The G&S Online Killing Power Formula: Take 2
By Gary Zinn
Guns and Shooting Online Owner/Managing Editor Chuck Hawks first published his version of a rifle cartridge killing power formula in 2005. After reading his article related to that formula The G&S Online Rifle Cartridge Killing Power Formula and List), I realized there were several implications and potential applications of the G&S Online formula that had not been adequately investigated. I addressed those in The G&S Online Rifle Cartridge Killing Power Formula: Implications and Applications.
The Hawks formula was developed to calculate the killing power of hunting loads, using downrange impact energy, bullet sectional density and bullet cross-sectional (frontal) area as the input variables. Calling the output variable of the formula for a given load "KPS" (Killing Power Score), the formula is:
KPS at y yards = (Impact Energy at y yards) x (sectional density) x (cross-sectional area), or simply KPS @ y = E @ y x SD x A
The Hawks article cited above includes a discussion of the logic of the formula and its independent variables, along with a list of KPS values for representative loads of a wide range of cartridges. The list contains 100 yard KPS values only, but a KPS can be calculated for any range, an important capability that sets this formula apart from others.
The KPS formula makes a lot of sense to me. My understanding of bullet terminal performance is that impact energy, sectional density and cross-sectional area are all quite important to terminal performance. The KPS formula combines these variables in a direct, easy to calculate way.
If anyone wonders about bullet weight, it is implicit in the KPS formula. This is because bullet weight in grains = SD x Diameter squared x 7000, where 7000 is the number of grains in a pound.
Bullet velocity is not neglected, because the energy generated by a given bullet at any particular range is partly a product of its velocity squared. Thus, energy implicitly includes velocity in the formula and energy at the point of impact is more relevant to determining the killing effectiveness of a hunting bullet than is velocity per se.
Whenever any of these variables change, the KPS number changes proportionally. For instance, if, between 100 and 150 yards, the energy of a bullet decreases by (say) 15 percent, the KPS will decrease by the same percentage (allowing for small variations due to rounding). Thus, KPS numbers generated from different data inputs (Energy, SD, or A) are directly comparable.
I have used KPS values in a number of studies of the ballistic performance of rifle cartridges and loads. Along the way, my understanding of the formula and its applications has evolved to the point where I feel that a sequel to my original article is in order. Hence this "Take 2" on the G&S Online Killing Power Formula.
Mass, weight, velocity, and kinetic energy
The subtle genius of the Hawks formula is that it uses downrange bullet energy as a dynamic variable in the calculation of KPS values. Here is a basic explanation of this.
In scientific terms, the kinetic energy of a moving object is a function of its mass and its velocity squared. The general mathematical expression of this is: E = (m, V sq.), where E is kinetic energy, m is the mass of the object and V is its velocity
Rather than drag readers through the physics relating mass to weight and the mathematical derivation of an operational kinetic energy formula, I will cut to the chase and state an equation that works for our purposes: E = (Wt. x V^2) / 450,000 where E is kinetic energy (in foot pounds), Wt. is bullet weight (in grains), and V^2 is bullet velocity (in feet per second) squared. This equation, though the constant in the denominator is rounded slightly, yields answers that fall generally within 1 or 2 ft. lbs. of energy figures generated by online ballistics programs.
There is a very important, but easily overlooked, relationship between what happens to velocity and energy over the course of a bullet's flight: bullet energy decreases more rapidly with distance than does velocity.
I will use the example of a .30-30 Winchester firing a 160 grain Hornady FTX bullet at 2400 fps MV to illustrate. Using the equation just stated to calculate kinetic energy, here are the results at the muzzle and at distances downrange where bullet velocity has decreased by 25 and 50 percent (labeled MV, V1 and V2, respectively; V stands for velocity and E for kinetic energy).
MV 2400 fps ME 2048 ft. lbs.
The point is that energy is a better variable than velocity for estimating the killing power of a bullet striking a game animal at a given distance downrange. Yet formulas developed to estimate killing power have generally used bullet velocity rather than energy in the analysis. This sucks, because of the divergence between velocity and energy over distance.
I compared the G&S Online Killing Power Formula with four formulas that have been developed for the same general purpose. (These are listed in the Addendum, below.) Perhaps my mathematical manipulation and interpretation skills have eroded over time, but I cannot discern that any of them treat bullet velocity and weight in such a way that kinetic energy emerges during application of the formulas.
