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Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles

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Modern Exterior Ballistics
The Launch and Flight Dynamics of Symmetric Projectiles.

I Want to Keep This Book. Thanks to the auspices of interlibrary loan, it's mine for three weeks. That's not enough. This book really has said everything there is to say about the subject, except for the classified data you would need to calculate trajectories of actual projectiles in the inventory.

Chap 1. A Brief History of Exterior Ballistics - nice shadowgrahs of near sonic and super sonic projectiles. Large drawings of several shell types showing how they are dimensioned.

Chap 2. Aerodynamic Forces and Moments Acting on Projectiles - you have to read this chapter if you are going to understand the symbols.

Chap 3. The Vacuum Trajectory-high school math with interesting examination of the effects of firing up hill or down hill.

Chap 4. Notes on Aerodynamic Drag - just what it says. There is a nice set of shadowgraphs showing a shell at successive mach numbers from sub sonic to transsonic to supersonic. Discussion of ogives, Meplats, burning tracer (provides a little thrust and more range), fins, and yaw.

Chap 5. The Flat-Fire Point Mass Trajectory-if you assume that the trajectory is fairly flat (such as with all small arms, rifles, and tank to tank combat), the effect of verticle motion on the down range motion can be neglected. Approximate the drag function by a simple (and useful) analytic function and you get some managable equations.

Chap 6. The Siacci Method for Flat-Fire Trajectories - More of the above, only the integrals are tabulated for several standard drag functions (about five pages for each drag function). Those tables plus pencil and paper and you too can calculate how far the 0.308 bullet travels and how fast it is going when it hits and at what angle it strikes.

Chap 7. The Effect of Wind on Flat-Fire Trajectories - head winds, tail winds, cross winds: pretty much what you would expect.

Chap 8. The Point-Mass Trajectory - the point mass doesn't yaw or pitch so we only have to worry about its three linear velocities. It only has to contend with the zero yaw drag function, air density, gravity and the Coriolis force. (These last three data sets are not classified). A 417 line basic program is included. With this you can calculate your rifle shots better than you can aim.

Chap 9. Six-Degrees-of-Freedom (6-DOF) and Modified Point-Mass Trajectories - The whole enchilada. Three linear velocities and three rotational velocities are considered. They are all coupled to each other by nonlinear functions, and the (ten or so) coefficients are functions of velocity and air density. Fascinating graphs of gyrating pitch and yaw of a 105mm projectile as it goes down range are included. Then come the modified point mass trajectories. This simplification reduces the degrees of freedom from six to four which allows some of the coefficients to be set to zero while still revealing considerable interesting behavior. The forth degree of freedom is angular velocity of the bullet along its long axis.

Chap 10. Linearized Pitching and Yawing Motion of Rotationally Symmetric Projectiles - ok, another simplification: assume that the pitching and yawing motions are small and linearize the functions. There is still a nasty differential equation, but you can (sort of) solve it and predict such things as the stability of the bullet as it goes down range without a numerical integration.

Chap 11. Linearized Swerving Motion of Rotationally Symmetric Projectiles - same idea as chapter 10, but applied to the position of the projectile's center of mass.

Chap 12. Lateral Throwoff and Aerodynamic Jump - what to do about unbalanced projectiles.

Chap 13. Nonlinear Aerodynamic Forces and Moments. - I don't know; I ran out of time.

This is ballistician's dream book. It describes all of the introductory and some advanced material pertaining to modern ballistics in wonderful detail, and the example problems are useful and illustrative. There is a wealth of information, such as drag coefficients and functions, form factors, and a beautiful derivation of the MPM yaw of repose that do not appear anywhere else in my experience. The experimental and computer generated plots are beautiful and informative.

Unfortunately, the book is also full of typographical and other errors, such as having the wrong variable in a place in several equations, having variables simply missing from equations or from an explanation in the text, having the wrong headings in some of the tables, repeating figure numbers on different figures, or even having multiple missing words in some explanatory paragraphs that make the sentences into hard to interpret giberish. Some of the citations,for instance in Chapter 9, do not appear in the Bibliography at all (Chapter 9 has references numbered up to 16 in the text, but only 10 appear in the Bibiolgraphy).

