Measuring Effectiveness of Cartridges: Hornady Index of Terminal Standards

First, let me wish the faithful readers of this blog a happy 2013.

Continuing our series on measuring effectiveness of cartridges, the next method we will study is another empirical method, this one originates from a cartridge manufacturer. We're talking about the Hornady Manufacturing Company (a well known manufacturer of ammunition and handloading components here in the USA) and the formula in question is the Hornady Index of Terminal Standards, otherwise known as H.I.T.S.

The Hornady Index of Terminal Standards is intended to be a guideline for hunters to compare cartridge and bullet combinations for any hunting situation around the world. The Hornady company has made an online version of the calculator available on their website, for the benefit of hunters. By looking at the Javascript code behind their webpage, it is not too hard to figure out what formula they are using. The formula is essentially:

HITS = (W² / 7000) * (V/D²) * 1/100

where
HITS = Hornady Index of Terminal Standards
W = Weight of the bullet in grains.
V = Impact velocity of the bullet in feet/sec.
D = Diameter of the bullet in inches.

The actual formula on their webpage does a bit of rounding here and there, but that doesn't change the result of the formula too much. Hornady claims that the HITS formula factors in bullet weight, sectional density, ballistic coefficient and impact velocity. Hornady then classifies the HITS value calculated by the formula above into one of four different classification types. The classifications are as follows:

H.I.T.S	Suitable For
below 500	Small game animals that weigh less than 50 pounds (22.67 kg.)
501-900	For medium-sized game animals weighing between 50 to 300 pounds (22.67 - 136.1 kg.) Suitable for Antelope, Deer, Black Bear, Caribou etc.
901-1500	Suitable for large and heavy animals which are generally not considered dangerous, weighing between 300 - 2000 pounds (136.1 - 907.2 kg.) This list includes elk, moose, red stag, American bison, zebra, kudu, giraffe and other such African plains game animals.
over 1500	Suitable for animals that are considered dangerous game i.e. animals that have no problem stalking the hunter. This would include lions, tigers, leopards etc.

Let's take the same rifle that we considered when we calculated the Thorniley Stopping Power a few articles earlier. We assumed a .30-06 rifle (such as the M1903 Springfield rifle or the M1 Garand rifle) firing a bullet weighing about 180 grains and .308 inch diameter moving at around 2900 feet/sec. Plugging the numbers into the formula above, we have:
HITS = (180² / 7000) * (2900 / 0.308²) * (1/100) = 1414.957 approximately

Looking at the classification for this HITS value, we see that this value falls in the third category, i.e. it can be used to shoot large and heavy animals that aren't considered dangerous e.g. elk, moose, red stag, zebra, kudu etc. This is similar to the conclusion we reached when we calculated the Thorniley Stopping Power value for this same cartridge/bullet combination.

Note that the HITS values are empirical and Hornady only classifies the HITS value into four different general categories. A HITS value of 800 doesn't mean that it is twice as powerful as a HITS value of 400, for instance. Hornady says that their HITS rating for their own ammunition is based on measuring impact velocities at 100 yards distance for rifle/muzzleloader/shotgun bullets and 50 yards distance for pistols/revolvers.

Measuring Effectiveness of Cartridges: Hatcher Formula

In our last two posts, we studied a couple of empirical formulae that were proposed by big-game hunters, the Taylor KO Factor by John "Pondoro" Taylor and the Thorniley Stopping Power Formula by Peter Thorniley. Today, we will study another empirical formula, this one invented by a military man, the Hatcher Formula.

The Hatcher formula was proposed by Major General Julian Hatcher of the US Army. He was originally a Navy man, before transferring to the Army. He worked his way up in the Army Ordnance department over several years. During the World War II time period, he served as Commanding General of the Ordnance Training Center at Aberdeen Proving Ground, Chief of the Military Training Division, Office of the Chief of Ordnance and later, Chief of Field Service, Ordnance Department. Due to the nature of his job, he became a well known firearms expert and after he retired from the military, he served as technical editor for American Rifleman magazine and wrote several books on firearms as well. One of his contributions to the literature was the Hatcher Formula, designed to measure the effectiveness of pistol cartridges. He also came up with a corresponding Hatcher Scale to put some meaning behind these empirical values.

Public domain image of Major General Julian Hatcher

The Hatcher formula was originally developed in the 1930s when Major General Hatcher was working in the US Army's Ordnance department. It uses the bullet mass, velocity, frontal area of the bullet and also a 'form factor' which depends on the type of bullet. Unlike the Taylor KO factor and Thorniley Stopping Power formula which only consider the diameter of the bullet in their calculations, the Hatcher formula uses the bullet cross-sectional area in its calculation. It also uses the bullet momentum formula (we studied this three posts back) as part of its equation. Additionally, unlike all the other formulae we have studied until now, this one includes the bullet type (jacketed, non-jacketed, flat point, round nose etc.) as part of its calculation. The Hatcher Formula is:

RSP = M/(2*g) * A * F

where:
RSP = Relative Stopping Power
M = Momentum of the bullet in foot-pounds/sec (momentum = mass * velocity where mass is in lbs and velocity is in feet/sec)
g = Acceleration due to gravity in feet/sec².
A = Frontal area of the bullet in square-inches
F = A bullet form factor that depends on the type of the bullet (see notes below)

In General Hatcher's original paper, he quotes the formula as RSP=M*A*F and prints a table of the calculated RSP values for a variety of common handgun bullet types. However, he calculates the momentum incorrectly as (kinetic energy/velocity), which ends up calculating a value of 1/2 of the actual momentum (since kinetic energy = 1/2 * mass * velocity²). He also incorrectly divides by g (acceleration due to gravity) when converting grains to lbs (no need to, because grains are a units of mass, not weight). Therefore I've updated the original formula to match the numbers on his original table and translated the equation to M/(2*g)*A*F.