Further, my understanding of these formulas is that three of the four use muzzle velocity as an input variable, rather than velocity at the point of impact, downrange. Therefore, the "scores" generated by these formulas actually tell us little about the killing power of a given bullet at 100 yards, 200 yards, or any other distance in particular.
The Hornady HITS formula explicitly uses velocity at 100 yards as an input variable. Therefore, HITS values indicate the killing power of a bullet at that range, but only at that range. (HITS and the G&S Online formula provide similar comparative results at 100 yards.) This is of some practical value, since many game animals are shot at distances of 100 yards, or thereabouts, but a HITS score will overestimate killing power at longer distances, because it does not account for the fact that energy decreases faster than does velocity.
This is not a limitation of the KPS formula, though, for bullet energy can be determined at any distance of interest, so that the KPS of the bullet at that distance can be calculated. This will be illustrated below.
Sectional density and cross sectional area
These are easy values to calculate for any particular bullet. Sectional density (SD) is the ratio of a bullet's weight in pounds to the square of its diameter, i.e., SD = (Wt. / 7000) / Dia.^2 where the numerator is bullet weight in grains, divided by 7000 to convert grains to pounds, and bullet diameter is in inches (e.g., a .24 caliber rifle bullet is .243 inches in diameter). Almost all bullet makers note the SDs of their bullets in reloading manuals, ballistics tables and the like, so the actual calculation is seldom necessary.
SD affects penetration, as all other factors being equal (bullet construction, for example) the bullet with the highest sectional density will penetrate deepest. Obviously, to kill cleanly, any hunting bullet must penetrate into the animal's vitals, so hunting bullet SD is important. For Class 2 game, a SD of .20 has long been considered about the minimum acceptable for medium range rifle cartridges, bullets with SDs of .26 or higher penetrate deeper in larger animals (Class 3), and heavy bullets favored for Class 4 game generally are in the larger calibers (.366" or greater) and have SDs in the neighborhood of .300, or better.
For instance, a normal power .308 Winchester load with a 150 grain bullet (SD .226) has enough terminal power and penetration potential for use against deer and any other Class 2 animal (up to about 300 pounds). However, a 190 grain (SD .286) or 200 grain (SD .301) .30 caliber bullet, in a full power .30-06 or .300 Winchester Magnum load, would be a better choice if one were hunting moose, which are among the largest Class 3 animals.
Cross sectional area (A) of a bullet is also a simple calculation, in that it is the area of a circle. The formula is A = (Dia. /2)^2 x 3.1416 or A = Radius^2 x 3.1416. The result is in square inches of area - e.g., the cross section area of a .243" diameter bullet is .0464 sq. in.
As bullet diameter increases, cross-sectional area increases in greater proportion. Here are some examples:
A for a .243" diameter bullet is .0464 sq. in.
A for a .308" diameter bullet is .0745 sq. in.; diameter and A are 27% and 61% greater, respectively, than diameter and A of a .243" bullet
A for a .375" diameter bullet is .1104 sq. in.; diameter and A are 22% and 48% greater, respectively, than diameter and A of a .308" bullet
A for a .458" diameter bullet is .1647 sq. in.; diameter and A are 22% and 49% greater, respectively, than diameter and A of a .375" bullet
The cross-sectional area of a hunting bullet matters, because a fatter bullet makes a bigger hole in the target (other factors, like bullet expansion ratio, being equal). The larger the wound channel, the more tissue is damaged. In other words, size matters.
In general, matching bullet caliber (and weight) to the size and "toughness" of the game animal is a good idea. For instance, it would not be prudent to hunt moose with a .24 or .25 caliber rifle and it would be silly, in my opinion to hunt Pronghorn antelope with a .300 or larger caliber rifle.
Baseline KPS values
Any killing power index system must have some baseline values that define minimums for the weight and/or type of game animals that might be hunted. For instance, the Hornady HITS system defines the following baseline values and ranges. The HITS classifications largely correspond with the Class game size/type classifications, which are noted in brackets.
HITS 500 or less: Small game (50 pounds or less) [Class 1]
The interpretation of this is pretty clear, so I will give just one simple illustration. A Hornady 150 grain .30-30 load has a MV of 2340 fps (20" barrel) and a 100 yard impact velocity of 1914 fps. Entering the bullet weight, impact velocity and bullet diameter into the HITS equation yields a HITS score of 649. Therefore, this load has comfortably adequate killing power for deer and other Class 2 game (at 100 yards), but not enough power to be used effectively against Class 3 game at 100 yards and beyond.