The bottom line: This is a GREAT BOOK if you are mathematically sound and don't mind deriving some of the equations yourself, and if you know enough to catch the errors and make the corrections. Don't get me wrong... I like the book a lot. It is well worth its price. Just be aware that you will have to work a little harder than usual to read and understand it.

[...]

book Modern Exterior Ballistics was published in 2004. I am an active ballistician and find that this book is, well, to be generous dated, very dated. The mathematical formulation is overdrawn to the point of being incomprehensible in places. First, the use of the old i,j,k unit vectors is pretty much been overtaken by direcitonal cosine matrices and column matrix representations of a vector. Why not write your mathematical formulations so that they are readily adaptable to computer code formulation. Computer codes enter vectors as column matrices, not in a i,j,k unit vectors. On page 183 and following he gives a BASIC computer program. While BASIC has generally fallen unjustifiably into disuse and been supplanted by C++ and MATLAB with FORTRAN clinging on here and there. But the version of BASIC given by McCoy is just not used today (line numbers? whew). No serious ballistician would use the Sicacci method for flat fire Trajectories. Computer versions written in even a Student version of MATLAB to say nothing of a professoinal version or in C++ renders this method obsolete now and for all time. This book was published in 2004; these methods were obsolete before the book came out. The book is a complete mis-mash and while it might make sense to the author, it would be really confusing to a beginning aero-ballistician trying to learn the trade. Aside from that McCoy give the drag data for a Sierra 168 grain bullet along with the drag coefficient vs Mach no. Unless he gives me the muzzle velocity, I can't do the trajectory model. He then gives data. On page 182 he gives the drag coefficient of a 155 mm M107 projectile, but where is the weight and muzzle velocity? Look, to calculate a point mass trajectory you have to know three things: (a) muzzle velocity or initial speed, angle of lanunch and weight of the projectile. A drag vs. Mach No. is also necessary when there is a high subsonic to supersonic speed range. Other than that, everything else is environmental. With all the graphs, tables thrown around helter skelter cannot these three items be given for an adequate results. Just a sloppy job all around. A much better book, also available from Amazon is Ballistics: Theory and Design of Guns and Ammunition by D.E. Caalucci. One of the reviewers says McCoys book is some kind of treasure. Look at Calucci's book and you will change you mind.

Math & Physics

First of all, I think it is important to identify exactly what is meant by some reviews indicating that the book is heavy on the mathematics and physics because it depends on your background. I'm a graduate student in Aerospace Engineering with an undergraduate bachelor's degree in Applied Mathematics. Much of my academic work in drag and aerodynamics is done on a calculus or partial differential equations level. The mathematics in this book do not exceed the high school or undergraduate calculus level, and most of it being accessible to someone with an Algebra II / Math Analysis education. Only some of the more advanced stuff is presented in integral calculus, but I think it could still be enjoyed by the curious shooter or ballistician with an algebra-level math background, although it may be challenging.

How to Use this Book

As far as uses, I approached this book for two reasons. First of all, it caught my eye as a shooter and enthusiast in ballistics several months before I bought it. I have yet to check out the sections on dynamics and spin and trajectories in great detail, but let me say this - it doesn't look like you should expect to be able to buy this book and calculate exact trajectories of your favorite .270 or .30-06 loads to get a leg up for next deer season. Unfortunately, just about no analysis or numerical solution can beat time and lead spent on the range for that.

Engineering Students

Secondly, I actually bought the book for research project I was doing on supersonic boundary layers for a graduate course.

