6.0  Equations of Bullet Flight

6.0  Equations of Bullet Flight

Many shooters and readers of earlier editions of the Sierra Reloading Manual have written to us over the years with questions about the theory and equations of exterior ballistics. We have referred those persons most often to the book Exterior Ballistics by McShane, Kelley, and Reno. This book, published in 1953, contained much information stemming from research prior to and during World War II. We have found it an excellent reference, and practically the only reference available to the general public. Modern work in ballistics has been done by the military departments and other government agencies, and results are not generally available to the public.

The book by McShane, Kelley, and Reno has long been out of print, and we have reports from several of our readers that it can no longer be found even in advanced technical libraries. Because so many of our readers need a basic reference on the theory of exterior ballistics, we have decided to include this section in this edition of the Sierra Reloading Manual.

This section will derive the differential equations of bullet motion, treating the bullet as a point mass with a ballistic coefficient. It will describe the drag model, and develop Siacci’s approach and Mayevski’s analytical descriptiolet trajectories.

We will generally follow the developments in McShane, Kelley, and Reno, even though there are more modern and up-to-date treatments of slender bodies moving in air. We have taken this approach for two reasons. First, some readers do have access to that reference, and our developments will therefore not be strange. Second, the drag function G 1 is historically and technically linked to the developments cited in that reference, and we will therefore avoid entangling ourselves in explanations of other technical approaches.

Treating the bullet as a point mass with 3 degrees of freedom, we neglect 6 degree of freedom effects on the spin-stabilized bullet such as Magnus forces and the dependence of aerodynamic drag and lift on  bullet precession. Also neglected are Coriolis effects related to rotation of the earth. The effects of altitude on air density and speed of sound, and in turn their effects on drag, are treated in the following developments.