The game size/type classifications used in the HITS system made sense to me and I also liked that they related to the CXP/Class game animal classifications. I decided to mimic the hybrid HITS - Class of animal categories when developing baseline values for KPS scores. Deciding on baseline KPS values has been a drawn out process, involving analyzing the ballistic data of a wide range of cartridges and loads, pondering what the numbers indicate and discussions with my muse, Chuck Hawks. (I almost resorted to using a Ouija board at one point!)
Here are the KPS baselines that I have settled on. Understand that these are ultimately judgment calls, so someone who has a different perspective might question particular baseline values. That said, I believe the baselines presented here to be reasonable, though not immutable.
KPS 12.5 or greater: Class 2 game, 50 pounds to about 150 pounds (e.g., average size NorthAmerican deer, pronghorn, feral hog)
KPS 15 or greater: All Class 2 game, but especially those between about 150 and 300 pounds (e.g., very large whitetail deer and feral hogs, mule deer, caribou, black bear)
KPS 30 or greater: Smaller Class 3 game, 300 to about 500 pounds (e.g., smaller elk, red stag, oryx, very large black bear)
KPS 32 or greater: Class 3 game between about 500 and 1000 pounds (e.g., trophy elk, average moose, kudu, zebra)
KPS 35 or greater: All Class 3 game, but especially those between about 1000 and 1500 pounds (e.g., very large moose, eland)
KPS 68.5 or greater: Thin-skinned Class 4 game (e.g., leopard, lion, grizzly bear)
KPS 88 or greater: Thick skinned dangerous game (e.g., Cape buffalo, rhino, elephant)
MPBR, KPS and effective range
The stage is finally set to demonstrate the full capabilities of the KPS system. I will use the Hornady 150 grain .30-30 load mentioned above as a first example. Vital statistics of the load include MV 2340 fps, BC .186 and SD .226. Here are the steps in evaluating this load.
Calculate the maximum point blank range of the load. In this case, the +/- 3-inch MPBR is 206 yards, which I will round to 205 (the nearest 5 yard increment). I always calculate +/- 3" MPBRs for hunting loads, +/- 1.5" MPBRs for varmint loads.
Build a ballistic trajectory table for the load, to get the energy of the bullet at 205 yards. This is 777 ft.lbs. (I recommend the ShootersCalculator MPBR and ballistic trajectory programs.)
Calculate the KPS of the load at MPBR. The KPS of this load at 205 yards is 13.1. (I have developed a spreadsheet program to calculate KPS and related values, see below.)
Interpret the result: Based on the KPS baseline values discussed above, this load is powerful enough to be dependably effective on average size deer and smaller Class 2 game out to its MPBR. It is not, however, powerful enough to be dependable on larger Class 2 game at that distance, because the KPS value is less than 15.
This leads directly into the topic of effective range of a hunting load. My definition of effective range is that it is the distance at which the killing power of a load falls to the relevant baseline KPS value, or the +/- 3" MPBR of the load, whichever is less. In this example, the .30-30 load has an effective range determined by its MPBR of 205 yards, in the sense that KPS is greater than 12.5 at that distance.
(The conditional +/- 3" MPBR distance restriction is based on my firm belief that one should never shoot at a game animal at a range beyond the MPBR of the cartridge and load being used, and closer is always better. I am strongly against the extreme range hunting fad. See Extreme Range Shooting.)
What would be the effective range of the load relative to a baseline KPS of 15? To determine this, we must start by calculating the energy level at which the load would just achieve a KPS of 15. (Rearrange the KPS formula to make E the unknown and set KPS equal to 15.) This works out to be 891 ft. lbs. of energy. With this result, we can scan down the energy data column of the ballistic trajectory table for the load, finding the energy level of 891 (or the next value greater) and reading the relevant yardage from the table. This is 170 yards for the example load, which tells us its effective range for larger Class 2 game is 170 yards.
Incidentally, this load does not have practical Class 3 capability. It requires an energy of 1783 ft.lbs. for a 150 grain .308" bullet to generate a KPS of 30. The load generates 1824 ft.lbs. of energy at the muzzle, which falls to under 1783 ft. lbs. within 10 yards. Further, the SD of the bullet (.226) is too low to assure adequate penetration in Class 3 animals.
