3.0
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Historical Summary
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Ballistics has existed as a technical art for thousands of years. The word ballistics itself can be traced to ancient Greeks who wrote about the design of throwing machines, called by a phonetically similar name, about 300 years B.C. Evidence of the development of throwing instruments and specialized projectiles dates back to Stone Age Man.

The development of ballistics as a true science began about the time that firearms were introduced into warfare in western Europe, near the beginning of the fifteenth century. That development has spanned the hundreds of years right up to the present day. The names of some of the greatest scientists and mathematicians in history are associated with the development of ballistics, including Leonardo da Vinci, Galileo, Isaac Newton, Francis Bacon, and Leonard Euler. From the earliest days of firearms, rulers were very much interested in ballistics for a simple reason—battles were won by armies with the best guns and gunners. The best scientific brains available were put to work on the problems of designing better guns (cannon, mortars computing accurate trajectories. In those days, just as at the present time, governments spent money willingly to gain more powerful and more effective armament. It is wd effort spurred the development not only of ballistics, but of mathematics and other physical sciences, particularly mechanics, dynamics, and aerodynamics.

A bullet traveling through the air is acted on by two distinct kinds of forces, the force of gravity and the force due to the air flow around the body. After a bullet leaves the gun barrel, the trajectory is completely determined by these forces. In the early days of ballistics science, nothing was known about the aerodynamic force, and very little was known about gravity. It seems strange now, but for about 200 years after firearms first appeared in western Europe in the early 1300’s, it was not known that the shape of a bullet trajectory is a curve. One of the earliee velocity slowed to near zero, and then fell more or less straight to the ground. Whenis notion was wrong, so another was conceived. This second notion described the trajectthe ground at the strike point of the shot (like two legs of a triangle), and these two lines were joined by a curve near the top of the trajectory.

In 1537 an Italian scientist known as Tartaglia wrote a book in which he said that the trajectory of a bullet was really a continuous curve. Tartaglia was a ballistics consultant to the Italian principality of Verona, and he was asked to determine what elevation angle for a gun achieved the maximum shot range. He directed some firing tests to determine this angle, and discovered that it was near 45 degrees. In this process he also noted that the shot trajectory was continuously curved.

An interesting observation today is that Tartaglia’s experiments resulted in an elevation angle near tion angle of 45 degrees maximizes the range of a projectile only if it is fired in a vacuum, where aerodynamic forces are absent and the trajectory shape is determined only by gravity. With today’s guns, the firing elevation angle which maximizes range is nearer 30 degrees than 45, because trajectory shapes are determined almost completely by aerodynamic forces at velocity levels achieved oday’s gury. This explains Tartaglia’s result. This result had an important significance in Galieo’s theoretical work which took place nearly a century after Tartaglia.

In 1636, Galileo published results from his famous experiments and was able to give a reason why the trajectory is a curve. Galileo was a ballistics consultant to the arsenal at Venice for a number of years. The famous experiment in which he dropped two cannonballs from the Leaning Tower of Pisa was only one of a long series of experiments having to do with the effects of gravity. Galileo determined that the acceleration of a falling body by gravity is a constant, and with this result he was able to show that a bullet trajectory was a type of curve called a parabola.

Galieo’s resuesult ich maximizes range is exactly 45 degrees, matching Tartaglia’s experimental result. This consistency between theory and exs.

Galileo’s work foundat neededd the very important effect of air drag. He neglected air drag because he thought it was a small factor compared to gravity, and there was absolutely no way available in his time to measure it.

Another hundred years went by before a way to measure velocity was invented. During 1740 in England, Benjamin Robins invented the ballistic pendulum. This was simply a pendulum with a large, heavy wooden bob. To make a velocity measurement, the bullet was weighted and the bob was weighted. Then the pendulum was positioned with the bob hanging motionless and the bullet was fired into it. By measuring the height of the pendulum swing resulting from the bullet striking the bob, the velocity of the bullet could be computed. Robins made a series of measurements with .75 caliber (12 gauge) musket balls, including measurements of velocity near the muzzle and at several ranges from the muzzle. Robins reported muzzle velocities ranging from a little over 1400 to a little under 1700 feet per second. These numbers were astounding, so much so that they were widely disbelieved. Even more astounding, however, were the measurements of velocity drop with range from the muzzle. In order to account for Robins’ measu that f aerodynamic forces on bullets, and the ballistic pendulum was used for many years to measure the effects of these forces.

