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.