The G&S Online Rifle Cartridge Killing Power Formula: Implications and Applications
By Gary Zinn
Guns & 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 that there are several implications and potential applications of the G&S Online formula that have not been adequately investigated. This is an attempt to explore some of these implications and demonstrate some basic applications.
This 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. I will call the output variable of the formula "KPS" (Killing Power Score). For a given load, the formula is:
G&S Online KPS at y yards = (Impact Energy at y yards) x (sectional density) x (cross-sectional area), or simply KPS = E x SD x A
The 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, a capability that is important in ways that I will discuss.
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.
In addition, 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 automatically 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.
Given that background, here are the most obvious and important implications of the KPS formula. (I will use real cartridge and load data to explore these implications.)
The faster a given bullet is driven from a firearm, the more energy and killing power it will have down range. (Duh!) To isolate the effect of bullet velocity/energy on killing power, I will examine two loads that use the same bullet, but with different muzzle velocities.
The bullet is a 140 grain Hornady XTP HP, loaded in the .38 Special and .357 Magnum cartridges. The .38 Special load has a muzzle velocity of 900 f.p.s., while the MV of the .357 Mag load is 1200 f.p.s. (These are real loads, as listed in the Hornady Handbook of Cartridge Reloading, 9th ed., 2012.)
The Hornady 140 grain bullet has a BC of .169 and a SD of .157; the cross-sectional area (A) of a .357" diameter bullet is .1001. (BC and SD values are as listed in the Hornady Handbook. Area = bullet radius squared x 3.1416; a comprehensive list of bullet cross-sectional areas by caliber can be found on the Expanded Bullet Cross-Sectional Area List)
My next data requirement is a ballistics table that shows the energy of these loads at selected ranges. I decided to get the energy values at the muzzle (0 yards), 17, 25 and 50 yards. (These are the typical target range distances at which these handgun cartridges might be fired; 17 yards is the closest yardage to 50 feet.)
I used the trajectory calculator at www.shooterscalculator.com to calculate the energy values. I input the BC, bullet weight and MV of each load and specified a total range of 50 yards, with one yard increments. From the trajectory table, I recorded muzzle, 17, 25, and 50 yard energy values. This completed the data needed to calculate KPS values for the loads. (Reminder: KPS = E x SD x A).
.38 Spl. - Hdy 140 gr. XTP, MV 900 f.p.s., BC = .169; (SD x A) = (.157 x .1001) = .0157
357 Mag. - Hdy 140 gr. XTP, MV 1200 f.p.s., BC = .169; (SD x A) = (.157 x .1001) = .0157
The differences in energy and KPS values generated by the .38 Special and .357 Magnum loads are entirely due to the higher velocity at which the .357 load is fired. While the MV of the .357 load is one-third greater than that of the .38 Special load, the energy and KPS value differences between the loads are larger. At the muzzle, the .357 load generates 78 percent more energy and KPS and at 50 yards the .357 load maintains an energy and KPS advantage of 62 percent.
In this case, the difference in energy and KPS values between the loads is roughly twice the difference in muzzle velocity. This indicates that changes in muzzle velocity have a multiplying effect on the killing power of a bullet; i.e., speed kills.
Bullet size (weight and diameter) also has a large effect on downrange energy levels and KPS values. The data sets below illustrate this.
The comparison here is between a 140 grain, .357 bullet (.38 Special) and a 200 grain, .430 bullet (.44 Special). Both loads are at 900 f.p.s. MV and the BC and SD values of these bullets are virtually identical. Thus, in the KPS formula the difference in size and weight of the two bullets is largely represented by the higher cross-sectional area (A) value of the .44 Special load.
.38 Spl - Hdy 140 gr. XTP, MV 900 f.p.s., BC = .169; (SD x A) = (.157 x .1001) = .0157
.44 Spl - Hdy 200 gr. XTP, MV 900 f.p.s., BC = .170; (SD x A) = (.155 x .1452) = .0225
The 200 grain, .430 bullet is 43 percent heavier than the 140 grain, .357 bullet and the heavier bullet generates 43 percent more energy, from the muzzle out to 50 yards. When this greater energy level is combined with the (SD x A) factor for the larger bullet, the result is that the .44 Special load gets KPS values that are an astounding 105 percent greater than those of the 38 Special load. Bullet size matters a lot when killing power is at issue.
The .38 Special / .357 Magnum / .44 Special comparison leads easily to a practical application of the KPS tool. Suppose I want to find a .357 load that can match the 50 yard killing power of the .44 Special load above. This can be worked out as follows.
First, rearrange the KPS formula to E = KPS / (SD x A). Then, solve for E when KPS = 7.40 (the 50 yard KPS for the .44 Special load) and (SD x A) = .0157 (the value for the 140 grain .357 Mag. load).