Anyone seeking a resource on that level should probably look to more mathematical material as noted below; however, "Exterior Ballistics" IS a great resource for the mathematically inclined shooter, amateur ballistician or undergraduate mechanical or aerospace engineer seeking a more broad and fine-grained understanding of what all goes into a projectile's flight. It is a great resource for anyone being introduced to compressible flow who is interested in practical examples of drag profiles of actual projectiles in flight. However, the caveat with many of the figures and graphs in this book is that because they are empirically derived, many of them do not provide a lot of insight with respect to specific physical phenomena that are going on. For instance, in some of the graphs depicting zero-yaw drag coefficient as a function of varying headshape for a range of Mach numbers - it is not possible to tell what exactly is causing net drag to drop - whether viscous drag is dropping in the boundary layer due to delayed separation, or whether changes in the shock geometry or wake behavior are lowering pressure drag. So there are some physical ambiguities there that require these figures be studied carefully.

As a preliminary let me just say that this book says NOTHING about boundary layers or compressible shock behavior; in fact, the author specifically disclaims this responsibility near the beginning of "Chapter 4: Notes on Aerodynamic Drag", citing:

"No attempt is made in this chapter to discuss the basic fluid dynamic and thermodynamic processes involved in the formation of boundary layers and shock waves. This book is about exterior ballistics, and space does not permit even a cursory review of the modern science of aerodynamics. References 9 through 12 are recommended to the reader who is interested in a more complete understanding of compressible flow, shock waves, and boundary layer theory."

I have no doubt that given McCoy's CV and background that he was extremely well versed in BLs, compressible flow and fluid dynamics; however this book doesn't attack those subjects academically. While this was admittedly frustrating to me, the resources cited in References 9 - 12 ARE excellent choices - specifically Shapiro's "Dynamics and Thermodynamics of Compressible Fluid Flow" (1953) and Schlichting's "Boundary Layer Theory" (1955) which I cited in my own research. Anyone seeking a graduate level discussion of compressible boundary layer theory should seek out these two resources.

Typographical Erorrs (da dun *TSH*)

Quite a bit has been made of the errors and typos in the text. All I can say is that while I did not read the book from cover to cover or go over it with a fine toothed comb, I only noticed one typo in my research and it was not substantial. Unfortunately, Bob McCoy died just about a month after the final print was sent to the publisher, and they have decided not to publish corrections at this point.

I did some research and discovered a web page where friends and family of the author have published corrections to the text:

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It's too bad the book is so pricey because it's just in a strange spot, conceptually speaking. It's a little too heady for the casual shooter, a little light on the math and fluid dynamics to be useful to the graduate student, and a little too expensive to be something you pick up just for fun. However, the material is solid and insightful.

As a ballastician, I must say that this book is a MUST HAVE for any professional in the field of exterior ballistics or the enthusiast with some knowledge of basic physics and high level mathematics. McCoy was a world renowned expert in the field of exterior ballistics throughout his 30 year career with the US Army Ballistic Research Lab (BRL) - this comes through in the pages of this book without a doubt. There are many excellent examples taken from McCoy's (and others) work with the BRL. All should take note, the book has many small typos peppered throughout both the text and figures. The errata pages can be printed off of several well maintained websites which can be found by searching on the title of the book.

Mr. McCoy covers the fundamentals and the more esoteric theories of ballistics. I recommend this book to the layman, serious shooter, and engineer. Although there are some editorial errors (book was published after Mr. McCoy's death), the principles are sound and well presented. A must read for anyone interested in shooting long range or working in the field of ballistic design. Of particular interest is the 6-Degrees-of-Freedom.

Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles by Robert L. McCoy is surely one of best books on exterior ballistics. Author gives clear explanations with many examples included. It is comprehensive covering of free flight dynamics of symmetric projectiles. The book includes historical topics on ballistics, as well as extensive physical and mathematical analysis of the motion of symmetric projectiles. My main objection is on chapter covering 6DOF motion of projectiles. It is not explained very well, some equations are just given, with no prior explanations. But generally speaking, this book is a necessary addition to any undergraduate or graduate course in flight dynamics.

