A Velocity-based Killing Power Formula
by Gary Zinn with Shane Hebert
Awhile back I received an interesting e-mail from Guns & Shooting Online reader Shane Hebert. I will share the essential text from the e-mail below, but will preview it by saying that Shane studied the killing power formula devised by Chuck Hawks, considered the relevant physics and did the math, and came up with an alternative killing power formula that gets results matching those of the original G&S Online (Hawks) formula.
The key difference between the two formulas is that the Hawks formula uses bullet energy (at the point of impact), sectional density (SD), and cross-sectional area (A) as the independent variables to calculate a killing power score (KPS). The Hebert formula uses bullet weight and velocity (at the point of impact) as the independent variables to get the same result. Thus the Hebert formula is simpler in a sense, for it uses one less independent variable. (Since bullet diameter is used in calculating both SD and A, and bullet weight in calculating SD, one can argue that the Hebert formula ultimately uses three fewer variables than the Hawks formula.)
Here are the two operational formulas:
Hawks formula: KPS = E * SD * A
Hebert formula: KPS = (((W / 7000) * V) ^ 2 ) / 81.851
where: KPS = Killing Power Score
E = bullet energy at impact (in ft.lbs.)
SD = bullet sectional density
A = bullet cross-sectional area (in sq.in.)
W = bullet weight (in grains)
V = bullet velocity at impact (in fps)
Understand that energy (E) and velocity (V) “at impact” can be at any distance of interest — 100 yards, 200 yards, or whatever. Also, the KPS concept is of most interest and usefulness in evaluating the performance of game hunting cartridges and loads, but it can be used to study the performance of varmint and handgun cartridges and loads, too.
Here is how Shane Hebert explained the development of the bullet weight—velocity based version of the KPS formula.
“Just for fun, I was looking at this (the Hawks KPS) equation. The first thing I noticed was that Sectional Density and Cross Sectional Area were similar... for Cross Sectional Area, you use radius = (diameter / 2). Use that to multiply with Sectional Density and you basically eliminate bullet diameter from the equation and it simplifies to an equation with mass (bullet weight / 7000) and a constant that's the ratio of Pi (from the Cross Sectional Area) to square area (the approximation from Sectional Density)... Pi / 4.
“Then multiplying that to the Kinetic Energy equation, you ultimately end up with mass^2 * velocity^2 (mass squared times velocity squared). That's the same as (mass * velocity)^2.
“Mass * velocity is momentum. Using mass in pounds and velocity in feet-per-second and not using some of the constants (450,000 and Pi/4, for example), you can get the same values scaled by (what I found to be) 81.851 (and some change) as the Killing Power Formula.
[Thus the equation] “(((weight-in-grains / 7000) * velocity) ^ 2) / 81.851 get(s) the same value(s) as the (Hawks) Killing Power Formula.”
(The expression (weight-in-grains / 7000 ) is necessary because the calculation requires the bullet weight to be in pounds; 1 pound = 7000 grains.)
Shane went on to note three additional points regarding his formula:
“First, this is an easier calculation to get the same value as the Killing Power Formula.
“Second, this lends additional insight into the calculation... it's highly correlated with momentum.
“The actual constant difference between the two I calculated is 81.8511135901176.”
Here are my thoughts regarding these three notes. First, if one were doing the calculations on a pocket calculator, the Hebert formula is indeed easier, in the sense that it requires only four individual calculations, while the Hawks formula requires eight. (I do not wish to take the space to explain this fully, so please trust me.) Conversely, the nested operations in the Hebert formula must be carefully done in the proper sequence, or the whole thing will blow up.
That the Hebert formula relies on momentum (bullet mass * velocity) as the prominent physical variable in determination of a KPS should recommend it to those who sweat the underlying physics. I am indifferent regarding whether downrange velocity or energy is used as the relevant variable leading to a useful measure of killing power, since energy derives from momentum.
Truncating the constant used in the divisor of the Hebert formula to 81.851 is quite practical.
This truncation sometimes results in a 0.1 or (rarely) 0.2 difference between the KPS values generated by the two formulas. When this happens, the KPS from the Hebert formula will usually be greater than that from the Hawks formula.
An example calculation
To see how the alternative formulas lead to the same result, consider a common .308 Winchester load, say the Hornady American Whitetail 150 grain InterLock SP (MV 2820 fps, BC .338, SD .226, A .0745 sq.in.). To calculate the killing power of this load at (e.g.,) 200 yards, first use a ballistic trajectory program to determine bullet velocity and energy at that distance. This is 2301 fps and 1763 ft.lbs., according to the Shooters Calculator (shooterscalculator.com) ballistic program. Then do the math:
Hawks KPS = E * SD * A = 1763 *.226 * .0745 = 29.6713
Hebert KPS = (((W / 7000) * V) ^2 ) / 81.851 = (((150 / 7000) * 2301) ^ 2) / 81.851 = 29.7017
which both round to 29.7.
Both Shane and I ran a number of test calculations to verify that the two equations get the same answers. We confirmed that they generally get the same results, to one decimal place, but occasionally they will differ by 0.1 or 0.2, as was noted above. Such a small inconsistency is of no consequence.
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. After considerable deliberation and discussion, Chuck Hawks and I settled on the following KPS baselines. 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 North American 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)
Additional information on KPS
Chuck Hawks and I have written quite a bit about the KPS concept, so I will not repeat that background information here. For anyone who is interested, here are links to three useful articles that provide essential information on the subject, and there are links in those articles to additional resources.
Rifle Cartridge Killing Power List https://www.chuckhawks.com/rifle_killing_power_list.htm
The G&S Online Killing Power Formula: Take 2 https://www.chuckhawks.com/killing_power_formula-2.html
Determining The Effective Killing Range of Rifle Cartridges https://www.chuckhawks.com/effective_range_rifle_cartridges.html
A special offer for readers
Using the KPS concept to its full potential involves a lot of number crunching. I have developed simple spreadsheet programs that help with this. A program based on the Hawks KPS formula generates 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.
Similarly, a program based on the Hebert KPS formula uses bullet weight and downrange velocity to generate KPS values.
I designed the programs to also display the sectional density and cross-section area of the bullet. These outputs come from the bullet diameter input in the energy-based (Hawks) approach, and of course are used in computing KPS via the Hawks formula.
One may also enter bullet diameter in the velocity-based (Hebert) spreadsheet, so that the SD and cross-sectional area of the bullet will be shown; these are informational only and are not necessary to calculate KPS values via the Hebert approach.
Finally, the spreadsheets also show the minimum energy (Hawks formula) or velocity (Hebert formula) that the bullet would need to generate to achieve the various baseline KPS values discussed above. One can then determine the maximum yardage at which a KPS of interest is attained by finding the relevant energy or velocity value in a ballistic trajectory table for the load.
Using either KPS program in conjunction with online ballistic trajectory and MPBR (maximum point blank range) 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 programs available to anyone who is interested in exploring the KPS technique in depth. Contact me at email@example.com to request either or both of the programs.
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(s).
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.
By developing a velocity-based version of the KPS formula, Shane Hebert has added to the usefulness of the KPS concept. Thank you for your contribution, Shane.
Copyright 2020 by Gary Zinn and/or chuckhawks.com. All rights reserved.
Copyright 2017 by Gary Zinn and/or chuckhawks.com. All rights reserved.