William
T. McDonald, Sc.D., and Ted C. Almgren
**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.