The ballistic pendulum was based upon laws of mechanics formulated by Sir Isaac Newton, who died about 15 years before Robins published his measurements. Newton ranks as probably the greatest scientist of all time. His work established physical laws and mathematical techniques which are the basis for several branches of science, ballistics included. Newton formulated the universal law of gravitation, which shows that gravity varies with altitude above the earth. This law is important in the computation of ballistics of high altitude rockets. He formulated the fundamental laws of mechanics (which we call “Newton’s laws” todaymany), which are the necessary mathematics for the computation of bullet trajectories.

Newton and a number of his contemporaries had also been interested in air drag. Newton himself performed some experiments on the drag experienced by pellets falling through air and through fluids. He was able to show that the drag on a pellet increases with the density of the air (or the fluid), the cross sectional area of the pellet, and the square of the velocity of the pellet. These were fundamental discoveries, and were true, but Newton’s experiments involved only low velocities, f sound (1120 feet per second in air), the drag would increase much more rapidly than his low velocity experiments predicted.

Between the middle 1700’s and the late 1800’s ballisticians devaccurately predicat it moves. Accurate measurements of drag only became possible in the late 1800’s when chronographs were invented in Germany and England.

Within the same period of time firearms design advanced rapidly, and the range and accuracy of artillery and shoulder arms improved considerably. Rifled barrels and elongated bullets were widely adopted in military small arms in the early 1800’s, and somewhat later in artillery. Percussion ignition also was widely adopted in the early 1800’s, and metallic cartridges and breech loading arms began to appear in the mid-1800’s. The theory of interior ballistics began to grow rapidly in the late 1700’s and smokeless powder appeared in the late 1800’s. These advances all increased the need for good exterior ballistics, so that accurate long range fire for both artillery and small arms could be obtained.

The theoretical drag investigations, which never developed as desired, did point out a major simplification in the experimental treatment of drag. This was the concept of a “standard bullet” which was deveas “standard,” precise drag measurements could be made for that bullet, and the drag deceleration of another bullet of that shape would then be related to the “standard drag” decelerationase, there would be little ballistics data available today.

This was the birth of the ballistic coefficient. It is the factor that relates the drag deceleration of an actual bullet to the drag deceleration of the standard bullet. The ballistic coefficient of a bullet is usually call
**
C
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, and it can be defined as follows:

Actually, this relationship is regarded as strictly true only if the actual bullet is an exact scale model of the standard bullet. It was found experimentally, though, that the relationship holds well enough, even when the bullet shapes are slightly dissimilar, to allow fairly accurate ballistic computations for the actual bullet.

It also turned out that the ballistic coefficient
**
C
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led to another advantage of major importance, which had to do with the method of computing bullet trajectories. It was mentioned earlier that calculus must be used to compute a bullet trajectory. The numerical computations necessary to calculate just a single trajectory are very lengthy and tedious when they must be done by hand, which was necessary until only a few years ago. The work involved in computing all the trajectories of interest would have been prohibitive by this method. About 1880 an Italian ballistician named Siacci discovered a way to greatly simplify this problem. He showed that the ballistics of an actual bullet could be computed from the ballistics of the standard bullet in a simple way with the use of the ballistic coefficient
**
C
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. By this method a single trajectory was computed for the standard bullet by the tedious methods of calculus, and then a trajectory for any actual bullet with known
**
C
**
was computed from the standard trajectory with just simple algebra. Siacci’s method shortcut a great amount of computation, and it has been used widely in ballistics ever since.

Between about 1865 and 1930 many firing tests were conducted almost worldwide to determine the drag characteristics of standard bullets adopted by different countries. Of particular note were tests made by Krupp in Germany in 1881 and by the Gavre Commission in France from 1873 to 1898, although many other tests were made. The Gavre Commission work was very comprehensive, including not only extensive firing tests going up to a velocity of 6000 fps but also a comprehensive survey of data available from tests in other countries. The Commission attempted to correlate all these data, and it published a composite drag characteristic for a certain standard bullet configuration. Unfortunately, the effects of varying atmospheric conditions during the many tests were not well understood at the time, and the Gavre drag function contained some errors. Shortly afterward, standard atmospheric conditions for the expression of drag data were adopted and used.

The Krupp test data turned out to be the basis for ballistics tables for small arms, especially sporting and target ammunition, right up to the present time. The standard bullet used by Krupp was a flat base design, 3 calibers long with a 2 caliber ogive head. After the Krupp drag data were published, a Russian Army Colonel named Mayevski constructed a mathematical model for the standard drag deceleration for this bullet. Colonel James M. Ingalls, U.S. Army, used Mayevski’s analytical model and computed and published his now famous Ingalls Tables this standard bullet has been a very fine model for use in computing the ballistics of most bullets for sporting use.

