regions. When the velocity of a bullet falls through one of these boundaries, Infinity automatically changes the BC to the value for the new region. In our current case, we do not know whether our test bullet starting at 2742 fps and ending up at 2549 fps crosses a velocity boundary for the bullet we are using.Yes, we could look to see and make a more educated selection of the one or two BC values that we would need to change, but if we change all five of the values in the sidebar we will be sure to be safe.
After we change the BC numbers in the sidebar to 0.5, we are ready to begin the iterative search procedure for the correct BC value for the first round fired. Table 2.3-1 summarizes the computations in the search procedure. To begin the procedure, click the ―Calculate‖ button on the bottom of the monitor screen. Infinity performs the first trajectory computation, the trajectory parameters appear on the screen, and we immediately scroll down to the final parameter values at 103 yds range. We find that the computed final velocity at 103 yds is 2564.8 fps for this first iteration (see Table 2.3-1). This is higher than the measured final velocity for this round 2549 fps, so the next guess for BC needs to be lower than 0.5.
At this point we have no idea how much to lower the next BC guess, but let’s try 0.4. We set the five BC numbers in the sidebar to 0.4 and click the ―Calculate‖ button for the second trajectory
computation. The computed final velocity at 103 yds for the BC equal to 0.4 is 2521.5 fps, which is too low compared to the measured 2549 fps. So for the third iteration, the guess for BC needs to be raised. Let’s try something halfway between 0.4 and 0.5; that is 0.45.
We change all five BC numbers in the sidebar to 0.45 and click the ―Calculate‖ button for the third trajectory computation. For this third iteration, we find that the computed final velocity at 103 yds is 2545.5 fps, which is closer but still lower than the measured 2549 fps. So, for the next iteration let’s raise the BC guess to 0.46.
Again, we change all five BC numbers in the sidebar to 0.46 and click the ―Calculate‖ button for the fourth trajectory computation. For this fourth iteration, we find that the computed final velocity at 103 yds is 2549.7 fps (as shown in Table 2.3-1). This is just a little higher than the measured 2549 fps. So, for the next iteration we must lower the BC guess just a little.
Table 2.3-1 shows the final three iterations, each of which follows the same procedure. In each iteration we change the BC guess by a smaller amount so that the computed final velocity
approaches the measured final velocity. The seventh iteration, which has a BC of 0.4583, produces a final computed velocity equal to the measured velocity, and this BC therefore is the correct value for this first fired round.
Table 2.3-1 Example of Ballistic Coefficient Iterative Search Procedure
Test Range Parameters:
Distance between chronographs: 103 yds Range altitude: 790 ft above sea level Temperature: 78º F Barometric pressure: 30.15 in Mercury Relative humidity: 80% Exterior Ballistics Program: Sierra infinity
Test Round 1: Initial velocity 2742 fps; final velocity 2549 fps
Iteration
BC
value
Computed final velocity
1 0.5 2564.8 2 0.4 2521.5 3 0.45 2545.5 4 0.46 2549.7 5 0.458 2548.9 6 0.459 2549.3 7 0.4583 2549.0
Test Round 2: Initial velocity 2751 fps; final velocity 2556 fps
1 0.4583 2557.6
3 0.455 2556.2
4 0.454 2555.8
5 0.4545 2556.0
Table 2.3-1 also shows the iterations for the second fired round. In this example we suppose that the second round has a measured velocity of 2751 fps and a measured final velocity of 2556 fps. To initialize for the second round, we momentarily return to the ―Operations‖ selection on the Infinity toolbar at the top of the monitor screen, select the ―Trajectory‖ mode of operation, and change the ―Muzzle Velocity‖ entry in the ―Trajectory Parameters‖ sidebar to 2751 fps. Again, we must ―Calculate‖ so that the ―Trajectory Variations‖ menu item is available. Then, we return to the ―Trajectory
Variations‖ selection on the Infinity toolbar, again select ―Ballistic Coefficients‖ on the dropdown menu. We verify that all five entries in the BC sidebar on the monitor have the value 0.4583 from the first round. Note that we have not changed any of the other Trajectory Parameters or Environment Parameters, since they all have the same values for our firing tests.