The solution is E = 7.4 / .0157 = 471 ft. lbs., which is the energy that a .357 load must generate at 50 yards to get a KPS of 7.4.
Next, use a trajectory calculator to find (by trial and error) the MV needed to get 471 ft. lbs. of energy at 50 yards, using the 140 grain, .357 bullet. This MV is 1380 f.p.s. and loads with this MV can be comfortably attained in the .357 Magnum cartridge. For instance, one reloading manual I checked shows that 7.7 grains of either Win 231 or HP 38 powder will generate 1378 f.p.s. of MV with the 140 grain XTP bullet in a .357 Magnum handgun.
The point of this illustration is that the KPS system can be used to compare or match the downrange performance of different cartridge/bullet loads.
Ballistic coefficient (BC) and sectional density (SD) also matter
For bullets of a given diameter and shape, ballistic coefficient and sectional density are variables that change together as bullet weight changes. Specifically, as the weight of bullets with a given diameter and shape is increased, the BC and SD of the bullets will also increase.
A heavier, longer bullet of a given caliber and shape will have a higher BC than that of a lighter, shorter bullet. (I leave it there, to avoid getting into a hopelessly complicated discussion of ogives, boat tails, meplats and how they affect BC.)
Sectional density is much simpler. SD = Bullet weight / Bullet diameter squared x 7000. (Bullet weight is in grains, diameter is in inches and 7000 is the number of grains in a pound.) Thus, the heavier a bullet of a given diameter, the greater its SD.
To isolate the effects of bullet weight on BC and SD, and ultimately on KPS, I use bullets of the same diameter and shape, but in different weights, fired at the same muzzle velocity. The bullets are 150 and 180 grain Hornady SSTs, fired at a MV of 2500 f.p.s. from a .308 Winchester rifle.
.308 Win. - Hdy 150 gr. SST, MV 2500 f.p.s., ME 2082 ft. lbs. BC = .415; (SD x A) = (.226 x .0745) = .0168
.308 Win. - Hdy 180 gr. SST, MV 2500 f.p.s., ME 2498 ft. lbs. BC = .480; (SD x A) = (.271 x .0745) = .0202
The 180 grain load exceeds the 150 grain load by 20 percent in bullet weight, sectional density, and muzzle energy, and by 16 percent in ballistic coefficient. The muzzle energy advantages of the heavier bullets can be explained simply: heavier bullets generate more energy at a given velocity.
At 100 yards, the energy of the 180 grain bullet exceeds that of the 150 grain bullet by 23 percent, and at 300 yards the heavier bullet has a 29 percent energy advantage over the lighter one. The higher BC of the heavier bullet serves to increase its energy advantage over the lighter bullet as the range extends.
These performance advantages of the 180 grain bullet culminate in the KPS values, with that bullet having a 48 percent greater KPS at 100 yards than the KPS of the 150 grain bullet. At 300 yards, the KPS advantage of the 180 grain bullet grows to 55 percent. The heavier bullet is more powerful out of the muzzle, and, due to its higher BC, increases its relative power advantage downrange.
Incidentally, it is easy to demonstrate the pure effect of BC on bullet power downrange. Here is the data workup for the .308 Win. load with 150 grain SST bullet, along with a workup for a load with a 150 grain RN bullet.
.308 Win. - Hdy 150 gr. SST, MV 2500 f.p.s., ME 2082 ft. lbs. BC = .415; (SD x A) = (.226 x .0745) = .0168
.308 Win. - Hdy 150 gr. RN, MV 2500 f.p.s., ME 2082 ft. lbs. BC = .186; (SD x A) = (.226 x .0745) = .0168
Study the differences in downrange energy and KPS values of the loads. The only difference in parameters between the loads is the BC values of the two bullets. Thus, the inferior downrange performance of the second load is totally due to the round nose bullet of that load having a much lower BC than the spitzer bullet of the first load.
Putting it all together with similar factory loads
The previous exercises were designed to examine the isolated effects of relevant variables on the downrange killing power of bullets. Examining a set of similar factory loads seems a useful culmination of these exercises, because such loads will differ somewhat in all of the relevant variables. (Except for cross-sectional area in loads with the same diameter bullet.)
I chose three Federal Fusion .308 Winchester loads to illustrate the effects of changing, interrelated variables on bullet performance. Fusion ballistics are for 24" barrels.
