III. Myths, Misconceptions and Miscalculations
If I haven't already gotten someone into a stew with what I've said so far, I'm certain to do so now. I want to examine some very popular myths, misconceptions and miscalculations of terminal effects with a view to demonstrating not only why they are not true, but also to account for the phenomena which caused them to be formulated. I think that's fair. If I can't explain to your satisfaction the observed phenomena that supports your favorite stopping power theory, I can't ask you to ditch it.
III.a. "Energy Dump", "Overpenetration" and "Hydrostatic Shock"
There is a myth to the effect that a bullet which remains inside a target is more effective (in terms of stopping or killing power) than one which completely penetrates. This myth is not new. Colonel Townsend Whelen writes in his very illuminating treatise, Small Arms Design and Ballistics, that "the thought at that time was that the ideal bullet should just shoot through the animal to its opposite side, and lodge under the skin without penetrating clear through, thus expending all its energy on the beast" (p. 137). The time he is describing is the latter half of the 19th century when the weapons were rifles "of .45 caliber, shooting a bullet of 350 to 550 grains and with a charge of black powder sufficient to give it a muzzle velocity of from 1300 to 1500 fps" (p. 136). Even in these early days of ballistics inquiry the significance of kinetic energy was being examined.
Unfortunately the conclusion reached by some is arrant nonsense. It is interesting that the 19th century model of "energy dump" required the bullet to completely pass through the body, but stop under the skin on the off-side; combining the features of an "energy dump" with lethal penetration and cavitation.
There are at least two contemporary variations on the "energy dump" premise. The principal argument seems to center on the concept of "overpenetration", which is essentially the same thought as expressed in the 19th century but with the added evidence of actual results from gunfights on the street (the chief culprit being the rather pointed 9 mm FMJ bullet). Bullets which "overpenetrate" do not stop opponents as readily as those that remain in the body. Therefore, if the energy isn't "wasted" on exit, the bullet is more effective. Right?
Not exactly. A bullet of a given construction and impact velocity will create a cavity of predictable dimensions over its path, whether it stops or penetrates completely. Therefore, if the hole created can penetrate all the way through, it causes more damage than if it stops at some point. The critical issue here is what sort of hole are we making, not whether it goes all the way through. "Overpenetration" is a misnomer. The ineffective stopping attributed to overpenetration is actually caused by "undercavitation".
I have a pet .41 Magnum load (170 gr Sierra JHC with a muzzle velocity of 1500 fps) that will probably penetrate more than 15 inches if it doesn't hit heavy bones, but the cavity created by this bullet is enormous, much larger and originating at a shallower depth than that caused by a 5.56 x 45 mm M193 military bullet. In short, it is much more lethal than the very lethal 5.56 mm. These two loads have essentially the same kinetic energy. There is no comparison between this cavity and the one produced by a Federal .40 S&W 180 gr HP, which is also a very effective "stopper". Am I to think that my .41 Magnum load is less potent because it penetrates farther?
If there were an ideal case from the standpoint of efficiency, I suppose it would be for a bullet which completely, but just barely, penetrated and fell to the ground. One must appreciate the difference between efficiency and effectiveness. (The engine tunings of dragsters are not efficient!) In this case the bullet has done all the damage that it can do to that particular target at that particular angle of entry. The problem here, of course, is that one cannot predict the exact size and toughness of the game encountered, or the exact range, which would have to be known in order to achieve the precise impact velocity required for ideal efficiency. All of these uncertainties drive bullet loads to exceed the minimum performance, and this is accomplished by designing a bullet that will create an adequate cavity while deeply penetrating over a wide range of impact velocities. Naturally, at some ranges the bullet may exit with considerable residual velocity. This wasted energy is irrelevant if the wound is adequate.
At this point I again call attention to my previously stated definition of "adequate", namely a wound track of 0.75 to 1.00 inches (19 to 25 mm) in diameter through heart, lungs or major arteries. A smaller hole through major arteries only (for example) will kill, but the game may run a significant distance and be lost. It is also infinitely more difficult to track wounded game without a blood trail than with one that looks as if it were painted with a roller, and entrance wounds rarely bleed much. A larger cavity at shallow depth may drop a game animal instantly, but what is gained by less than 25 to 100 yards at the risk of an inadequate wound if a difficult angle is involved or major bones intervene or the bullet self-destructs unpredictably? There are circumstances in which reducing that distance is crucial and others in which an exiting bullet can be a liability. It is not a cut and dried issue. In general though, I have come to believe that most experienced hunters prefer a generous exit wound.
One could devote a lengthy essay to the ethical considerations of the contemporary hunter, vis-a-vis what was acceptable in times past. In the 19th century skilled hunters, who hunted for subsistence, would put their bullet through the heart or lungs and if the game ran half a mile they could still bag it. For most of us, times have changed. We lack the tracking skills of our forebears and the moral justification to use marginally effective weapons. Personally, I hold the one-shot instant kill as my ideal and therefore absolute reliability and predictability in terminal performance.
Tactical considerations concerning penetration are a different matter entirely. Bullets such as the Glaser Safety Slug were designed primarily with tactical considerations in mind, rather than optimal wounding. Glasers are designed not to penetrate aircraft or sheetrock walls, or to exit from the body of the target and endanger someone immediately behind. Glasers and similar such designs create extremely effective wounds if they do not have to penetrate deeply.
For those who yet doubt, consider a hypothetical example of two projectiles with equal kinetic energies. One is a 1 lb gel-filled bag launched at 60 fps. The other is a 490 gr broadhead arrow travelling at 225 fps. Both have a kinetic energy of 55 ft-lbs. Which would you rather be hit by? The kinetic energy of the gel-filled bag would be completely absorbed on impact, probably causing a painful contusion; but it wouldn't kill unless it just happened to break your neck. The arrow would easily and completely penetrate a human torso from any angle and continue travelling with residual energy, but prove highly lethal.
"However," one might object, "we're only considering a mere 55 ft-lbs. What happens when the energy levels are comparable to that of a high powered rifle?"
An excellent question. Assume 2000 ft-lbs of kinetic energy, about that of a .308 cal, 165 gr Ballistic Tip fired from a .30-`06 Springfield to a game animal at 250 yds; well within its effective range for lethal dispatch on large game. This equates, in strict energy terms, to the kinetic energy of a 50 lb bag of fertilizer when dropped from a height of 24.86 ft. That sounds awful and I daresay it would clobber most living things, but its the kinetic energy of the falling bag of fertilizer, not necessarily the effect of the falling bag itself, which is under consideration. The actual momentum transfer of the bullet is slightly less than that of the gel-filled 1 lb bag previously described, and if one were wearing body armor to slow the absorbtion of energy at impact, that is what would be felt - a good solid thump. With the body armor one's body absorbs all of the kinetic energy without harm, without it the bullet penetrates and exits using only a portion of its kinetic energy but delivers a lethal wound. Two "foot-tons" of kinetic energy does not equate with the impact of a full-size sedan. It is not the energy itself that kills, it is the character of the work done by it.
To be fair, in this case the body armor helps to absorb the energy, in lieu of one's flesh and bone, and to distribute the force exerted over a larger area. I cannot defend the premise that game animals or unprotected people can absorb 2000 ft-lbs of kinetic energy instantaneously (and this is essential) with complete impunity. Moreover, I can think of no hunting bullets for rifles today which do not provide adequate penetration under most circumstances against the game for which they are intended. But there are frangible bullets composed of compressed metal powders which disintegrate upon impact, expending all of their kinetic energy without penetrating, and these are not suitable for hunting, although they may inflict wounds which are ultimately fatal. Varmint bullets are also demonstrated miserable performers on big game, although under ideal conditions (missing all heavy bones on a clear broadside lung shot) they can kill spectacularly against lightly bodied deer and antelope. Hit the shoulder or angle from the rear and its ruinous.
Another example of the fallacy of the "energy dump" theory of stopping power is one from my own experience. I once shot a thin metal screen with my 41 Magnum (210 gr. HP, 1350 fps, 850 ft-lbs) at a very close range (do not try this at home!). The bullet was completely stopped; it lay on the ground in front of the mesh screen. The screen apparently had completely absorbed all of the kinetic energy of the bullet. However, the screen suffered no serious damage; there was only a faint depression where the bullet had struck and glanced. What actually occurred is that the screen "gave" against some tall grass when impacted and caught the bullet like a catcher's mitt. When I attempted to reproduce this spectacular behavior for a friend of mine, the screen was supported so that it did not give and was easily perforated by the bullets.
This last example from actual "testing" graphically demonstrates why the energy dump premise is fundamentally flawed. These examples also serve to illustrate the profound differences between kinetic energy and momentum. Kinetic energy is an exponential function of velocity, momentum is not. However, one should not assume from this that kinetic energy means nothing, or is somehow unrelated to wounding potential. Clearly, the naked human torso (and that of even large and tough game) is far more capable of absorbing a low velocity impact than the kinetic energy of a rifle bullet with a corresponding momentum.