McCoy's book is phenomenal if you seek the essentially pure theory and mathematical characterization of "real," differential equation-based ballistics. The book starts with the simplest form of ballistics calculations, the basic parabolic gravity-only point-mass trajectory everyone encounters in a basic physics course, and then slowly works up from there, all the way to the full six-degrees-of-freedom model (three positional dimensions of motion plus the three rotational dimensions of the spinning, wobbling projectile,) accounting for gravity, frontal air resistance, crosswinds, lifting forces, forces due to the projectile shape, Coriolis force (the consequence of Earth's rotation,) and even rocket forces if the "projectile" is actually a self-powered one. McCoy goes into the simplifications one can make to compute with only a point-mass projectile model (no rotational motion) and I most liked this approach for my own uses. He divides the work into chapters, with each new one adding a level of generality to the model(s) involved. This organization makes it very easy for the reader to choose the level of complexity and accuracy most desired. I was very impressed with McCoy's development of envelope curves, those being the curves of absolute flight limits of a projectile, giving the total interior region that a fired projectile could possibly hit.

This book provides an immense background of everything needed to develop your own ballistics programs and could even provide enough material to do some original research in the field, but I would definitely contend that this book isn't for everyone. In particular, you need to have a very strong mathematics background, including linear algebra and differential equations, and one would do best with some numerical analysis background as well. McCoy does give some detail about various numerical methods one could use to advantage when doing the calculations with a computer, but not quite enough to actually get you started if you weren't already significantly familiar with standard single-step Runge-Kutta and multi-step differential equation solvers. Further, this book does contain a lot of typos and small errors buried deep in the equations, so you need to be able to follow the development well enough to catch them. Essentially, you need to be able to understand the methods used in higher math quite well, use McCoy's initial modeling, and then re-derive his equations on your own, with his derivations as guidance. Also, I think the typesetting of equations is done poorly in general, so it is often somewhat difficult to read them and keep his notations straight.

This book is well worth its price, and the knowledge contained in it is probably nearly unavailable anywhere else these days. Ballistics is an old, but very technical and complicated field, and few people study it extensively any more, so I think it has become somewhat extinct in academic circles. This is unfortunate, for it is magnificent and rewarding, and McCoy's book is an opus dedicated to preserving this wonderful field of hard-won knowledge. The study of precise ballistics modeling and trajectory prediction was the first "big" problem that spawned the development of the digital computer, and the first true calculus problem ever solved by any mathematician was the shape-of-minimum-resistance ballistics problem, solved by Sir Issac Newton. McCoy's book is a grand survey of mid-20th century ballistics knowledge, complete with graphs, photographs, high-speed shadowgraphs from the ranges at Aberdeen Proving Ground, and references to the original authors and studies done by the military and other technical institutions.

You will not regret the purchase of this book, especially if you want to understand the real theory and know how to calculate real ballistics trajectories for all kinds of projectiles at various speeds, not just watered-down, oversimplified, inaccurate, textbook-friendly versions of them. I'm even considering buying myself a second copy, just to be sure I will always have a mint-condition copy available!

Who this book is intended for: Physicists. Who this book is not intended for: gun enthusiast looking to improve their shooting skills and/or gain a basic understanding of ballistics.

With that being said, I'm sure it's THE amazing book on Exterior ballistics my fellows rave about... however, my eyes were bigger than my stomach. MUCH bigger. :(

I'm a graduate student (unfortunately not in physics) and could only hope to skim the text. The real pearls are hidden for only those who speak rich mathematical language (ie. would you rather read a page full of equations instead of text?) Now that I have this book all I need is a lab, 30+years in ballistic analysis, and +20-50 IQ pts...
I'd recommend several texts before this for the gun enthusiast: Understanding Firearm Ballistics; With Winning in Mind: The Mental Management System; Hatcher's Notebook, Revised Edition (Classic Gun Books Series)Rifle Accuracy Facts

This is a great book, but full of typos and edition errors of all types.
The errata ( you can find it in Internet ) is 20 pages long.
The book, as it stands now, is of very limited use.
Better wait the Dover edition.