## Friday, December 28, 2012

### 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)
1100   Lead Flat Point (Large Flat)
1000   Jacketed Softpoint (unexpanded)
1350   Jacketed Softpoint (expanded)
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/secapproximately.

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.

1. I cannot duplicate your results with the formula.

To use the .45 ACP example:

185 grs = .02643lbs
.452/2 = .226
1,000 ft/s
1.35

.02643 * 1000 * pi * (.226^2) * 1.35 = 5.73

Even if we use the diameter (which does not represent cross-sectional area):

.02643 * 1000 * pi * (.452^2) * 1.35 = 22.9

Also, momentum in American units is foot-lbs/sec, not foot-lbs (which measures energy).

1. Hi Nathaniel,
Thank you so much for your feedback. I did a little more reading and it looks like Gen. Hatcher did indeed quote the formula as M*A*F, but he made a few errors in his calculations when publishing a table of RSP values of common handgun bullets. For instance, he stated in his paper that he calculates momentum as (kinetic energy/velocity) which actually gives half the actual momentum value. He also seems to divide by the gravitational acceleration when he doesn't need to do that. Therefore, I've updated the formula to correct his mistakes, so that the calculated values now agree with the table that he published.

It also turned out that the source I'd copied the form factors for various bullet types, had divided the original form factor values by 1000 (0.7 instead of 700, 1.35 instead of 1350 etc.), so I updated that table as well. I fixed the momentum unit to foot-lbs/sec instead of foot-lbs in the one spot where I'd typo'd it (Good catch!)

I also updated the article to show how these numbers were obtained and gave you props for pointing out the mistakes. Thank you so much for all the useful feedback.

2. In the english(British Engineering) system of units lbs are units of force and are fundamental. The unit of mass is the slug and is a derived unit. If Hatcher was trying to obtain correct units of mass in the BE system for use in a momentum or kinetic energy expression he was correct in dividing the weight of the bullet by g. To obtain a mass in slugs one would divide the bullet weight in grains by 7,000 and again divide that result by g = 32.2.