The rate of energy transfer to the target is vastly more important than the quantity of energy transferred. This is the technical definition of power. Anyone sunbathing on a clear summer day at the beach will receive an irradiance equivalent to over 4600 ft-lbs every minute! Eventually, this bombardment by extremely high velocity particles will result in sunburn, but the body can withstand the energy it receives because it is spread over a large area and arrives at a relatively slow rate (compared with bullets). The power and intensity (power per unit area) is much less than ballistic events.
The other popular contemporary misconception results from the assumption that the kinetic energy of the bullet is "transferred" to the target, thereby somehow killing it through "hydrostatic shock".
I don't know where this term originated, but it is pseudoscience babble. In the first place, these are dynamic - not static - events. Moreover, "hydrostatic shock" is an oxymoron. Shock, in the technical sense, indicates a mechanical wave travelling in excess of the inherent sound speed of the material; it can't be static. This may be a flow related wave like a bow shock on the nose of a bullet in air or it may be a supersonic acoustic wave travelling through a solid after impact. In terms of bullets striking tissue, shock is never encountered. The sound speed of water (which is very close to that of soft tissue) is about 4900 fps. Even varmint bullets do not have an impact velocity this high, let alone a penetration velocity exceeding 4900 fps.
Some people use "shock" in the colloquial sense to describe a violent impact, but it is confusing, especially in connection with the term "hydrostatic" and lends undeserved quasi-scientific merit to the slang. It also tends to get confused with the medical expression attending trauma. We are not describing any medical shock. The word shock should never appear in a gun journal.
Before I become too dogmatic and overstate the situation, let me concede that there may be some merit to the idea that hydrodynamic (not hydrostatic) impulse created by bullets which have a high kinetic energy and generally exhibit violent cavitation, can cause some secondary effects due to pressure on the nervous system or heart. It is possible to kill manually by nerve "strangulation". In this case actual damage to the central nervous system is not caused, but the signals governing the heart or diaphragm are shut off, resulting in instantaneous unconsciousness or even death. Certain rare sports fatalities have been definitely attributed to a swift blow which interrupts the cardiac rhythm. Acoustic pressure on the spine can also cause temporary paralysis. These phenomena may account for the rapid effectiveness of some high-velocity hollow-point pistol bullet wounds, especially in cases in which the victim is not mortally wounded and recovers consciousness within a few minutes. Several special handgun loads have been designed with no regard whatsoever to penetration (e.g., the THV bullet) in order to achieve this result. Unfortunately, this is an unreliable mechanism of incapacitation, generally obtained at the expense of effective penetration. No bullet yet designed will produce this effect even 10% of the time. Many of the bullets designed to utilize this effect can be defeated by common barriers, such as glass, sheetrock, and even clothing. Doing this deliberately by hand, even with a profound understanding of the mechanism and vital points, is extremely uncertain; using the passage of a pressure wave from a bullet to accomplish this falls into the freak event category. Such is never an acceptable mechanism for the hunter.
The point that I have attempted to press here (perhaps in a rambling fashion) is that complete penetration is not something to avoid in the hunting field. In fact there is good evidence that through and through wounds cause collapse quicker in many instances, especially lengthwise shots.
On the other hand, as I have alluded to previously, some contemporary bullet designs (Nosler Ballistic Tip and Remington Bronze Point) as well as some renowned performers from years past (e.g., the original 130 gr. load of the .270 Winchester) achieve a high percentage of instantaneous kills by blowing to bits and never exiting the game. I find this interesting in view of the current obsession with avoiding bullets in which the lead cores separate from the jacket. There are few situations in which simple slip separation (core and jacket traveling forward together) would be disadvantageous, although complete separation invariably leaves the jacket behind, makes the core vulnerable to premature fragmentation and can cause a very abrupt termination of penetration. But returning to the issue, the successful frangible bullet designs nevertheless always penetrate to the vitals and have never been regarded as reliable for rear raking shots requiring deep penetration or against very tough heavy game and most knowledgeable authorities prefer bullets which exhibit modest cavitation with deep penetration because of their flexibility in the field.
III.b. Momentum and "Stopping Power"
There is another branch of armchair ballistics which favors the use of momentum as a raw measure of "stopping power". I confess that I was long enamored of this view because it seemed to give more credit to lower velocity big-bore cartridges that were strong performers than the kinetic energy story told, but momentum alone is equally inadequate to describe terminal behavior and the arguments given in support of that view are as full of blue sky nonsense as any claim made of kinetic energy.
The old African hunters talked of "stoppers" which would cause a charging pachyderm to halt its forward progress in its tracks and stand stock still in a daze. This situation applied only to (if you carefully note the details) the truly huge weapons such as the 8 and 4 bore blackpowder rifles and later .577 and .600 Nitro Expresses when fired into the spongy skull of an elephant, but when the brain is closely missed. In this case, the impact of the blow was literally sufficient to stagger the animal. It is to be noted, however, that these ancient blackpowder weapons could only be used by men of considerable size and strength and even they dreaded pulling the trigger. Frederick Courteney Selous says that using such weapons crippled him and Sir Samuel Baker foreswore such things as soon as he developed guns and loads capable of dependable penetration, preferring the far less punishing 10 bore!
Unless you hunt elephant with these antique monsters, and I don't know of anyone alive who does, then "stopping" an animal in this sense isn't a matter of concern. From what I've read, one cannot depend upon a similar effect on the remarkably tough Cape buffalo until something like a .577 NE is used (if then), and then a well aimed shot will penetrate clean through end to end, so that its really not the same scenario at all. Proper stopping of a charge depends on shot placement.
Similar rules apply to human targets. The human frame is so lightly constructed that any "stopper" class weapon will easily penetrate through even after encountering major bones. Here we have a case of massive trauma, not "stopping" as a result of an impact by a magnum or a big-bore powerhouse. Again, the force of the blow is comparable to the recoil of the gun. It has to be. That's physics. In either case the bullet is acting under high acceleration loads over a short distance. Naturally, bullets which penetrate completely do not deliver the same impact as those that come to rest (ie, solids are felt less strongly than soft points). There are always anecdotal accounts (invariably told by those who have never shot anyone) of people being hurled off their feet by a .45 ACP using hardball when pathetic little .38 Special bullets are shrugged off without effect. In actuality, there are quite as many incidents of people shrugging off .45 slugs as for many lesser calibers.
It is hypothetically possible for the impact to cause a pressure pulse of such magnitude through the body of the target, that it is instantly killed (notice I deliberately avoided the word "shock"). This happens, for example, when you shoot a chipmunk with a .30-`06, even if the chipmunk wears body armor. It doesn't happen often (if ever) against big game and humans with ordinary (ie, shoulder fired) weapons (NOTE: I'm not referring here to hits against the central nervous system, or to any hit which results in a penetration; only to hits which could kill exclusively from the force of the impact). I haven't yet seen any cases in which a wound that did not reach to the vital organs resulted in death - except as a result of septicemia.
What this means is that if all of the momentum of a high-powered rifle bullet were delivered by a non-penetrating blow, the damage inflicted on any game larger than a small varmint would be relatively insignificant.
I've asked people who have been shot to describe their sensations of being shot, physical reactions, etc. The common response is that the impact feels precisely like the impact of a punch. It is not any more impressive. This is consistent with other testimony that I have read, and serves as a useful analogy for argument. A punch is not staggering unless it also causes debilitating damage, usually to the brain, but alternatively to the lungs, etc. - or in the case of the victim being surprised, off balance, etc. The impact of a bullet behaves according to similar constraints, all things being relative (ie, a .458 Winchester Magnum causes relatively more of an impact to a man than to an elephant because of the size difference, assuming the force of the impact is absorbed in both cases).
Many readers will doubt the truth of this assertion. Some will say, "I have actually seen a deer do a complete somersault!". Many people have, but it wasn't caused by the force of the impact. For the skeptical among you, if you require proof that the spectacular flips often reported are merely an apparent effect of the bullet impact, I challenge you to rent a video of dangerous game hunting in Africa and observe the effect of bullet impact on downed, already dead, animals when the insurance shot is made. This practice isn't carried out on deer and it separates the effect of the bullet from the effect of the living animal. Lions and even buffalo may seem to flip high in the air, but if they are dead the bodies only quiver ever so slightly; its barely visible, and these are shots through the most massive shoulder and spinal bones, delivering the greatest resistance and impact with very powerful rifles.
Alternatively, you can perform a penetration test into wet phonebooks that completely absorbs the impact in a manner very similar to a game animal. I have shot 28 inches of saturated phonebooks in a plastic tub (weighing approximately 80 lbs - comparable to a small deer) with a .340 Weatherby Magnum firing a 225 grain bullet at 2900 fps (at ten feet). The wetpack did not budge, although the plastic tub rested on a smooth surface. More recently, that same load was used to kill a springbok in Namibia. My first shot seemed to crush the diminutive antelope in its tracks. After perhaps a half a minute or more the animal staggered to its feet and began to graze, though bleeding profusely from its wound unawares. I shot it lengthwise of its body. It did not move at all this time, but after a few seconds it pitched over sideways, stiff-legged and dead. Same load, but the numbed beast did not feel the second impact, even though the bullet did not exit. I have conducted the same test into a wetpack using a .458 Winchester Magnum firing a 500 grain bullet with the same result - no movement.
