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Modeling the Probability of Incapacitation: A Perspective
Modeling the Probability of Incapacitation: A Perspective
By Charles Schwartz
Possibly putting the proverbial ‘cart before the horse’ in an earlier article (April 2016), I made reference to and even relied upon a mathematical relationship that quantified the probability of incapacitation, or P[I/H], of a projectile striking human anatomy to illustrate the ability of a test round mentioned in that article. Without realizing it at the time, I suspect that I may have ‘lost’ some readers with the reference and reliance upon the technical concept and would like to take the opportunity to both correct that oversight and provide a perspective of just what it means. The concept of incapacitation probability, also referred to as ‘stopping power’ by many, is extremely controversial to say the least. While I am certain that this article will never resolve the decades-long arguments that rage to this day over the debate of which caliber, projectile design and construction is ‘best’, I hope that it will serve to offer another perspective from which the debate can be framed insofar as the ability to provide an ‘apples to apples’ comparison of ammunition through the algorithms offered. So, to start off, a little ‘history lesson’ seems to be in order.
During the 1960s, the Biophysics Division of the US Army Ballistic Research Laboratory at Aberdeen Proving Grounds served as the ‘garden’ in which the genesis of modeling the mathematical probability of incapacitation took root. Based upon the WDMET (Wound Data Munitions Effectiveness Team) wound data collected during the Vietnam War, these mathematical models sought to quantify the lethality of munitions against the human anatomy. The WDMET wound data base was first assessed by assigning the wound data to 16 functional groups (discrete levels of functional disability) which referred to an “on the average” decrease in the combat effectiveness of an enemy combatant. Additionally, only six discrete post-wounding times ranging from 30 seconds to 5 days were considered in the assessment of the WDMET wound data. Generally speaking, as post-wounding time increases, the detrimental effect of any given wound accumulates and the wound is subsequently assigned to a functional group with a higher level of incapacitation. Relying upon the incremental kinetic energy expenditure of a projectile from a penetration depth of 1 to 15 centimeters, or ΔE15, these discrete disability groups were then used to produce a logarithmic equation that predicts the probability of incapacitation given a random munition strike to the thoracic/abdominal cavity of an enemy combatant. At the time, the only way to obtain the correct ΔE15 value for a specific type of ammunition was to fire the ammunition in question into calibrated ordnance gelatin and then, using high frame-rate photography, determine the instantaneous velocity of the projectile at both one and fifteen centimeters depth. Correctly analyzed, the amount of kinetic energy lost by the projectile over that distance (which is actually 14 centimeters or about 5½”), or ΔE15, can then be determined and used in the equation to produce a predicted P[I/H] for the ammunition being evaluated. Of the P[I/H] models produced by the Biophysics Division of the US Army Ballistic Research Laboratory at Aberdeen Proving Grounds, the one that most deserves our attention was created by Dr. Arthur J. Dziemian in 1960.
Unfortunately, terminal ballistic testing in calibrated ordnance gelatin is a messy, complicated and inconvenient exercise made even more technically burdensome by the need to employ the high frame-rate photography needed to determine the loss of kinetic energy by the projectile during its penetration of the medium. Obviously, what is needed is a dimensionally-based approach in the form of a terminal ballistic model with the ability to calculate the loss of kinetic energy with respect to instantaneous penetration depth of a bullet. Of course, this is where the quantitative model found in Chapter 3 of Quantitative Ammunition Selection allows mere mortals like you and me (who lack access to an expensive, taxpayer-funded and equipped laboratory) to use the Q-model to predict the probability of incapacitation for our chosen self-defense ammunition.
While the coefficients for the BRL P[I/H] model were declassified quite some time ago, they can prove extremely difficult to locate. The coefficients for the two tactical conditions that describe what the average armed citizen is likely to encounter in the field during a shooting, that is, an assailant whose ability to engage in a dynamic assault within a 30-second timeframe or to act defensively within a 30-second timeframe, are located along with the complete BRL P[I/H] equation in Chapter 10 of Quantitative Ammunition Selection. Because non-expanding projectiles, such as full metal jacket bullets, typically do not change their diameters or lose weight during the penetration event, they do not require testing in water before being analyzed in the models. On the other hand, expanding bullets, such as jacketed hollow point bullets, must be tested in water for expansion and weight retention before evaluation in the models since they are designed expand upon impact and may lose some weight during that process.
So, how do we know that using the Q-model is an accurate way to determine the ΔE15 for use in the P[I/H] equation for a specific type of ammunition? The answer to that question can be found in research on the matter conducted by the British military in 1964. Evaluation of the then-new M-16 and its cartridge, the 55-grain 5.56×45 (M193) FMJ, was being conducted by the British military. In that research, the 148-grain 7.62×51 (M80) FMJ and the 55-grain 5.56×45 (M193) FMJ were filmed at 12,000 frames per second as they passed through calibrated ordnance gelatin blocks allowing the researchers to calculate the projectile velocity and energy at any point in the gelatin block. The gelatin test results were also verified against the performance of the same ammunition in deeply anesthetized adult ewes weighing 150-200 pounds such that the bullets’ trajectories crossed the mid-axillary line of the thoracic cavity of the animals between the 4th and 5th intercostal space so that the test bullets’ trajectories included both the lungs and the heart of the test animals.
