In Quantitative Ammunition Selection, Charles Schwartz presents an accessible mathematical model that allows armed professionals and lawfully-armed citizens to evaluate the terminal ballistic performance of self-defense ammunition using water as a valid ballistic test medium. Based upon a modified fluid dynamics equation that correlates highly (r=+0.94) to more than 800 points of manufacturer- and laboratory-test data, the quantitative model allows the armed professional to generate ballistic test results equivalent to those obtained in calibrated 10 percent ordnance gelatin. Using data generated from water tests, the quantitative model accurately predicts the permanent wound cavity volume and mass, terminal penetration depth (+/- 1cm), and exit velocity of handgun projectiles as these phenomena would occur in calibrated 10 percent ordnance gelatin and soft tissue.
A retired law enforcement professional, Schwartz provides a concise explanation of the relevant principles of mechanics, fluid dynamics, and thermodynamics pertaining to the model and its derivation.
The quantitative model is clearly presented with illustrated computational examples that provide guidance to the armed professional in every aspect of the model's application.
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About the Author
Charles Schwartz earned a bachelor's degree in psychology with a minor in physics from Ohio State University. He worked for the US Department of Justice after graduating with honors from the Federal Law Enforcement Training Center, Glynco, Georgia. His interests include dangerous game hunting, scuba diving, astrophysics, and classic science fiction.
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Quantitative Ammunition Selection
By Charles Schwartz
iUniverse, Inc.Copyright © 2012 Charles Schwartz
All right reserved.
While many U.S. law enforcement agencies have policies that authorize their sworn law enforcement personnel to select and carry approved non-issue firearms and ammunition while off duty or during assignments that require the use of non-issue firearms, very few provide guidance to their personnel during the process of ammunition selection. After earning a bachelor's degree in psychology from Ohio State University, I entered the profession of law enforcement and found myself shortly thereafter to be in need of guidance in the selection of off-duty ammunition from a relatively diverse list of approved ammunition. Lacking guidance during the selection process, I made a selection from the agency's list of approved non-issue ammunition and carried it faithfully in my off-duty pistol.
More than two decades after that defining moment in my law enforcement career, an extensive background in statistical analysis developed over the course of my career and a life-long fascination with physics that culminated with a minor in that discipline has allowed me to devise the quantitative soft-tissue penetration model that is presented in the following pages. Having personally experienced the demand for such a predictive instrument, I believe that there is a genuine need for a valid, translational, mathematical model that will allow armed professionals, under- funded agencies, and lawfully-armed citizens to test, evaluate, and select self-defense ammunition that is the most appropriate to their respective tactical environments.
Once a handgun has been selected, the selection of appropriate self-defense ammunition becomes the next important consideration for the armed professional. Because it is a decision upon which the armed professional's life may depend, the selection of self-defense ammunition is a deeply personal choice. It is often fraught with a vast array of confusing options and considerations and requires thoughtful deliberation and research in order to ensure the selection of an appropriate projectile design. For these reasons, the need for an easily understandable process for self-defense ammunition selection becomes apparent.
Model Design Constraints
During the prior century, several mathematical models were offered with the intent of qualifying and quantifying the effects of the ballistic wounding mechanism (tissue permanently damaged through direct contact with the projectile). Some of these models were comprised of arbitrary collections of variables and contrived mathematical arrangements capable of producing only equally arbitrary, dimensionless yields. Dimensionless yields are of little practical utility to the armed professional because they offer no meaningful context for objective comparison and lack the technical intricacy necessary to describe correctly the ballistic wounding mechanism and its effects. While some of these older, technically-nuanced models offer valuable insight into the nature of the ballistic wounding mechanism, they often exceed the grasp of those lacking the technical proficiency necessary to comprehend and apply such esoteric models.
A valid quantitative model should be the unified product of the correct analysis and interpretation of empirical data and the fundamental laws of physics. It should also exhibit a high degree of correlation when compared to a large body of manufacturer- and laboratory-generated test data. Compared to more than 700 points of independent test data, the quantitative model exhibits a correlation of r = +0.94.
Ultimately, a valid predictive instrument must originate from conceptually-sound equations and yield its results in real units of measurement, not arbitrary or dimensionless numbers. In this case, the mathematical model accepts its input variables and yields its results in SI (Système International d'Unitès) units of measurement, a unitary system that is easily convertible to English units of measurement when necessary. The model should also describe in plain language a method of testing and evaluating self-defense ammunition with a minimal logistic and technical burden. In addition to adhering to these requirements, the model must remain easily understandable, utilize definite, uncomplicated input variables, and offer an operational procedure that is easy to learn and quick to execute. The quantitative model presented in this book meets all of these requirements.
With this objective in mind, the purpose of the mathematical model is to provide the armed professional with an understandable predictive instrument that will generate test data that is directly comparable to test data obtained through the technically-burdensome procedure of conducting ballistic tests in calibrated 10 percent ordnance gelatin. Toward that end, the following chapters provide a concise explanation of the relevant principles of mechanics, fluid dynamics, and thermodynamics that pertain to the model and its derivation as well as several clearly illustrated examples meant to demonstrate the implementation of the model.