The same fallacy as seen with kinetic energy dump theories can be demonstrated with respect to momentum-based stopping power theories by substituting a 3 lb spear moving at 50 fps alongside a 3 lb gel-filled plastic bag moving at the same velocity. Now we have two projectiles with exactly the same masses and momentum. Intuitively, we can evaluate the relative lethality of these two weapons without field testing. Incidentally, these both produce the same momentum as a .458 Winchester Magnum loaded with 500 grain bullets. The .458 Winchester Magnum is a "heavy hitter". Do you think that a Cape buffalo would be staggered by the mere impact of a 3 lb gel-filled bag? A haymaker punch (15 fps), using about 50% of the body weight of a 180 lb man would have a kinetic energy of about 300 ft-lbs, but a momentum over eight times greater than that of a .458 Winchester Magnum or .450 Rigby Nitro Express; indeed a hard swift jab (50 fps) would have a kinetic energy of still only about 350 ft-lbs, but a momentum almost three times greater than the standard in stopping class rifles! I wouldn't even dream of punching a Cape buffalo.
III.c. Thresholds of Wounding Potential Based on Kinetic Energy
I think it was the sage and revered (and great favorite of mine) Col. Townsend Whelen who first proposed the idea that modern sporting arms ought to deliver at least 1000 ft-lbs of kinetic energy on the target for quick dispatch of deer and 2000 ft-lbs for larger species such as elk (Craig Boddington, American Hunting Rifles, Safari Press, 1995, pg. 20). This guidance has been reiterated for probably half a century or more now, but the world of smallbore ballistics today is very different than in the days when the esteemed Colonel was at the forefront of modern military small arms development with the U. S. Army Ordnance Department. Moreover, here is what he had to say in his own treatise on the subject of "Killing Power" over sixty years ago:
- "The killing power of a bullet in flight depends entirely upon the average size of the wound it makes in the animal, and upon nothing else. The size of the wound in turn depends upon the size, weight, construction, and shape of the bullet, and the velocity with which it strikes, and upon no other details. ... We frequently see it stated that the killing power of a cartridge depends upon its energy, and tables of the properties of cartridges often give the energy of each. Now energy depends upon the weight of the bullet times its velocity, and on nothing else, and thus can have only a very distant bearing on our subject." (Townsend Whelen, The Hunting Rifle, Stackpole Sons, 1940, pg. 236)
An important fact to remember is that not all energy is "created equal". What this ultimately means is that a kinetic energy value used as a measure or threshold for lethality is practically meaningless. The character of the work done by a certain quantity of kinetic energy will be dependent upon the mass, construction and velocity of the projectile. In other words, 1000 ft-lbs of kinetic energy generated by a slow-moving rock is not as lethal as that of a bullet. Furthermore, the damage actually caused by a lesser amount of kinetic energy may easily exceed that caused by a greater quantity of kinetic energy! Expressed differently, kinetic energy has "quality" as well as "quantity". This is easier to understand in terms of heat energy, which has temperature (degrees F or C) as well as quantity (BTUs or Joules). Kinetic energy is governed by similar laws.
As further evidence of this fact, observe that when terminal ballistic experiments are scaled the velocity is held constant. Kinetic energy, mass and the dimensions are scaled, but velocity is not. In like manner pure water at standard pressure boils at 100° C, regardless of quantity. A small amount of water does not boil at a lower temperature than a larger amount. The heat required to bring a quantity of water to a boil is directly proportional to the mass of the water (just as the kinetic energy is proportional to the volume of displacement by a bullet), but the character of the work done on the water by that heat energy is determined by the temperature it produces. It is velocity, not kinetic energy, which is the quantity of greatest interest in the terminal ballistics of small arms.
Since a knowledge of the velocity and projectile construction is essential to evaluating the character of the kinetic energy and its wounding potential, simply relying on a quantity of energy can be quite misleading. The way in which a sporting bullet (say, a 7 mm 140 gr spitzer boat-tail at an impact velocity of 3000 fps) expends its first 1000 ft-lbs of kinetic energy on a target (from 2797 ft-lbs to 1797 ft-lbs) will little resemble the way in which it expends its last 1000 ft-lbs (at an impact velocity of 1794 fps, where it will most likely fail to deform and simply drill straight through causing a neat little hole with negligible cavitation). In the former case, a lung shot would result in a wide wound track and a gaping exit wound as it exits the body at 2405 fps, but cause rapid collapse; in the latter case even a lengthwise shot which fully absorbed the energy of the projectile would probably mean a lost game animal because of the low probability of causing rapid hemorrhage. Interestingly, in the former case probably 20% or more of that kinetic energy would be lost to deformation of the bullet, whereas in the latter case all of it would be delivered to the target. However, that same 1000 ft-lbs of energy delivered by a .41 caliber 280 gr LBT-WFN flatnosed hard-cast bullet at 1268 fps would quickly drop a bull elk with the same lung or lengthwise shot because its larger diameter and strong flat nose would create a large diameter and deep wound even after smashing through heavy shoulder bones. (Incidentally, this misunderstanding is not confined to the ballistics of sporting arms. I have encountered the notion in recent years in my profession of a tank killing threshold of 10 MJ.)
A popular term among some gun buffs is the "foot-ton", a magical quanta of kinetic energy that is supposed to translate into all sorts of killing authority. Aside from the problem described above in assigning an arbitrary kinetic energy level for lethality against a type of game, there is the matter of unit definitions. If you like to think of it as the energy required to raise a one ton block a distance of one foot, that would be correct (again, not necessarily the same as being crushed by that falling block!). Forget the comparison to automobile impacts.
However, simply because a quantity of kinetic energy is not, in and of itself, enough to describe the wounding characteristics of our weapons does not imply that kinetic energy is not a valid measure of ballistic performance. We need not be reactionary or suppose that someone got it wrong and that what we need is a better "formula".
At the polar extreme from this viewpoint of using 1000 ft-lbs or 2000 ft-lbs to judge the killing capacity of a cartridge is a truly novel and disturbing conception highlighted in an article entitled "Stopping Power: A Skeptical Look at "Foot Pounds" as a Means to Measuring Your Rifle's Ballistic Energy" (Lee Saunders, Petersen's Rifle Shooter, June 1998, pp. 58 - 62) and the subsequent very eloquent letter to the editor (Mail Room, Petersen's Rifle Shooter, December 1998, pg. 5) by Joseph A. Schetz, a professor from Virginia Tech, with the author's rebuttal, which is that kinetic energy is simply the arbitrary fabrication of some 18th century mathematician, not based on physics, and is fundamentally incorrect.
Its as if the cancerous pseudoscience of gun writers has spread to corrupt even the hallowed precepts of true science. I shouldn't make it seem as if the author of this particular article were alone in his assumptions. The history of popular terminal ballistics in the 20th century saw several examples of this kind of crackpot science, such as Elmer Keith's ridiculous invention of "pounds-feet". What is most astounding about this latest outrage against science and clear reasoning is that the (long since departed) editors of the magazine didn't know enough themselves to prevent its publication. I expect this sort of thing in cyberspace, but I expect a higher standard from publishers (incidentally, the present editorial staff has a much more scientifically founded perspective). Men like Townsend Whelen knew their basic science and would not have made such errors, nor permitted them to be published in their journal. It troubles me that our knowledge has diminished so much in 70 years.
When gun writers attempt to describe terminal ballistics in terms more technical than "wallop" they take on the mantle of science and bear the responsibility to their readership to convey an accurate discussion of the mechanisms involved. Science does not merely belong to scientists nor only in the realm of the scientific journal. It is truth on a fundamental level. There are no "everyday" meanings to terms such as velocity, momentum, kinetic energy and impulse. They are not slang or jargon used to describe nebulous, ill-defined concepts. They hold precise meanings. To carelessly misuse scientific language is to render a disservice to the readership, even though it be predominately composed of non-technical readers.
Before going any further let me make an apology to the reader. This article and the letter exchange is now ancient history. It was not my intent then nor is it now to pillory anyone. I wrote my own letters to the editor of the magazine at the time in hopes of inspiring a more fastidious editorship in terms of technical matter. I feel justified in rehashing the argument in cyberspace because it represents a viewpoint with which I suspect many reasonably educated shooters would sympathize - and I don't mean my viewpoint! I understand that much of what I am pressing here seems arcane and unimportant to most shooters. But the integrity of such concepts is the fundamental underpinning to all ballistics, the technology on which we depend. Is it necessary for the average shooter to understand all these concepts and be thoroughly conversant? No, it is not. However, when they are discussed, the discussion needs to be scientifically correct.