During gelatin testing, the 148-grain 7.62×51 (M80) FMJ fired at a velocity of 3,130 feet per second was found to have a ΔE15 of 969 foot-pounds of energy for a predicted P[I/H] of 89%. Assuming stable, nose-forward flight through the test medium or a human body, the Q-model predicts a ΔE15 of 965.3 foot-pounds of energy for the same ammunition; within one-half of a percent of the actual test value for a predicted P[I/H] of 87%. Fired at 2,625 feet per second, the 148-grain 7.62×51 (M80) FMJ was found to have a ΔE15 of 670 foot-pounds of energy for a predicted P[I/H] of 86.5%. The Q-model predicts a ΔE15 of 680.1 foot-pounds of energy for the same ammunition and a predicted P[I/H] of 84%. The 55-grain 5.56×45 (M193) FMJ fired at a velocity of 3,280 feet per second was found to have a ΔE15 of 630 foot-pounds of energy for a predicted P[I/H] of 86.1%. For the 55-grain 5.56×45 (M193) FMJ, the Q-model predicts a ΔE15 of 514.3 foot-pounds of energy for the same ammunition and predicts a P[I/H] of 81.1%. The Q-model provides a reasonably accurate prediction of both ΔE15 and P[I/H] assuming stable nose-forward flight through an enemy combatant or armed violent felon.
In the accompanying graph, eight different projectiles, ranging in size from 9mm NATO and .45ACP ball ammunition to the 155mm artillery shell are compared using the Q-model. The red and green curves represent the P[I/H] for the 30-second ‘assault’ and ‘defensive’ tactical conditions respectively.
As can be seen in graph, the 9mm NATO 124-grain FMJ and the .45ACP 230-grain FMJ are relative pipsqueaks in the grand scheme with respective P[I/H]s of 64.8% and 66.1%. The 7.62×51 NATO 147-grain FMJ demonstrates its superiority with a P[I/H] of 86.05% over the 5.56×45 NATO 55-grain FMJ with its P[I/H] of 81.10%. At 2,850 feet per second, the .50-caliber BMG M2 projectile weighing 706.7 grains with a P[I/H] of 92.8% and an expenditure of nearly 2,400 foot-pounds of kinetic energy during its first 5.5 inches of travel through a human body pales only against the 30x173mm autocannon ammunition fired by the GAU-8/A autocannons mounted on US military aircraft like the A-10 Warthog and the AH-64 Apache helicopter.
With a projectile weighing 4/5ths of a pound and having an impact velocity of 3,500 feet per second, the 30x173mm projectile possesses a P[I/H] of 98.33% even without the high-explosive charge detonating upon impact!
Used in conjunction with the Q-model, the BRL P[I/H] equation can also be used to confirm that the US military was correct in fielding the 5.56×45 NATO 55-grain FMJ (3,281 fps) as a cartridge having equivalent lethality to the Soviet 7.62×39 123-grain FMJ (2,329 fps). Using the Q-model to determine the ΔE15 of the 5.56×45 NATO yields a value of 514.647 foot-pounds and a value of 517.773 foot-pounds for the Soviet 7.62×39. When these ΔE15 values are applied to the BRL P[I/H] equation the 5.56×45 NATO demonstrates a P[I/H] of 81.06% and the Soviet 7.62×39 exceeds the P[I/H] of the 5.56×45 NATO by a very small fraction of a percent with a P[I/H] of 81.12%. In terms of lethality, the 5.56NATO and the 7.62Soviet cartridges are a statistical ‘dead heat’.
Finally, as if being able to predict the ‘first-round’ lethality of any projectile is not enough, it is also possible, using the cumulative binomial function, to calculate the cumulative P[I/H] of any number of subsequent projectiles striking the combatant/assailant within the specified time frame of 30 seconds. Using the ‘one-shot P[I/H]’ value of 66.1% computed above for the .45ACP 230-grain FMJ, it is possible to calculate the P[I/H] of any number of .45ACP 230-grain FMJs striking a combatant/assailant in rapid succession. Two such projectile strikes results in a ‘two-shot P[I/H]’ of 88.5%, three rapid projectile strikes results in a ‘three-shot P[I/H]’ of 96.1% and, if one were to discharge an entire GI magazine-full (7 shots) into an attacker very rapidly, the resulting ‘seven-shot P[I/H]’ would be 99.95%! It is also possible to alter the 30-second P[I/H] timeframe to a longer or shorter timeframe by using the equation-
P[I/H] = 1 – (1 – P[I/H])(T÷30)
-and simply entering the desired new time frame in seconds (T) and completing the computation using the original ‘one-shot P[I/H]’ value.
It is no longer necessary to shoot non-expanding and expanding ammunition designs into calibrated ordnance gelatin to measure maximum penetration depth, instantaneous velocity, wound cavity volume, crushed tissue mass and lethality. Using the equations set forth in Quantitative Ammunition Selection, it is now possible to do so by simply firing test ammunition into water and, relying upon the average recovered diameter, mass and impact velocity, obtain a valid measure of all five performance variables.
Quantitative Ammunition Selection is available domestically and internationally in hardcover, paperback and eBook formats and may be purchased at www.quantitativeammunitionselection.com. Just select the appropriate link found on the lower portion of the page for the format that you want.