Chapter TwoFundamental Principles
In this chapter, the principles of mechanics, fluid dynamics, and thermodynamics will be discussed as they pertain to the development and application of the terminal ballistic performance model. This is intended to provide the armed professional with an informed perspective of the phenomena involved in projectile motion through a homogenous fluid or hydrocolloidal medium.
Through comparative analysis, the dynamic equivalence (in terms of respective density and internal speed of sound) of water and calibrated 10 percent ordnance gelatin as ballistic test mediums will also be established.
The first principle of mechanics relevant to the mathematical model is Newton's first law of motion which states that, once in motion, an object of mass, m, will remain at the same velocity, v, until it is acted upon by an exterior force, F.
Corollary to this, an object at rest remains at rest until it is acted upon by an exterior force. Both conditions illustrate the concept that is qualitatively known as the property of inertia.
This law of mechanics is illustrated through the simple example of a ball thrown through the air from one person to another. After the ball is thrown, the effects of gravity and air resistance dictate its path, or trajectory. Gravity acts upon the ball causing it to accelerate (or fall) toward the Earth and the force of frictional resistance from the air (which is a fluid) acts upon the surface of the ball causing it to decelerate immediately after it is thrown. If these forces were not present, the ball would continue to move in a straight line at the same velocity forever, or until another exterior force acts upon it.
The second principle of mechanics that pertains to the mathematical model is Newton's second law of motion, expressed by the equation
F = ma,
which states that an exterior force, F, acting upon an object having a mass, m, will induce in that object an acceleration, a, proportional to the object's mass, m, until the exterior force ceases to act upon the object.
Conversely, if a directly opposing exterior force, -F, acts upon the object of mass, m, moving at an initially acquired velocity, vo, a deceleration, -a, proportional to the mass, m, of the object results, subsequently reducing the object's initially acquired velocity, vo. Given a sufficient period of time, [increment of t], a reversal of the object's velocity, -v, will eventually occur.
Using the prior example of the ball thrown from one person to another provides an excellent illustration of both aspects of this physical law. The force exerted upon the ball as it is thrown from one person to another accelerates the ball to a velocity that is proportional to the force exerted upon it by the hand from which it is thrown.
Conversely, as the person to whom it was thrown catches the ball, the hand used to catch the ball must exert a force upon it that opposes the ball's forward motion, causing it to decelerate and come to a stop. If the ball is to be thrown back to the person who first threw it, a successful return requires that an equal and opposite force must be exerted upon it, causing an acceleration that is the opposite of the first and imparting an equally opposing velocity (and trajectory) to the ball.
The third principle of mechanics that applies to the mathematical model is Newton's third law of motion, which states, "To every action there is an equal and opposite reaction." This relationship is expressed by the equation
mv = mv,
and is known as the law of conservation of momentum.
Since an exterior force, F, applied to a mass, m, produces an acceleration, a, over a period of time, [increment of t], (the time rate of change of velocity),
F = ma = m([increment of v]/ [increment of t]),
it may be deduced that force also causes a change in the momentum of an object, [increment of p], over a period of time, [increment of t]. This quantity, the time rate of change of momentum, is known as impulse and is expressed as
F = [increment of p/[increment of t].
During the acceleration or deceleration (the time rate of change of velocity) induced by the action of an exterior force upon an object, the distance traveled by the object during its acceleration or deceleration may be expressed by the equation
S = V2/2a,
which, with proper modification in a subsequent chapter, will permit the prediction of the terminal penetration depth of a projectile.
Once an object of mass, m, acquires a velocity, v, it possesses both momentum, p,
p = mv,
and kinetic energy, EK,
EK = &fra12;mv2.
While a projectile in motion possesses both momentum and kinetic energy, the penetration of a transient projectile through a homogenous fluid or hydrocolloidal medium constitutes an inelastic collision mandating that it be treated as a momentum transaction. Therefore, a momentum-based analysis of projectile motion is the most equitable approach in constructing a terminal ballistic performance model.
Although it may be possible to devise a mathematical model based upon the expenditure of a projectile's kinetic energy as it traverses a medium, there is nothing to be gained from the pursuit of such an unnecessarily complex approach.
The first principle of fluid dynamics relevant to the terminal ballistic performance model is the effect of viscous and inertial drag force components upon a projectile as it passes through a homogenous fluid or hydrocolloidal test medium. Viscous (or frictional) drag is the predominant form of drag encountered by projectiles moving through fluids and gases at low velocities. It is produced by the direct contact of the fluid or gas acting upon the surfaces of a projectile. Inertial drag is the predominant form of drag encountered by projectiles traversing fluids at relatively high velocities. It is produced by the fluid's inertia in opposition to the projectile's motion.
Flow field regimes are quantified and qualified by a Reynolds number, Re, which is a dimensionless parameter that is expressed as the ratio of inertial drag forces to viscous (or frictional) drag forces encountered by a projectile as it traverses a fluid medium. The relationship is described by the equation
Re = ρVD/μ,
where ρ is the density of the medium in grams per cubic centimeter, V is the velocity of the flow field in centimeters per second relative to the transient projectile, D is the diameter of the projectile in centimeters, and μis the dynamic viscosity of the medium expressed in grams per centimeter·second, or poise, an SI unit of measurement.