Table 2.3-1 shows the sequence of iterations for the second round. The first iteration with a BC guess of 0.4583 produces a computed final velocity of 2557.6 fps, higher than the measured final velocity of 2556 fps. The second iteration with a BC guess of 0.457 produces a computed final velocity of
2557.1, closer but still higher than the measured 2556 fps. The third iteration with a BC guess of 0.455 produces a computed final velocity of 2556.2 fps, even closer but still a little higher than the measured 2556 fps. The fourth iteration with a BC guess of 0.454 produces a computed final velocity of 2555.8 fps, which is lower than the measured 2556 fps. Since the results of the third and fourth iterations equally straddle the measured 2556 fps, the fifth BC guess is chosen as 0.4545, halfway between 0.455 and 0.454. This final iteration produces a computed final velocity equal to the measured 2556 fps, so this value is the correct value for the second bullet fired.
The same procedure should be followed for each of the remaining eight bullets in the test series that we fired at the test range. In this way, we will derive the measured BC values for all ten bullets and can apply statistical analysis to this limited sample of test bullets for this type of bullet at velocities between the initial and final values.
The computations in this example have been explained in some detail, so that the reader can repeat these calculations step by step if he or she uses Sierra’s Infinity program. If a different exterior
ballistics program is used, the detailed steps of the procedure should be changed because any other program will function a little differently, but the basic method will not change. The idea is to find a BC value that makes the computed final velocity equal to the measured final velocity for each bullet tested. This will be an iterative, trial and error procedure. This example will still be useful as a guide even if a different exterior ballistics program is used, because BC values very close to the ones produced by using Infinity should result. This can serve as a check on the procedure developed for any other program.
2.3.1.2 Important Precautions and Points to Consider
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.
2.3.2 Initial Velocity and Time of Flight Method
This method, illustrated in Figure 2.3-2, uses two chronometers (time measuring instruments), which measure the time of flight t12 between screen 1 and screen 2, and the time of flight t13 between screen 1 and screen 3 for each round fired. The screen separation distances d12 and d13 are measured precisely and very accurately, and these distances remain the same for all rounds tested. Screens 1 and 2 are located close to the muzzle of the gun and are separated by at least 10 feet. These screens then provide an accurate measurement of the initial velocity of each round, which is obtained by dividing separation distance d12 by time of flight t12 for each round. This initial velocity is valid at a point halfway between screens 1 and 2, and this point is then the reference point for
computations performed by Infinity. As in the previous method, a blast shield with a small hole for bullet passage usually is used to protect screens 1 and 2 from muzzle blast, muzzle flash and powder gases exiting the muzzle.
Figure 2.3-2 Test Range Setup for Initial Velocity and Time of Flight Method for BC Measurement
For each round fired the recorded measurements are the times of flight t12 and t13. Because the reference point for the trajectory calculations to be performed in Infinity is halfway between screens 1 and 2, corrections must be applied to the range distance and the bullet time of flight for each round. The corrected range distance for all trajectory calculations is the distance d13 minus half the distance d12; that is, the distance from the center point between screens 1 and 2 to screen 3. The time of flight for the trajectory calculation for each round is the measured time t13 minus half the measured time t12; that is, the bullet time of flight from the center point between screens 1 and 2 to screen 3. When the firing test data have been obtained, Infinity is initialized with the measured altitude,
atmospheric conditions and the corrected range distance for all rounds. Then, for each round fired, an iterative search for the correct BC value takes place in the same manner explained in the previous method (Section 2.3.1.1). In each iteration a BC value is guessed, a trajectory is calculated using the measured initial velocity out to the corrected range distance, and the calculated time of flight is inspected. The correct value of BC has been found for the test round when the calculated time of flight matches the corrected measured time of flight as closely as possible.
The precautions and points for consideration described in Section 2.3.1.2 for the initial velocity and final velocity method also generally apply to the present method. There is one significant difference. There is one less systematic error source in the initial velocity and time of flight method. There is no final velocity chronograph in this method, so the very sensitive systematic error source associated with measuring the screen separation distance for the final velocity chronograph does not occur. However, any measurement error in the separation distance between screen 1 and screen 2 will cause a highly sensitive systematic error in the BC values. A measurement error in the distance between screen 1 and screen 3 will cause a less sensitive systematic error in the BC values determined by this method. Any digital instrument that measures the travel times between the screens should contribute little or no errors in the measured BC values, because elapsed times can be measured very precisely and accurately.