.308 Win. - Fed 150 gr. Fusion SP, MV 2820 f.p.s., ME 2649 ft. lbs. BC = .414; (SD x A) = (.226 x .0745) = .0168
.308 Win. - Fed 165 gr. Fusion SP, MV 2700 f.p.s. (24\'94 bbl.), ME 2671 ft. lbs. BC = .446; (SD x A) = (.248 x .0745) = .0185
.308 Win. - Fed 180 gr. Fusion SP, MV 2600 f.p.s. (24\'94 bbl.), ME 2702 ft. lbs. BC = .503; (SD x A) = (.271 x .0745) = .0202
The most interesting differences between these loads are the KPS values at 200 yards and beyond. At 200 yards the KPS of the 180 and 165 grain loads are, respectively, 29% and 14% greater than that KPS of the 150 grain load. At the MPBR distances, the 180 grain load has a KPS 34% greater than the 150 grain load. The 165 grain load has a KPS advantage of 16% versus the 150 grain load.
The results I have emphasized would seem to imply that the heavy load is better than the lighter ones. This is true if one wants to get maximum downrange power from a .308 Winchester, but that is not always the primary goal.
The 150 grain load gets a 15 yard longer MPBR than the 180 grain load, which means that it has a flatter trajectory. It also kicks noticeably less. 150 grain .308 hunting bullets, such as the 150 grain Fusion, are normally designed for optimum terminal performance (expansion and penetration) on Class 2 game (deer, antelope, sheep, goats, etc.).
Coupling this with KPS values that are strong enough for any Class 2 game animal, the 150 grain .308 Win. load is a very good choice for most hunting situations. Perhaps this is why the .308 Winchester is such a popular caliber and why the 150 grain load is likely the one most used by hunters.
THE MOST IMPORTANT IMPLICATION: The larger diameter and/or heavier the bullet, the less downrange energy it needs to achieve a given KPS value
Realizing this implication of the killing power formula was a Eureka! moment. Understanding this implication quickly led me to conclude that I could use it to quantitatively compare the downrange performance of different sizes and weights of bullets, driven at different velocities.
The key to such comparisons is specifying a KPS value of interest. Suppose I wish to determine the energy level at which three different sizes and weights of hunting bullets would have a KPS value of 15.
I start by rearranging the KPS formula to read: E = KPS / (SD x F). Dividing the KPS value, 15, by the (SD x A) product of each of the three bullets specified below tells me the energy level at which each would generate a KPS of 15.
I can now determine and compare the ranges at which cartridges in these calibers and bullet weights get KPS values of 15. I will use Federal factory loads in .243 Winchester, .270 Winchester and .30-30 Winchester to illustrate. (The .243 and .270 loads use JSP spitzer bullets and the .30-30 load uses a RN bullet. MV values are for 24 inch barrels.
I used an online ballistics program to build a trajectory table for each load. I read down the energy column of the table to find the energy value that is closest to the target energy value I calculated above; then I read the yardage at which this energy value occurs.
Here are the results for the three loads: .243 Win., 100 gr. bullet = 195 yards; .270 Win., 130 gr. bullet = 510 yards (!); .30-30 Win., 150 gr. bullet = 215 yards.
The KPS value I used here is not arbitrary. I believe a KPS of 15 is the minimum power level that a hunting bullet should have, in order to promptly and humanely dispatch a deer or similar sized quarry at a given range (assuming a solid vital area hit).
Put another way, I am using a KPS value of 15 as a baseline power level that a deer load should have and the range at which a given load can generate a KPS of 15 is the "effective killing range" (EKR) of the load. In that sense, I have determined that the effective killing ranges of the example loads are 195 yards for the .243 load and 215 yards for the .30-30 load.
I knew from experience that the traditional .30-30 loads have definite downrange power limitations, due to low BC bullets driven at moderate velocities. In addition, I have long suspected that the .243 Winchester has effective range limitations for hunting deer, because of its relatively small diameter, light weight bullets. Now, I have a way of quantifying the range limits of cartridges and loads.
The .270 load is a very different case. The calculated EKR of the example load, 510 yards, is obviously a ridiculously long shot at a deer and I am not condoning taking such a shot.
For the .270 Winchester and other powerful, flat trajectory cartridges, a +/- 3" MPBR range is the absolute range limit I would endorse (and that only under the most favorable conditions for setting up and executing the shot). The MPBR for this .270 Win. load is 295 yards.
My concepts of a baseline killing power score and effective killing range for given cartridge/load combinations are more fully explained in the companion article, Determining The Effective Killing Range of Rifle Cartridges: A Proposal.
The G&S Online Killing Power Formula is a crescent wrench for the armchair ballistics analyst. Using it can help us understand how and why given loads behave as they do downrange.
It gives us a tool for comparing the performance of different cartridge/load combinations. It even enables us to calculate the range at which a given cartridge/load is dependably effective on a particular size/type of game animal. These are very useful capabilities!
Copyright 2017 by Gary Zinn and/or chuckhawks.com. All rights reserved.