Had this article restricted its scope to the question of whether kinetic energy was a sufficient measure of terminal performance, we might have seen an insightful and interesting study. Unfortunately, it strayed into the realm of "Things I Don't Know". [We all need to know the limits of our knowledge and practice more learning than expounding. I have been as guilty as anyone.] If the average shooter doesn't know what is wrong with the following tidbits then this country has more serious problems than confusion about terminal ballistics:
- "The upshot is that the kinetic energy formula is neither correctly labeled as to resulting units, nor particularly accurate in describing projectile energy. I get the feeling that it is used very little outside the ballistics field. [emphasis added] If it were, it would likely have been changed long ago... In the KE formula we have something that is provably wrong in regard to the foot-pounds label..." (pg. 62)
The author displays an appalling incomprehension of junior high mathematics and general science, confusing a quantity squared with one doubled and the operation of addition with multiplication, using the terms energy, "impulse energy" (his own invention), momentum and force interchangeably, confusing rate with duration, and then has the incredible arrogance to unequivocally assert that 300 years of scientific inquiry is deluded, but that he perceives the truth of projectile motion. This article is so unspeakable that I go into hysterics when I read it. (To his credit, the author is a thoughtful experimentalist and his improvised ballistic slide demonstrates what I had earlier claimed, which was that a light jab produces as much impact and physical translation as a .375 H&H or a 12 ga. slug). Again, nobody thought it might be worthwhile to ask the local junior high science teacher if there might be a problem (or better still, remembered their junior high science classes!). I guess I expect too much and perhaps I am being too critical, but whether the average Joe understands the ins and outs of physics is not the issue - its the attitude that accredited science is no better than hip-pocket hooey that bothers me.
Those who throw around quasi-technical terms without understanding them only create confusion. Velocity is not impulse. It is not like impulse. Kinetic energy is not momentum and velocity "combined". Momentum will not describe "the load with the hardest thump". There is no room in true science for a private opinion about a better definition of energy, and those who ask "Is the ft-lb an accurate KE label?" should not be published.
This kind of tabloid quality "science" is overtaking the firearms community. In the age of bioengineering, quantum electronics and relativistic physics, the firearms community is becoming mired in a level of scientific ignorance comparable to Medieval Europe. The truth is not marketable but crackpot theories about better formulas for kinetic energy warrant feature articles. Falsehood and error need to be corrected. Those of us who care about the quality of the literature and the accuracy of the inquiry into terminal ballistics bear the responsibility to repudiate the nonsense and to authoritatively instruct concerning the facts.
Since this has proved to be a pitfall for some I will unravel the mystery. The definition of energy is based upon physical laws. A ft-lb is a valid unit of kinetic energy - by definition. There is nothing to prove. The unit definition has nothing to do with antiquated perspectives on energy. Kinetic energy is calculated as (1/2) mass times velocity squared. But pounds are actually a unit of force (i.e., weight), which is mass times acceleration (due to gravity in this case). So, to get kinetic energy we must divide by 32.174 ft/s2, which reduces the velocity squared terms of ft2/s2 simply to feet. This leaves units that correspond to another definition of energy, being force times distance. Its really only confusing in the old English system of units because we normally think of pounds as mass rather than as force; in metric its obvious that all forms of energy are the same thing because they are all in Joules or kg-m2/s2.
Just in case somebody doesn't know, foot-pounds are a real quantity and can be converted into BTUs, Joules, kilowatt-hours, calories, ergs, electron-volts or any other measure of energy as you please. All of these resolve down to the same fundamental quantities of mass times distance (divided by time) squared. Not all energy is the same, but all energy has the same fundamental units. Kinetic energy was not invented for the delight of gun writers. The different definitions of energy are based upon inter-related physical laws, none of which have been overturned since God created the universe, let alone in the last century.
Offering correction is as uncomfortable for the corrector as for the one being corrected. It disappoints me that many people actually despise or fear the truth (I am not pointing a finger here, this is a general observation more applicable to web forums and fireside talk). It is a subtle thing, typically taking the form of a resistance to let go of cherished misconceptions and an egoistic tendency that we all share to be knowledgeable. I have been proved wrong many times in my life and while the experience is not always a comfortable one, I am happier for being corrected. Not everyone is made happier by correction.
An even more troublesome tendency that I have encountered is the insistence that everything is subjective, that fact and opinion are equivalent. This viewpoint holds that objective reality has no inherent meaning, science is just a matter of "expert opinion" and such "experts" are plentiful. If, as some have contended, all measured data such as penetration depth and wound diameter (even in game animals) provide us no absolute knowledge, if any degree of uncertainty removes all hope of understanding , then truly we have embraced superstition (or a weird form of agnosticism) and are equally well served by consulting an astrologer about the performance of our weapons.
III.d. Optimal Game Weight (OGW) Formula
The OGW formula was published in the April 1992 issue of GUNS magazine and has since appeared in several references. It is purportedly the result of careful examination of the various contributions of "kinetic energy, momentum, bullet sectional density, bullet diameter, bullet nose configuration, impact velocity and a number of other criteria" (pg. 62) to terminal effect. The author, without elaborating on his methodology, settled on the following formula:
OGW (lbs) = Velocity (fps) 3 x Bullet Weight (grs.) 2 x 1.5 x 10-12
This is, of course, nothing more than kinetic energy multiplied by momentum, then multiplied by some constant to arrive at the desired weight range (momentum is actually velocity times mass, not velocity times weight, but the two are related by a constant). There is absolutely nothing magical about the game weights derived by this calculation; they are entirely the result of a subjective selection of the constant (i.e., 1.5 x 10-12 divided by the accelration due to gravity), although the choice of this constant may be based upon sage judgement, drawing on years of field experience. The basic premise of the formula is worthy of closer examination. The OGW formula attempts to combine in one measure the separate contributions of kinetic energy and momentum, or perhaps the two schools of thought: fast and disruptive versus slow and deep. This is well intentioned; however, simply multiplying the two values is an unacceptable method of deriving a composite effect.
For instance, a certain load may have a very high velocity and therefore a high kinetic energy, yet have a very light bullet weight and a correspondingly low momentum. How will it perform? The most useful description of its performance would be found by separating its component functions, cavitation and penetration, and analyzing these in relation to the game in question. From experience, we know that very lightweight, small-bore ultra-velocity loads are poor performers against large game. Yet an 85 grain .243 cal light game bullet traveling at 3500 fps would have an OGW rating of 389 lbs at the muzzle! Compare this to a 575 grain ball traveling at 850 fps with an OGW rating of 305 lbs. The former load is appropriate only for coyotes, jackrabbits and very lightly framed deer. The latter load is for a 16 bore howdah pistol intended to stop charging tigers!
Important caveats are in order. The author included an exhaustive list of cartridges and loads, but the distinctions of OGW are only applicable between bullet weights and velocities. The model itself makes no consideration whatsoever of the effect of sectional density, bullet diameter or nose configuration, although these were "considered" in its development. Most glaringly, bullet construction is neglected. A 150 gr. bullet moving at 2800 fps is identical in this analysis, no matter whether it be a .264 caliber RWS H-Mantel or .358 caliber Remington Core-Lokt. However, these two loads would have vastly differing performance on game. The model assumes you have selected a reasonable bullet weight and construction for the application in mind (in fairness to the author, Matunas makes this point clear, but it does beg the question of just what this model does tell you if you must already know the answer before you begin).
What the OGW Table really attempts to be describe is the approximate heaviest weight of animal that could be reliably killed from any angle using modern high-powered cartridges (not 19th century big-bore blackpowder weapons or even early 20th century weapons), assuming the bullet selected was reasonable. Strictly speaking, a true optimum game weight would be that size of beast which this load killed most effectively, and any game animal either heavier or lighter would be less effectively dispatched, so the use of the term optimum is misplaced.
Unfortunately, by making velocity a third order term it wildly exaggerates the effect of this component in terminal behavior, which (as will be shown in Part IV) has surprisingly little meaningful effect for deforming bullets. Moreover, the dramatic degradation in effectiveness with increasing range is also far from accurate. A typical high velocity bullet is shown to lose half its effectiveness between the muzzle and 250 yards; in practice no meaningful loss of performance is experienced for most high velocity rifle cartridges and indeed some performance improvement may be observed with many bullets of conventional construction at moderated velocities! Worse, it suggests that if one uses light for caliber bullets, then the most effective employment of these projectiles is at close range and very high velocity - in direct opposition to all conventional wisdom! Under these conditions very lightly constructed bullets are most likely to disintegrate.
Comparisons between different calibers and bullet weights in this analysis, as suggested by the author, are an absurdity for reasons outlined previously. They are simply not valid.
III.e. Taylor Knockout (TKO) Formula
I almost hate to comment on this one because it happens to be a favorite of one of my favorite gun writers, a man of outstanding skill and a reputable hunter whose guidance in such matters should not be taken lightly (and I don't refer to Taylor!). Taylor himself was also a man of unimpeachable experience and his views on rifles and calibers, especially for dangerous game, is taken as gospel on the subject.
However, this formula has got to go.
