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.

Measuring Effectiveness of Cartridges: Taylor KO Factor

In the previous two posts, we saw how some people obtain a figure of merit for a cartridge by measuring the kinetic energy and the momentum. While these two methods have a basis in physics, the next method which we will study in this post is more of an empirical formula. This is called the Taylor KO Factor (where the KO stands for Knock Out). This term is also sometimes called the Taylor Knock Out Formula or simply abbreviated as TKOF.

The inventor of this formula was a famous 20th century big-game hunter and ivory poacher named John Howard "Pondoro" Taylor. Born in Dublin, Ireland, he developed a passion for hunting and decided to become a professional hunter in Africa. As a result of this, he became an expert in hunting with various rifles and cartridge combinations. In a career spanning over thirty years, he is credited with hunting over 1,000 elephants (though many of these were illegally hunted) as well as thousands of other African big game like hippo, rhinos, lions, cape buffalo etc. He received the nickname "Pondoro" (meaning "lion" in some African languages) from some of the locals, because of his lion hunting skills. Allegedly he was so busy hunting in remote African jungles that he didn't realize that World War II had broken out (he signed up for the King's African Rifles regiment after he finally got the news!)

John "Pondoro" Taylor (1904-1969)

John Taylor wrote quite a few books on the subjects of big game hunting and African hunting. In one of his books, African Rifles and Cartridges, published in 1948, he makes mention of a formula he came up with to test for cartridge effectiveness when hunting big game.

The story behind his formula is that during his long hunting career, Taylor had observed that some cartridges were more suitable for stopping elephants than others. While he admitted that many cartridge types would work at killing an elephant when aimed accurately at an elephant's brain, he was more concerned with situations where he missed the brain and the elephant would become enraged and charge at him. He wanted to evaluate cartridges that could stun an elephant, even if the bullet didn't hit a lethal spot, reasoning that a "knock-out" blow on the elephant would give the hunter enough time to reload and follow up with a more accurately aimed shot. It was really meant to calculate the effectiveness of solid big-bore bullets. John Taylor himself used this formula to make the point that big-bore bullets were more effective at stopping larger game than the lighter and faster bullets available at that time.

His formula is an empirical one and is defined as:

where:
m_bullet = Mass of the bullet in grains
v_bullet = velocity of the bullet in feet per second
d_bullet = diameter of the bullet in inches.
The dividing by 7000 is because his formula converts grains to pounds (1 pound = 7000 grains).

The TKOF obtained by this equation is a dimensionless number, as there isn't really a science behind it and it is merely a figure of merit for comparing different cartridge types. A higher TKOF value indicates better stopping power for the cartridge. For people who like to work with metric units, the calculation is defined as:

TKOF = m * v * d / 3500

where m is in grams, v is in meters per second and d is in millimeters.

Consider a NATO standard 5.56x45 mm. cartridge. The bullet from this cartridge normally weighs 4 grams (62 grains), has a velocity of 940 meters/sec (3100 feet/sec) and a diameter of 5.70 mm. (.223 inches). Using these values in the above formula, we get TKOF = 6.12 approximately.

The following table lists TKOF values for some common cartridges:

(Figures taken from wikipedia)

TKO Factor	Name	Mass (gr)	Velocity (fps)	Bullet Diameter (in)
19.6	.308 Winchester	168	2650	0.308
147	.50 BMG	660	3050	0.510
4.72	.380 ACP	95	980	0.355
6.20	.38 Special	158	770	0.357
8.56	.357 Sig	125	1350	0.355
24.9	.300 Winchester Magnum	180	3146	0.308
4.64	5.45x39mm	49	3000	0.221
35.5	.338 Lapua Magnum	250	2940	0.338
20.8	7.62×54mmR	181	2580	0.312
70.3	.458 Winchester Magnum	500	2150	0.458
29.8	.480 Ruger	325	1350	0.475
19.9	.44 Magnum	240	1350	0.429
12.3	.45 ACP	230	830	0.452
20.8	.30-06 Springfield	170	2850	0.308
10.4	.40 S&W	165	1080	0.400
11.3	.357 Magnum	158	1400	0.357
14.9	.30-30 Winchester	150	2250	0.308
7.31	9mm Parabellum	115	1250	0.355
6.12	5.56 x 45 NATO	62	3100	0.224
1.33	.25 ACP	50	750	0.251
1.33	.22LR	30	1400	0.222

Per the above table, we can see that a .44 Magnum has better stopping power than a .45 ACP as it has a larger TKOF value, but a .308 Winchester is considered nearly equivalent to a .44 Magnum in stopping power since their TKOF values are close to each other. Similarly, it suggests that a 7.62x54mmR is equivalent to a .30-06 Springfield and .25 ACP is equivalent to .22LR in stopping power, while a .50 BMG outdistances everything else by a very wide margin.

Unlike the kinetic energy and momentum formulae, the Taylor KO Factor takes the bullet diameter into account as part of the calculation. It tends to favor big-bore heavy solid bullets and is really meant for big-game hunting. If we were to calculate the kinetic energy and momentum of 7.62x54mmR and .30-06 Springfield, they would both suggest different stopping powers for the cartridges, but per the Taylor KO Factor calculation, these two are pretty close to each other in stopping power.