The values for bullet form factor for some bullet types are defined as:
    F                      Bullet Type
  700     Fully Jacketed Pointed
  900     Fully Jacketed Round Nose
  1050   Fully Jacketed Flat Point
  1100   Fully Jacketed Flat Point (Large flat)
  1000   Lead Round Nose
  1050   Lead Flat Point
  1100   Lead Flat Point (Large Flat)
  1000   Jacketed Softpoint (unexpanded)
  1350   Jacketed Softpoint (expanded)
  1250   Lead Semi-wadcutter
  1100   Hollow Point (unexpanded)
  1350   Hollow Point (expanded)

In an earlier version of this article, your editor had accidentally quoted the numbers as 0.7, 0.9, 1.05 etc. instead of 700, 900, 1050 etc. Apologies for that and thanks to reader Nathaniel Fitch for pointing it out in the comments below (boy, do I have egg on my face now :-))

Because the type of bullet is part of the calculation, cartridges of a particular caliber meant for a single firearm can have different RSP values because they have different bullet types. For example, for a .45 ACP bullet which has a mass of 185 grains and moving at 1000 feet/sec, we compute a RSP value of 65.661 if the bullet is a Lead Round Nose bullet, but 88.642 for a Hollow Point (expanded) bullet. How do we get these numbers, you ask?
Weight of bullet = 185 grains.
We know that 1 lb = 7000 grains.
Therefore, mass of bullet in lbs = (185/7000) = 0.0264285 lbs approximately
Velocity of the bullet = 1000 feet/sec
Therefore, Momemtum of the bullet (M) = 0.0264285 * 1000 = 26.4285 foot-lbs/sec

Diameter of the bullet = 0.451 inches. Therefore, radius of the bullet = 0.451/2 = 0.2255 inches
Frontal area of bullet (A) = pi * r² = 3.1415927 * 0.2255² = 0.160 inches² approximately

Now, let's assume acceleration due to gravity (g) = 32.2 feet/sec² approximately.

For a lead round nose bullet, the form factor bullet F = 1000 from the table above.
Therefore RSP for this bullet is calculated as:
RSP = M / (2*g) * A * F = 26.4285 / (2 * 32.2) * 0.160 * 1000 = 65.661

For a hollow point (expanded) bullet, the bullet form factor F = 1350 from the table above.
Therefore RSP for this bullet is calculated as:
RSP = M / (2*g) * A * F = 26.4285 / (2 * 32.2) * 0.160 * 1350 = 88.642

Special thanks go out to reader Nathaniel Fitch for pointing out the errors in an earlier version of the article. His comments are posted below. Give him a big round of applause folks!

For self-defense purposes, the Hatcher scale recommends that the RSP be between 50-55 for effective stopping power. Values of RSP beyond 55 lead to diminishing returns, as the increase in stopping power is offset by the extra recoil strength that must be managed by the user. Per the Hatcher scale, values below 30 give a user a 30% chance of stopping the target in one shot. For values between 30 and 49, the chance of a one-shot stop rises to 50%. For values above 50, the chance of a one-shot stop rise to 90% per the Hatcher scale. Most .45 ACP cartridge types have a RSP value over 50, while 9 mm. Luger cartridges are mostly between 30 and 40. This means Hatcher's formula tends to favor .44 Magnum and .45 ACP over 9 mm. Luger for stopping power.

While the Hatcher formula does not consider factors such as bullet penetration, it is considered a fairly decent formula to determine the effectiveness of pistol ammunition.

Measuring Effectiveness of Cartridges: Thorniley Stopping Power

In our last post, we looked at a formula called the Taylor Knock Out Factor, which was developed by a big-game hunter with extensive experience with African wildlife. In this post, we will look at another empirical formula which was developed by another hunter, this one had extensive experience with wildlife in both Africa and North America. His name is Peter Thorniley and he developed the Thorniley Stopping Power Formula.

The Thorniley Stopping Power Formula is similar to the Taylor KO Factor we studied in the previous page. It is calculated as:

TSP = 2.866 * v * (m/7000) * sqrt(d)

where:
TSP = Thorniley Stopping Power
v = velocity of the bullet in feet per second
m = mass of the bullet in grains
sqrt = square-root function
d = diameter of the bullet in inches.

Since this formula uses the square-root of the bullet's diameter (unlike the Taylor KO factor formula, which uses the bullet's diameter without taking the square root), the values are on a different scale than the Taylor KO factor numbers. Like the Taylor KO factor, the values obtained by the TSP formula are empirical.

The Thorniley scale is as follows:

Thorniley Stopping Power	Suitable For
45	Antelope
50	White-tail Deer, Mule Deer etc.
100	Black Bear
120	Elk, Moose, Kudu, Zebra etc.
150	Lion, Leopard, Grizzly Bear, Brown Bear
250	Hippopotamus, Rhinoceros, Cape Buffalo, Elephant

The values in the table above are based upon Peter Thorniley's long experience as a hunter.

Let's say that we have a .30-06 rifle (such as the M1903 Springfield rifle or the M1 Garand rifle). Let us assume that this rifle fires a bullet weighing about 180 grains and .308 inch diameter moving at around 2900 feet/sec. Plugging the numbers into the formula above, we have:
TSP = 2.866 * 2900 * (180/7000) * sqrt(0.308) = 118.61 approximately.

Looking up the TSP value on the table above, we see that a .30-06 rifle can be used to hunt antelopes, deer, black bears, elk, moose, kudus, zebras etc. (since 118.61 is pretty close to 120), but probably not such a good idea against lions, grizzly bears, hippopotamuses, rhinoceroses, elephants etc.