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Author’s Note)
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The Ingalls Tables were published in the U.S. Army Artillery Circular M. This document is relatively well known compared to U.S. Army Artillery Circular N. Artillery Circular N contains the theory from which the Ingalls Tables are computed. The importance of Artillery Circular N has seemingly been neglected through the years, probably because it is intensively mathematical. In the Preface to the 1890 edition of Artillery Circular N, Ingalls credits Colonel Siacci of the Italian Artillery for the origins of the subject matter, and the work of one of Siacci’s associates, a Captain Scipione. These works included the closed form solutions to the equations of bullet flight described in
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Section 6.0
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of this article. This author (WTM) is grateful to Mr. John Villarreal of Broken Arrow, OK for making us aware of the existence of Artillery Circular N.
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Once the concept of the standard bullet was adopted, the one remaining problem was how to determine the ballistic coefficient
**
C
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for each actual bullet. One method of determining
**
C
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is firing tests. This method will be more fully described later, but the basic idea is to measure the ballistic properties of the actual bullet, compare them with the ballistic properties the standard bullet would have if fired at the same muzzle velocity, and determine the right value of
**
C
**
to make them match. Tests should be made at several muzzle velocities to see how well the standard drag model fits the actual bullet.

Another method of determining
**
C
**
by shape comparison was developed and published by Wallace H. Coxe and Edgar Beugless, ballistics engineers at the DuPont Company in the 1930’s, and it has been widely used ever since. The method estimates the ballistic coefficient related to the drag model of the Ingalls Tables. It f shapes for which ballistic characteristics are known. When a match is made as nearly as possible, the chart provides a number from which
**
C
**
can be quickly calculated. In Sierra’s experience, this procedure gives a ex point shapes, or boat tails (which reduce tail drag). However, the Coxe-Beugless method was a great step forward in the 1930’s and it is still widely used because of its simplicity, and because the ballistics data resulting from this method of determining
**
C
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are reasonable for most hunting situations.

Ballistics research intensified during World War II for small arms, artillery, and aerial bombs. A new technical breakthrough entered the ballistics scene about that point in time, the electromechanical analog computer. The electromechanical computer marked the birth of the modern age of computers and computer science. By today’s digital computer standards, it was slow and not very accurate, but it could solve calculus problems, and so it was applied to bthemselves emancipated from tedious manual computations, and this opened up new vistas for ballistics research. The electromechanlem for aerial bombs dropped from high-flying WW II bombers, and this was a major step forward in ballistics.

The small arms ballistics research during and after the War tended toward more accurate mathematical models of bullet flight, enabling more accurate trajectory computations, especially as computers improved in speed and numerical accuracy. It had been apparent for many years that a single standard drag model (i.e., the Mayevski model) could not serve with a high degree of accuracy for all bullet shapes. Even with the most modern digital computer, of course, it isn’t practical to have a separate drag function for each type of bullet tail bullets), and to have a separate drag function for each family

In 1965 Winchester-Western published a set of ballistics tables (see list of references at the end of this article) based on four standard drag models for four families of bullets, defined as follows:

**
G
**
**
1
**
drag function — for all bullets except those in the categories below

**
G
**
**
5
**
drag function — for low base drag bullets (boat tails or tracers)

**
G
**
**
6
**
drag function — for flat base, full patch, sharp nose bullets

**
G
**
**
L
**
drag function — for hollow point lead nose bullets

The drag functions
**
G
**
**
1
**
,
**
G
**
**
5
**
, and
**
G
**
**
6
**
**
**
had been developed in earlier research at the U.S. Army Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland. These four drag models were considered adequate to cover the vast majority of sporting bullets.

Drag functions for standard bullets have been called by the letter
**
G
**
for many years in recognition of the extensive work done by the Gavre Commission in France. The drag function
**
G
**
**
1
**
**
**
is almost identical to the drag function used in the Ingalls Tables. Ballistic coefficients for the Sierra bullets can be used with the Ingalls Tables without error.

It should be mentioned in passing that ballistic coefficients always relate to a specific standard drag function, since they express the relation between the drag deceleration of actual bullets and the drag deceleration of a particular standard bullet. Each
**
G
**
function described above has its own particular standard bullet. For example, if firing tests show that the
**
G
**
**
5
**
drag model most closely matches the drag characteristics of some actual bullet, then its ballistic coefficient (call it
**
G
**
**
5
**
) relates to
**
G
**
**
5
**
. Any attempt to specify a constant ballistic coefficient
**
C
**
**
1
**
**
**
related to
**
G
**
**
1
**
**
**
would be unsuccessful, because
**
G
**
**
1
**
**
**
just does not model the bullet drag.