The magnitude of a Reynolds number qualifies the flow field surrounding a transient projectile as being either laminar or turbulent and determines whether viscous or inviscid equations are applicable to the analysis of the flow field relative to the passage of a transient projectile. Laminar flow fields are characterized as having uniformly smooth, consistent motion with no lateral currents or flow disruptions. Turbulent flow fields are characterized as having random flow irregularities and chaotic flow vortices that exceed the intrinsic viscous dampening properties of the medium being traversed.
At impact and transient projectile velocities of 300 feet per second to 1,700 feet per second, the Reynolds number of flow fields relative to the passage of typical, non-expanding and expanding service-caliber projectiles through water ranges from 500,000 to 16,000,000. Reynolds numbers of this magnitude indicate that turbulent flow field regimes predominate over this range of projectile velocities. Accordingly, inviscid equations are appropriate for modeling the behavior of high- and low-velocity service-caliber projectiles traversing homogenous fluids, such as water, and thixotropic (the material property of liquefying when subjected to pressure) hydrocolloidal mediums, such as calibrated 10 percent ordnance gelatin.
The second principle of fluid dynamics that pertains to the mathematical model is the production of dynamic pressure, PD, arising from the impingement of a projectile in motion upon a hydrocolloidal soft tissue simulant (calibrated 10 percent ordnance gelatin) as compared to the dynamic pressure produced by a projectile traversing a homogenous fluid medium (water). The dynamic pressure, expressed as force per unit of area, produced by a projectile's motion through a medium is responsible for driving projectile expansion and ultimately dictates the terminal penetration depth of the projectile. In order to establish the dynamic equivalence of the two mediums, it is possible to compare the respective dynamic pressures produced in both mediums using the equation
PD = &fra12;ρV2.
Since the respective densities of water (999.972 kg/m3) and calibrated 10 percent ordnance gelatin (1,040 ± 20.00 kg/m3) have already been established, it may be demonstrated with little difficulty that the dynamic pressures produced by identical projectiles at identical velocities are 4.00 ± 2.00 percent greater in calibrated 10 percent ordnance gelatin than in water. While the difference between the pressures produced within the two mediums is not insignificant, it is also not enough to preclude the use of water as a valid test medium when the totality of its properties are weighed against those of calibrated 10 percent ordnance gelatin.
The first thermodynamic principle relevant to the mathematical model is the internal speed of sound, Vs, within each respective test medium. This physical property is inversely proportional to the density, ρ, of the medium and directly proportional to the compressibility of the medium, expressed by the adiabatic (relating to an "isolated" system into which no additional energy is introduced from an outside source) bulk modulus, K, which has its units in pressure (pascals or newtons per meter2).
It is possible to compare the speed of sound in water and calibrated 10 percent ordnance gelatin through the established relationship that is described by the Newton-Laplace formula,
Vs = √(K/ρ).
By applying the values for the density of water, 999.972 kg/m3 at 39.2°F/4°C, and the adiabatic bulk modulus (K) of water, 2.24 x 109 Pa, to the prior equation, the speed of sound through water may be calculated as:
Vs = √(2.24 x 109 Pa/999.972 kg/m3)
Vs = 1,496.68 mps or 4,910.38 fps.
The density of properly calibrated 10 percent ordnance gelatin at 39.2°F/4°C is 1,040 ± 20.00 kg/m3. Since water is the predominant component (90 percent) of calibrated 10 percent ordnance gelatin, it is reasonable to expect that the adiabatic bulk modulus (K) of calibrated 10 percent ordnance gelatin, 2.32 x 109 Pa, is very close to that of water. The calculation for the speed of sound within calibrated 10 percent ordnance gelatin is then:
Vs = √(2.32 x 109 Pa/1040 ± 20.00 kg/m3)
Vs = 1,493.78 mps ± 14.36 mps or 4,900.85 ± 47.11 fps.
Through this comparative analysis, it becomes apparent that both water and calibrated 10 percent ordnance gelatin have a virtually identical internal speed of sound, the difference being approximately fifty feet per second or approximately one percent.
Although all fluids exhibit varying degrees of compressibility, compressibility effects such as transonic drag rise, which are dependent upon the speed of sound within a medium, may be considered negligible when transient projectile velocities do not exceed one third of the speed of sound within the medium (1,635 feet per second in water and calibrated 10 percent ordnance gelatin). For this reason, the terminal ballistic performance of handgun projectiles in water, calibrated 10 percent ordnance gelatin, and soft tissue may be accurately modeled as occurring within an incompressible subsonic flow regime.
Excerpted from Quantitative Ammunition Selection by Charles Schwartz Copyright © 2012 by Charles Schwartz. Excerpted by permission of iUniverse, Inc.. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
2 Fundamental Principles....................4
3 The Model and Its Derivation....................15
4 Model Implementation and Examples....................21
5 Modeling Exotic Projectile Performance....................31
6 A Practical Test Method....................40
7 Myths and Misconceptions....................48
8 Barrier Effects....................56