One more example and I will wrap this up. A Federal Premium .270 Winchester load with a 140 grain Nosler AccuBond bullet has a MV of 2950 fps (BC .496, SD .261). The MPBR is 290 yards, with an impact energy of 1824 ft.lbs. at that distance. KPS of the load at 290 yards is 28.7.
Clearly, this load has plenty of power for Class 2 game out to its MPBR, but it also has Class 3 capability, within limits. Using the KPS system, we can figure out what are those limits.
The load would generate enough energy (1910 ft. lbs.) to score a KPS of 30 at 255 yards, while it would achieve a KPS of 32, with 2037 ft.lbs. of energy, at 210 yards. Therefore, the load is adequate for use against smaller Class 3 game out to about 250 yards and against most Class 3 animals out to 200 yards, or a bit further. Only the KPS system can give us this information; other killing power calculation methods have no such capability.
A special offer for readers
Using the KPS concept to its full potential involves a lot of number crunching. I have developed a simple spreadsheet program that helps with this. The program is primarily designed to generate a KPS for any bullet diameter, weight, and downrange energy combination one might wish to analyze. All that the user need do is input bullet weight, diameter and energy at a chosen yardage and the program will return the resulting KPS value.
But wait, there is more! I designed the program to also display the sectional density and cross section area of the bullet, and the minimum energy that it would need to generate to achieve the various baseline KPS values discussed above.
Using the KPS program in conjunction with online ballistic trajectory and MPBR programs makes it easy to determine the range and power capabilities of a given load, or to compare different cartridge and load combinations. Much better than doing a lot of number punching on a calculator.
As a courtesy to Guns and Shooting Online readers, I will make the KPS calculation program available to anyone who is interested in exploring the KPS technique in depth. Contact me at email@example.com to request the program. I have program masters in both Excel (for PCs) and Numbers (for MACs) format. Be sure to specify whether you need the Excel or Numbers version of the program.
Chuck Hawks has written:
"Killing power is the most difficult factor to estimate, as there is no definitive scientific formula to apply. Various systems have been created to estimate the killing power of rifle cartridges, with varying results in terms of accuracy. Unfortunately, many such systems have no correlation with reality at all. One of the best, in terms of positive correlation with reality, has proven to be the G&S Online Rifle Cartridge Killing Power Formula. Not only is it generally consistent with results in the field, it can be used to compare any load at any range."
That pretty much sums up the case for the G&S Online Killing Power Formula and the KPS analytical procedure. I would add that the KPS system has analytical capabilities that other killing power formulas with which I am familiar cannot touch.
Addendum: Other killing power formulas of note
The four formulas below are the most notable of several "killing power" formulas. The first three predate the G&S Online Killing Power Formula, while the Hornady HITS formula is roughly contemporaneous with our killing power formula.
(In these formulas, Wt. is bullet weight in grains, Vel. is velocity in feet per second and Dia. is bullet diameter in inches; "sqrt. Dia." means the square root of diameter. To the best of my understanding, the Taylor, Thorniley and Matunas programs use muzzle velocity, while the Hornady program explicitly uses velocity at 100 yards.)
Taylor KO Factor = (Wt. x Vel. x Dia.) / 7000
Thorniley Stopping Power = 2.866 x Vel. x (Wt. / 7000 ) x sqrt. Dia.
Matunas Optimal Game Weight = Vel.^3 x Wt.^2 x 1.5 x 10^-12
Hornady H.I.T.S. = (Wt.^2 x Vel.) / Dia.^2 / 700,000
This is not the time or place for a detailed discussion of the few merits and many limitations of these formulas. I note them here because I made explicit reference to the HITS formula in this article and indirect reference to the others; in particular, I had all four formulas in mind when I referred to the deficiencies of using bullet velocity rather than energy as a key independent variable.
Anyone interested in learning more about these formulas can find references via an internet search. As a starting point for researchers, useful articles on the first three formulas are on the Firearms History Blogspot.
Chuck Hawks wrote an informative article on the Hornady HITS system (Hornady H.I.T.S. Guideline for Rifle Cartridge Killing Power.)
Copyright 2019 by Gary Zinn and/or chuckhawks.com. All rights reserved.