I'm sympathetic to the motivations which brought about its creation. The "smallbore cranks" were a cult phenomenon at the time, preaching vehemently about high velocity and kinetic energy. A number of this following ventured to Africa, and like their predecessors in the heyday of blackpowder "express" cartridges, experienced miserable failures in the field, sometimes with fatal consequences to the shooter or guides. Taylor was attempting to counter this "scientific" kind of argument with a kind of scientific methodology. Applying his many years of experience to the problem (and it must be confessed, his biases as well), he developed a formula which favored the kind of bullets and cartridges he knew to work reliably:
- TKO = Bullet Weight (lbs) x Impact Velocity (fps) x Bullet Diameter (in)
Regrettably, this formula is as misleading as any kinetic energy figures or OGW or any other I've seen. For example, a hand-thrown baseball would have roughly twice the TKO of the standard nitro express load. I doubt if anyone would argue that bouncing a baseball off the noggin of an elephant would produce any positive result. Taylor himself acknowledged that there wasn't any appreciable difference in the killing performance of the various .400s, .416s, .450s, .465s, .470s, .475s, and .500s on dangerous game when loaded with reliable bullets of sound construction. But his TKO formula (as generally interpreted) exaggerates any difference that might exist because it makes the bore diameter equally as important as the velocity; thus a .488 caliber .475 Jeffery No. 2 is seen to be 7% more potent than a .458 caliber .450 NE even though they both have the same ballistics. The comparison becomes even more exaggerated between a .450/.400 NE and a .500 NE in which the larger bore is calculated to be 55 % more potent, even though Taylor regards them as being very similar in killing performance. In fairness to the author, the TKO value is generally misinterpreted (notice that the table he provides only includes loads for solid bullets). Taylor himself said of it:
- "I do not pretend that they [TKOs] represent "killing power"; but they do give an excellent basis from which any two rifles may be compared from the point of view of the actual knock-down blow, or punch, inflicted by the bullet on massive, heavy-boned animals such as elephant, rhino and buffalo". (African Rifles and Cartridges, pg. xii)
- "There seems to be a lot of misunderstanding about this word "shock"; men seem to be under the impression that it implies killing power. But that is erroneous." (African Rifles and Cartridges, pg. 58)
Elaborating, the author indicates that this stunning effect truly applies for the most part to near misses of the brain on elephant, enabling a more leisurely dispatch with a follow-up shot (possibly of lesser caliber) or, especially, permitting the shooting of other nearby elephants, while the first is down. Such tactics are no longer permissible and were never ethical in my view (Taylor was a self-acknowledged poacher). Indeed, whether his TKO is true even in this sense is a highly contentious matter, disputed by some very experienced African hunters (I will not pretend to be highly experienced in this regard, but I have seen a Cape buffalo shot between the eyes with a .500 NE which did not produce any effect whatsoever). But the point here is that Taylor never offered this formula as an indicator of killing or even "shocking" performance for hits on the body. That is an American extrapolation of thought. Taylor includes TKO values for everything down to the .256 Mannlicher, but not with a view to offering the relative merits of one small-bore or medium-bore against another for general hunting use - its to show how puny these are relative to the big-bores for stopping an elephant. Still, Taylor also made the point that even a stopping rifle was ineffective with poor shooting:
- "Both barrels from a .600 in the belly will have little more apparent effect on [an elephant] than a single shot from a .275 in the same place." (African Rifles and Cartridges, pg. 59)
American hunters and gun writers use terms like "stopping power", "shock" and "killing power" to describe how quickly a deer (elk, antelope, etc.) falls when hit. Practically no one hunts elephant anymore and I can't remember the last time I saw an article on that subject. Promotion of the TKO is indicative of the careless way in which any quasi-scientific method is seized upon, even though the originator of it may reject that purpose to which it is put (though, again, I am not endorsing or placing validity on Taylor's TKO calculation, even for the purpose he intended).
Incidentally, if there is a "knockout" effect it will almost certainly be a function of bullet shape, presented area and velocity. Bullet mass will not matter greatly, but a separate calculation would be necessary to assess whether sufficient penetration was provided.
III.e. Lethality Index Formula
John Wooters is another old salt who ought to be listened to reverently. I love Wooters, even if he is a bit of a curmudgeon (when one has graduated to the plateau of veneration upon which he resides, one can be forgiven for curmudgeonhood). No doubt frustrated with the ceaseless litany of kinetic energy figures, he concocted a formula for what he felt was a better measure of a bullet's effectiveness on game. Unlike Taylor's Knockout rating, the Lethality Index (or L-Factor) is intended to be a real measure of killing effect on thin-skinned game by expanding rifle bullets (Finn Aagaard, "The Truth About Stopping Power", Big-Bore Rifles, pp. 8 - 11). The author's formula is described below:
- LI = Kinetic Energy (ft-lbs) x Sectional Density x Bullet Diameter (in)
To be fair to Wooters, let me assert that indeed, large diameter bullets do make more cavernous and effective wounds than smaller diameters, kinetic energy certainly is a valid component in the measure of the wounding, and bullets with a high sectional density tend to penetrate deeper and expand without coming apart (all things being equal). The trouble is: all things aren't on equal footing and "quickie" formulas that don't even take bullet performance into account cannot be considered meaningful measures of terminal effect. To be meaningful and scientifically sound (ie, correct and true), a formula or theory must be founded on carefully collected test data, not "gut feelings", prevalent perceptions and anecdotal evidence (which is little better than hearsay). Again, to be fair, this formula is an assessment of the potential of a specific cartridge-load combination and its components are assessed at the muzzle, but (as I will demonstrate in Part IV) the downrange performance of otherwise identical loads can be very different and sectional density, in particular, is an unreliable indicator of bullet performance.
Not to pick on Wooters alone, but I am suspicious of any lethality, killing or stopping formula which just goes up and up without end. What exactly does this mean? Something can only be so dead. Does a value twice as high as another mean that it kills twice as quickly, twice the game weight, two animals standing side by side, or with twice the probability? Just what is conveyed by these figures? Even their creators cannot tell you. After all the computations and three (or four!) significant digits of precision, it all comes down to a "touchy-feely" sort of thing.
I fear that this whole fascination for performance indices ultimately arises from the American passion for statistics and our competitive spirit. It is sports stats on a different plane. "My rifle load is better than your rifle load and I can prove it!" As a young person, I was as guilty as anyone of making all sorts of comparative calculations of kinetic energy and momentum for various cartridge-load combinations. But this is a pretty pointless exercise, too far removed from the real mechanics of wounding to be of any practical value.
III.g. Knock-Out Value (KOV) Formula
Well, here is what appears at first blush to be a step forward. Invented by South African pundit Chris Bekker, whose column "The View from the Veld" appears in the UK-based quarterly webzines Sporting Rifles and Deerstalker, the Knock-Out Value (KOV) formula is based on a simplified (and again, slightly erroneous) "terminal momentum" calculation:
- KOV = "Terminal Momentum" (lb-ft/s) x Sectional Density x "Mushroom Factor"
- Where,
- "Terminal Momentum" = Impact Velocity (ft/s) x Retained Bullet Weight (lbs)
- Sectional Density = Original Bullet Weight (lbs) / Bullet Diameter (in)2
The author has presented more than one version of this formula in his writings at various times (for instance, dropping the "Mushroom Factor" and using muzzle values for bullet weight and velocity), but the above is said to be the proper form. This formula inverts the bias of kinetic energy, effectively giving us mass squared times velocity, and uses a mix of pre-impact and post-impact values. It looks pretty good for all that. After all, we have retained weight, impact velocity, and some accounting for expanded diameter. So what is wrong?
Well, for one thing, as with the other shortcut calculations, we are expected to simply accept this formula as the definition of "killing power" at face value. Why should it be "terminal momentum" multiplied by sectional density? Why not kinetic energy multiplied or divided by expanded frontal area, for instance? From where did this insight arise? Where is the physical evidence justifying a "Mushroom Factor" assessed to three or four significant digits for a plethora of bullet configurations? To paraphrase Jerry MacGuire: "Show me the data!" The only evidence that the author offers to support its validity is in the form of comparisons to other popular "indices" and "factors" or to kinetic energy alone. No documentation or analysis is provided to show how these relationships in the KOV were derived. If there were any objective analytical basis for this formula then the author would have presented that data instead of simply making unsubstantiated claims about the correspondance between the KOV and actual field experience. When the support of real world field experience is drawn upon for evidence, it is entirely subjective, simply the conventional wisdom, thusly: "Everybody knows that the 9.3 x 62 mm Mauser would be far superior on Cape buffalo than the .300 Remington UltraMag, and the KOV shows it to be 28 % better". Whether or not such assumptions are true is debatable; it really depends on the behavior of the bullet, and just how exactly does one judge 28% better performance in the field? Deeper? Larger hole? Fewer steps until it collapses? Was it actually 32 % better or only 24 % better? See the problem? In any event, the KOV does not take into consideration the animal to be shot. Why then a Cape buffalo for the comparison? Why not a much smaller animal such as a springbok? I seriously doubt if the 9.3 x 62 mm Mauser can be shown to be 28 % better on a springbok. What then can such a measure really tell us?
This sort of reasoning does not qualify as scientific argument or evidence. None of this mathematical dexterity demonstrates anything but that one can multiply several numbers together and end up with a result that subjectively and qualitatively satisfies one's preconceived notions about how things work - most of which are based on popular notions of terminal ballistics (better known as "conventional wisdom" and frequently incorrect). By a careful (or haphazard) selection of terms and their arrangement, one can make almost any cartridge-load combination seem far superior to others. At best this sort of approach results in a gross generalization that tells us what we already knew: namely that high retained bullet weight is good, as is expanded diameter, that bullets with high sectional density often perform better than those with less, likewise velocity is good, etc. However, some of these also have a downside.
Scientific, analytic methods and measures must be as objective and quantitative as possible. Consequently, theories of terminal effect must be evaluated in quantitative terms, meaning that dimensions of wounds must be evaluated. None of these formulaic approaches can be defended with a study of field results, nor have their creators (including Taylor) ever attempted such an analysis. Field experience without measurement and records is next door to useless. It is no better than armchair opinion or hearsay. Science is not founded on faith in the analytical powers of an individual, it is founded on fact which can be examined and tested by any individual. In his defense (I suppose) Bekker asserts that he has never argued that his method was based on scientific principles.
In constrast with the OGW which never even considered the caliber or the TKO which looked only at nominal caliber, the KOV at least considers the expanded diameter; however, the expanded diameter and impact velocity do not occur together in time. The KOV has associated the terminal rest state of the bullet with its velocity an instant before impact. This is a bit like suggesting that a person can walk away from a head-on collision with a 10-ton cement truck at 100 mph in exactly the same state that one steps into the vehicle as it is sitting in the driveway at home. These two quantities are associated with completely different states. Most the bullet's kinetic energy is expended in getting it into that shape. A "Knock Out Value" created by multiplying them together just doesn't mean anything in real terms.
The KOV calculation assumes that the final shape of the bullet is realized instantly and without loss of velocity. In actuality, the velocity changes instantly with each micromilimeter of penetration as the bullet gradually expands to its maximum expanded diameter, then assumes its smaller fully deformed diameter that it will hold when the hydrodynamic pressure falls below the effective flow stress of the bullet (at a velocity of roughly 1900 to 2100 fps for most rifle bullets). Though transpiring in a matter of time on the order of a fraction of a millisecond, this is not an instantaneous process and the interval between impact and the termination of hydrodynamic deformation in the bullet dominates the mechanics of the penetration event.
The following figure illustrates the point. It shows a series of curves describing the fraction of the kinetic energy at impact which is expended during the hydrodynamic deformation phase (of the bullet) for various retained bullet weights, assuming a hydrodynamic termination velocity of 1900 fps. It will be quickly observed that, for 3000 fps impacts, between 60% and 80% of the kinetic energy at impact is expended by the time the bullet has assumed its final form. Even at impact velocities only slightly above the hydrodynamic termination velocity, the energy of the bullet deformation and resultant cavitation amounts to between 10% and 55% of the total kinetic energy. If we care about retained weight and expanded diameter, we ought to care about the interaction which resulted in those dimensions.
However, even if one knew the termination velocity for deformation in the bullet (which could be anything from 1400 to 2300 fps), any statement about the character of the wound (or the impact velocity) based solely on the final expanded diameter and retained weight would amount to mere speculation. It is the process by which the bullet assumes that final form that dictates the dimensions of the wound. I have tested many bullet designs and there is no obvious correspondance between the appearance of the recovered bullet and the hole it produced during its penetration. Two bullets having the same recovered appearance may create very different wounds based on their impact velocity and design characteristics. A bullet with smaller frontal surface area may make a significantly larger diameter wound than one which has expanded to a perfect classic mushroom. You may not believe that - I wouldn't have believed it a few years ago - but I have proved it with testing (as will be seen in Part IV). In fact, if I may generalize, I have observed that the bullets recovered with the smallest expanded diameters frequently produce the largest cavities (not that they are the best overall performers). This is why, in my mind at least, both testing and careful observation and measurement of field performance are infinitely more meaningful than homemade formulas based on no physical evidence or pontifications safely removed from the contrary facts.
Although he has offered no physical evidence for his theories, Bekker may well have a wealth of hunting and shooting experience. That can be incontravertably said for Pondoro Taylor and John Wooters and others who have attempted to create a mathematical expression consistent with their experience. I do not wish to be perceived as casting doubt on the substantive creditable field experience of anyone. Still, these methods do not work.
In the final analysis, the KOV is little better (nor any worse) than the TKO, OGW or any of a score of other formulas trying to get at the answer. They all fall short of telling us more than we already knew. I don't wish to be unduly criticizing the KOV or Bekker. These formulas are merely the manifestations of the baises we have all held as hunters. The KOV looks like it has a little more thought behind it than many such calculations, but precisely for that reason (and as a springboard to elucidate what is really happening during bullet deformation) I have given its features a more studied treatment.
What most or all of these shortcut attempts suffer from at the very outset is old-fashioned human prejudice - the pre-conceived notion on the part of the creator concerning what the answer ought to be. We are all guilty of this. Not even the best of us is free from prejudice (which is why objective methods must be adhered to), but if we seek the truth, we must be prepared to be divested of our cherished beliefs, not always see them upheld. Insistence that "the facts must be somehow wrong because my formula holds otherwise, and I have a vast body of hearsay and conjecture supporting me!" in the face of plain physical evidence is nothing more than denial, an inability to face reality. Theory must bend to empirical evidence. A better understanding will follow.
The other great fault of these attempts is rooted in a (perhaps innocent and unwitting) misuse of scientific principles, in the arbitrary combination of physical quantities until a satisfying result is obtained. Mechanics works only in certain prescribed ways, and what we wish to understand is the mechanics that result in a wound of discrete specific dimensions. Pseudoscience doesn't advance our understanding and those of us who know better need to improve the understanding of those who genuinely seek to know, yet stumble here and there with the details.
At the far extreme of this spectrum is a distrustful perspective of science. Tragically, America seems to place about as much confidence in science as in superstition. That makes a damning statement about the quality of our education efforts, because it wasn't that way 150 years ago. I frequently encounter the attitude on cyber forums that physics and biology cannot really answer our questions about terminal performance, that it all comes down to some kind of personal thing, like faith, or that it is far too complex to ever understand or quantify. That is nonsense. The truth isn't "up for grabs", no matter how poor our grasp of it may be.
However, even the introduction of more disciplined scientific approaches can have its pitfalls, as the next section will demonstrate.
III.h. When Good Physics Goes Bad
Anyone who has persevered this far is either with me or else hopelessly confused but doggedly staying in there. For the hopelessly confused I now offer some hope. Probably I should have included this discussion in the Mechanics of Terminal Ballistics, but I wanted that discussion to be general and easily accessible. Of necessity, we must now get down to the details.
1. Relationships of Force, Momentum and Energy
I am appalled by the articles that I see published which evince an ignorance of basic science. Just for my own sanity and peace of mind, lets review a few fundamentals.
Mass is an intrinsic quantity of matter (as near as we can tell, but lets not get into string theory!) which describes the inertia of an object. Inertia is the property of an object that resists changes in its motion. Mass and weight are not the same thing, but we normally associate the two. Weight is actually a force exerted on a mass in a gravitational field. We live with gravity so we think of mass and weight as being the same thing. It is vital in mechanics to recognize that they are distinct. This distinction was not perceived at first by scientists (understandably) and so the original English system of units is based on the fundamental quantities of force (pounds), length (feet) and time (seconds), rather than more properly on mass (kilograms), length (meters) and time (seconds), as was done in the late 18th century when the metric system was developed (yes, its over 200 years old!). Force is actually a derived quantity, not a fundamental quantity, but that's water under the bridge, so lets not brood about it. [This conundrum has led to the introduction of pounds-force, lbf, and pounds-mass, lbm, where 1 lbf = 32.174 lbm-ft/s2, but this convention has not removed the confusion, only resulted in NASA Mars probes falling from orbit! Vive le Systeme Internationale!]
Velocity is the rate of change in the position of an object with respect to time. Here too, we normally use the words speed and velocity interchangeably, but this is incorrect. Speed is the magnitude of the velocity, but velocity is a vector quantity which means that it carries with it an implicit direction of movement. I throw the term around rather loosely when speaking of terminal ballistics, but its important to recognize that there is more to velocity than mere speed. Because next we get to acceleration.
Acceleration is the time rate of change of velocity. Note that this is any change in the velocity, either in magnitude or direction. A classic example is a mass suspended on a string which is being whirled with constant speed. It is being subjected to a constant acceleration because its velocity direction is constantly changing as it turns around its center of rotation. [Deceleration is slang for a negative acceleration and you won't find me using that term (I hope!)]. Obviously then, acceleration is also a vector quantity. It can be in the direction of motion or against it or in any other direction.
Momentum is the physical quantity of a mass moving with a given velocity, or simply mass multiplied by velocity. The units in the English system are lb-s, not lb-ft/s as some might suppose (remember, pounds are a unit of force, not mass). Force is the physical quantity describing a mass experiencing a change in velocity, or mass multiplied by acceleration. Clearly then, the time rate of change in momentum is the same as force. Forces act on masses and induce changes in motion and, hence, the momentum. More on that in a moment.
Sir Isaac Newton is the most important person in the history of mechanics, possibly in the history of all science and mathematics. He not only established the rules of mechanics in terms of its fundamental physical quantities, he also invented the calculus, which is essential for solving the problems of mechanics. It is useful to understand the difference between a derivative and an integral for the following discussion.
A differential quantity is an infinitessimally small slice of a whole. A derivative is a rate of change between two differential quantities. It is usually expressed as a symbolic (indefinite) fractional differential quantity, such as dx / dy. Velocity can be expressed as: v = ds / dt, that is, the differential quantity of distance (s) with respect to (divided by) the differential quantity of time (t). Acceleration can be similarly expressed as: a = dv / dt. Since acceleration is the time derivative of velocity and velocity is the time derivative of distance, then it is clear that acceleration is the second time derivative of distance, a = d2s / dt2. Sometimes we speak of the differential quantities of mass, distance, velocity and acceleration as dm, ds, dv, and da, but it is important to note the distinction between a differential quantity and a derivative. Not all derivatives are taken with respect to time, for instance.
An integral, on the other hand, is the sum of the differential quantities integrated over some boundary. For example, the total distance (s) covered by a bullet can be described using calculus by performing the time integration of the velocity, taking each instantaneous velocity (ds / dt) and multiplying by each differential time quantity dt and summing at each step. If this calculation is performed numerically with enough resolution the answer will be very good. It can be performed analytically for an exact answer if the expression describing ds / dt is known and has an analytical solution (not always the case).
Energy is a term that gets abused more than any other scientific term. Crackpots love energy. Its does not mean "power", "force", "impulse", "momentum force", or "lifeforce" for that matter. Energy is a physical quantity describing the activity state of matter. It occurs (or manifests itself) in many forms. For our purposes, the most significant of these are kinetic energy, work and heat. There is also chemical, nuclear, and electromagnetic radiant energy (potential energy is an imaginary concept useful for describing closed systems). Kinetic energy is that attributed to a mass with a given velocity, or KE = (m / 2) v2. Work is the energy expended by a force acting over a distance, W = Integral (F ds). Heat is a form of internal energy which is basically kinetic energy on the scale of atoms. Heat can be expressed as the mass multiplied by the specific heat of a material and its absolute temperature, U = mCT. Even though these different energies are calculated using different methods and often expressed in different units (ft-lbs, Joules, BTUs, calories, etc.), they all resolve back to the same fundamental units of mass x length2 / time2 and are totally interchangeable. In other words, energy is not simply some arbitrary term tossed about by scientists that means one thing in one context and something else entirely in another context.
Power, in a scientific sense, is the time rate of energy change, transfer or usage, expressed as the derivative de / dt. If any term gets abused more than energy it would have to be power. Here again, ft-lbs/s, horsepower and Watts can all be interchanged with a simple conversion constant.
2. Bogus Ballistics
Freshman level physics textbooks are notoriously bad sources for an understanding of ballistics. I can sympathize with the confusion that many have in this regard. One would like to believe that the college level science class is free from error. Sadly, the late American penchant for making things accessible to the student has led to some regrettable abuses of the truth.
Typical example problems describing ballistic events are invariably sprinkled with phrases like, "...neglecting air resistance...", "...assuming perfectly elastic (or rigid) bodies...", and "...a collison between perfectly plastic bodies...". Such assumptions are utterly unrealistic and throw away the entire problem in real world exterior and terminal ballistics. One could more sensibly neglect gravity than air resistance! The acceleration due to gravity is only 32.174 ft/s2, but the (negative) acceleration due to air resistance is on the order of 1700 ft/s2 or more for high velocity rifle bullets. The equations of motion governing ballistic flight of projectiles are a set of highly complex, coupled second order differential equations that have no analytical solution, so the textbook manufacturers don't want to discuss it. Its easier to give the student a false impression of reality with a problem that is simply solved.
Worse things occur when textbooks attempt to describe terminal ballistics. All real bodies are imperfect, plastic (inelastic) and non-rigid. Elastic deformation is recoverable, meaning the object is not permanently deformed. Plastic deformation is permanent. All ballistic events of interest involve large plastic deformations. Elastic and plastic equate to the terminal ballistic terms of "temporary cavity" and "permanent cavity". Rigidity is the completely imaginary property of not deforming, either elastically or plastically, when loaded. There is no such thing as a rigid body, but the assumption is often used in textbooks for simplicity when (typically static) loads are small in relation to the strength of an object. It ought to be pretty clear that rigidity is not an assumption that makes much sense in terminal ballistics. Furthermore, any degree of penetration implies plastic deformation, which is not described by the most commonly offered relations of elastic mechanical behavior (such as Hooke's Law), and which involves a significant expenditure of strain energy. Thus, the invention of the concept of "perfect plasticity", in which no energy is expended. This abomination violates the Second and Third Laws of Thermodynamics and should never even be discussed.
Newton's Second Law of Motion, also known as the principle of Conservation of Momentum, states: The time integral of the change in momentum (or, the impulse) of a system is equal to the time integral of the resultant of all applied forces acting on the system, or: I = Integral (F dt). This has been restated in the more familar form that the resultant of all the forces acting on a system is equal to the integral of the mass multiplied by the acceleration, or Integral (F) = Integral (ma). This is the most important expression in all mechanics. Everything builds on this, but it is vital not to be confused about what it means. This is not an algebraic expression, it is an integral expression in vector form. Different rules apply.
Newton's Second Law is often violated in textbook sample problems. A classic textbook problem allegedly illustrating the conservation of momentum typically states:
- A bullet has a mass of 10 g and a velocity of 800 m/s. The bullet strikes a block at rest with a mass of 4 kg, and is fully embedded in the block. Assuming that the bodies are perfectly plastic and that the block rests on a frictionless surface, find the final velocity of the block. [Solution: ~ 2 m/s]
This is rubbish and does not demonstrate Newton's Second Law. The effects of time and force are completely lost! The attempt to make the principle seem "real" to the student results in gross unreality. Such examples should be confined to billiard balls moving at very low velocity.
But even here we have a problem. The classic freshman physics class demonstration of the principle of conservation of momentum is the rack of billiard balls suspended by wires. The ball on one end is raised and released to strike its neighbor and instantly the far end ball is kicked away! Very cool! Ever wonder though why it is that the balls do not go on clacking forever? They return to rest in just a few seconds! What they didn't tell you in school is that there is more to mechanics than conservation of momentum. In fact, the gross way that conservation of momentum is typically described, as the resultant of all the gross mass-velocity interactions, is simply untrue. But Newton didn't express his second law in these terms. It is clear in the case of the clacking billiard balls that something "steals" a piece of momentum with each cycle until the whole system returns to rest. [In fact, there exists a finite number of balls such that if one is clacked into its neighbor on one end there will be no movement by the ball on the other end due to the accumulation of small "robberies" of energy along the way.]
Some have argued that this is entirely due to the frame or the wires or contact with the Earth, but this is simply not true. The result would be no different if the same experiment were repeated in deep space, far from any gravitational field and not contacting any other body, with a series of hard spheres interconnected by springs. Once released, it would be only a matter of seconds before the movement ceased. This is a closed system, from which the energy is not escaping as radiant heat fast enough to explain why it stops moving (and heat is not a term in Newton's equation anyway - however, this serves to illustrate why energy methods are the only methods of analysis that really prove dependable in some situations). Ah! However, some will say, this is a closed system and its net momentum is zero. Indeed, but that misses the point. The momentum of the cycling masses within the system will still degrade to zero and the laws of motion apply to them as surely as to anything else. Moreover, if you built a device that permitted the momentum of these interconnected masses to be transmitted in all directions (due to their interconnections - as is true in atomic matter) then the linear momentum of an impact would not be preserved for an inelastic event, even in space far from any gravitational influence. Understand what I am saying. I am not contending that Newton's Second Law is invalid; quite the contrary. What I am saying is that real mechanics is not simplistic. Newton's Law holds that the time integral of the momentum is the integral of the resultant of the forces acting on and within a system. I am trying to emphasize that those forces depend on the system; the motion of the balls depends on the microstructural makeup of the balls. If the forces vary with physical properties then the mechanics of motion cannot be assumed to be invariant.
That thieving something is the resistive forces in the atomic structure of matter. Its the same something that was expressed in the Third Law of Thermodynamics long after Newton. All movement encounters resistive forces, such as friction. There are forces akin to friction between the atoms in every body. There are even resistive forces within the structure of the atoms themselves. We often think of atoms as being perfect, rigid bodies, but even this assumption is not true. So, the hard fact is, that even on the most fundamental level the simplistic view of the conservation of momentum is not perfectly true. It is almost true for low velocity impacts between elementary particles (certainly not true for relativistic impacts).
So Newton was wrong? Absolutely not! Its only that we often do a sloppy job of accounting for all the forces which act on and within a system. If you neglect any force, then the answer is wrong. Maybe its only slightly wrong and you don't care, but in the case of terminal ballistics involving large plastic deformations we simply cannot neglect the complex interactions of the materials. Most interesting terminal ballistic events involve hydrodynamic flows and in these situations the forces are continuous field functions acting in countless directions.
Recall that energy may be expressed as force acting over a distance. One may then say that a certain quantity of energy, say 100 ft-lbs, is equivalent to raising 100 lbs up against gravity a distance of one foot. Here we see the difference between pure science and engineering. Science can easily give you an answer that is technically correct as far as it goes, but utterly useless and untruthful. In the real world, the example I just gave is true only if the weight is raised at quasi-static velocity, barely moving. It matters how fast you do something. Fast work wastes energy. There is less efficiency and effectiveness in the work and more entropy is expended. Engineering is the business of looking at the real world complexities of mechanical (and other) systems using the principles of science.
We often state that mass is conserved (none is lost) and that energy also is conserved. Properly speaking, mass-energy is conserved. Mass and energy can be converted one into another and both quantities must be accounted for together to preserve the whole truth of mechanics. For ballistic events at non-relativistic speeds, theory holds that no mass is converted into energy, but the kinetic energy is converted to strain energy (causing both elastic and plastic deformation) and this energy is finally converted into thermal internal energy when the body assumes a rest state. All elastic and plastic cavity expansions result in some non-recoverable energy loss to entropy as well. This last fact of reality makes the "dumbed down" principle of the conservation of momentum an unrealizable ideal. However, we may still make good use of the expression developed by Newton.
As an aside, the most dramatic demonstration of this elastic energy dissipation that I have seen was on a late night infomercial for some high tech golf clubs. The heads were made from a metallic glass that is composed of zirconium with a half dozen other metals mixed in to fully occupy all the spaces in the crystal lattice. Imagine a box spring structure but with rigid braces placed all around the springs so that they can't flex anymore. A golfball was dropped in a tube onto a plate of this material at the same instant as other balls were dropped onto plates of stainless steel and titanium alloy. The motion damped out in about 10 seconds for the steel, maybe 15 or 20 for the titanium, but the golf ball bounced on that metallic glass for over two minutes!
A worse case of the bullet striking the block example was brought to my attention some time back, in which the conservation of momentum was again reduced to mass times velocity resultants, but the problem then went on to compare the initial kinetic energy of the bullet to the final kinetic energy of the moving block and concluded by stating that the difference in these energies was absorbed by deformation and resistive forces. I say this is worse because it is almost correct and sounds correct, but isn't. You don't start with the idea that gross body motions must be preserved and then declare any discrepancies in energy to be resistive losses. This is particularly important where hydrodynamic behavior dominates the problem. The correct solution is to treat the interaction at a scale of resolution appropriate to the deformation, perform basic calculations of the fundamental equations of motion and visco-plastic deformation while numerically integrating, account for the energy explicitly at each step, and then describe the gross body motions. Doing this manually is out of the question, which is why the defense sector uses high powered computer workstations and finite element or finite difference methods.
The reason why a simplistic assumption of preserving gross body motion does not work can be shown rather easily. If we describe an impact between two perfectly rigid bodies, m and M, the larger one of which is initially at rest, we may express the force exerted by the smaller mass on the larger at impact as:
F = m d2s / dt2,
and the force exerted by the larger mass on the impacting mass as:
F = M d2s / dt2
(Note: the forces are equal in magnitude and opposite in direction and the accelerations are not the same).
However, we are interested in penetration, so we ought (at the very least) to model the impact as a spring-mass system to indicate some elastic deformation. Let's concentrate on the larger mass (the target), recognizing that what applies to the target also applies in its own way to the projectile. According to Hooke's Law the force applied to a spring in elastic deformation is equal to the spring constant multiplied by the deformation distance. For a spring-mass system, the total force exerted by and on the projectile is given by:
F = M d2s / dt2 + K s
Observe that already the acceleration on the mass is reduced by the quantity Ks / M. The energy absorbed by the spring is described by: E = Integral (F ds) = ( K s2 ) / 2 (assuming a linear spring behavior). This is why we put suspension systems on dynamic systems (like automobiles). If the acceleration on the mass is less, it doesn't move as far or as fast. Its that simple. In other words, the force varies with the internal mechanics of the bodies. Newton isn't violated.
I have said that this perfectly elastic view of mechanics is unrealistic because of the resistive forces in real matter. So, lets add some nonrecoverable energy losses. Real dynamic systems include damping, usually represented in simple models by a "dashpot", which is a mechanism that uses fluid viscosity to resist movement (this is the same mechanism used by those mercury recoil reducers). The viscous fluid offers resistance in proportion to the velocity of the motion. In consequence, the total force is expressed by:
F = M d2s / dt2 + D ds / dt + K s
The energy dissipated by the damper is given by: E = Integral (F ds) = Integral ( D v ds ). The acceleration is reduced by: ( Ks + Dv ) / M. Since in each case the input is identical, we must conclude that the acceleration on the mass in the case of the viscous damped spring-mass system is less than the acceleration experienced by the rigid mass. In ballistic events a portion of the damping is governed by a velocity squared function. Therefore, neglecting the damping and elastic dissipation of energy results in an overestimate of imparted motion.
The May/June 2002 issue of Petersen's Rifle Shooter has an article entitled, "Where Do the Foot-Pounds Go? - Relating Kinetic Energy to Hunting Cartridge Effectiveness", which is much improved over the pseudoscience of four years before, and even begins to address the conversion of kinetic and strain energy into residual internal energy; however, it introduces one misunderstanding worth clearing up. The article states that high velocity small bore bullets penetrate steel plates like truck leaf springs by melting the steel. This sort of mythology is prevalent in my field and seems plausible, but even shaped charge jets moving at 9000 m/s (29,500 fps) do not melt the steel armor plates that they penetrate, nor is the soft pure copper jet melted (this is not conjecture, it has been demonstrated). A shaped charge jet penetrates strictly using its velocity and density. The dynamic pressure vastly exceeds the strength of the steel and the penetration velocity exceeds the sound speed, so the steel effectively has no strength at all. Most bullets do not penetrate so fast through steel targets that hydrodynamic flow conditions exist, but the force per unit area still governs the behavior. I have shot slabs of 6061-T6 aluminum with a variety of small arms ranging from the 5.56 x 45 mm M193 and M855 (SS109 type) bullets up to the .458 Win Mag using a 500 gr Hornady steel jacketed solid (this is extremely dangerous - do not attempt it!). At no stage in any of these penetration tests did melting occur, either in the target or in the bullets (aluminum, copper and lead have much lower melting points than steel). The bullets were recovered and exhibited mechanical strain, which would not be evident if they had melted. The target exhibited similar behavior. Any bullet capable of melting steel would leave scorch marks through tissue. When was the last time you saw that? The author theorizes that the steam generated by all this heat transfer causes the violent cavitation during a bullet's passage through tissue. One need not guess at it. We know that dynamic pressure is responsible for cavitation in hydrodynamic penetration. Bullets actually deliver very little energy: 2000 ft-lbs is 2700 J or just enough energy to light a 100 W bulb for 27 seconds. I don't fault the author of that article for the mistake, as even sources that ought to be authoritative in the defense field make similar errors with regard to penetration mechanics (I still regularly see references to shaped charges producing plasma jets, and that sort of misinformation).
Real world penetration is best described as a fluid mechanics problem. For inviscid hydrodynamic flow, the dynamic pressure is (according to Bernoulli's Law) proportional to the fluid density and the square of the velocity: P = ( rho / 2 ) v2. The force exerted on the bullet at the stagnation point is equal to the dynamic pressure multiplied by the presented area of the projectile in the direction of the flow, F = P A. The first thing that this tells you is that the mass of the bullet has no direct bearing on the diameter of the cavity created - rather, that is governed by shape and velocity. The kinetic energy expended during penetration is given by the force exerted over the distance of travel by the bullet, or: E = F ds. So, the differential penetration depth, ds, is simply the kinetic energy divided by the force acting on the projectile, ds = E / F. Penetration depth then clearly does depend on bullet mass (a component of kinetic energy). This expression could be integrated very easily (and conveniently) in terms of velocity.
Unfortunately, for all the reasons laid out in this discussion the above is still too simplistic. Living tissue is not a Newtonian fluid. It is solid, with fluid characteristics in high velocity impact events. It obeys non-linear behavior laws. It is viscous and exhibits second-order damping effects, as well as compressibility and elasticity (fluids are inelastic and, notwithstanding popular mythology to the contrary, all fluids are compressible), so calculations of the permanent cavity created by the bullet will be complex. We need a velocity dependent drag function and a constitutive relation governing the strength, elasticity, compressibility and failure mechanisms of the target material(s).
So, where do we go from here? Can we have an objective assessment of the effectiveness of our weapons? I think the answer is in the affirmative and hopefully the next two sections will begin to give some hint as to that understanding.
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