Sunday, December 30, 2012

Measuring Effectiveness of Cartridges: Optimum Game Weight Formula

We've studied several empirical formulae in the last few pages: The Taylor KO Factor, the Thorniley Stopping Power Formula and the Hatcher Formula. In todays article, we will study another formula called the Optimum Game Weight formula.

This formula was devised by Edward A. Matunas and was first published in the April 1992 issue of Guns magazine. The author claimed that this formula was devised based on studying the effects of kinetic energy, momentum, bullet sectional density, diameter, bullet nose shape and other criteria. However, the author did not elaborate on how he managed to derive his formula.

The Optimum Game Weight formula is defined as:

OGW = V3 * W2 * 1.5 * 10-12

OGW = Optimum Game Weight in lbs.
V = velocity of bullet in feet per second
W = weight of bullet in grains

For hunting bullets, the constant used in the equation is 1.5 * 10-12. For varmint bullets, it is recommended to use 5.0 * 10-13 instead. What the OGW value tells us is the approximate maximum weight in lbs. of an animal that can be reliably killed by a particular cartridge. It is assumed that the hunter has selected an appropriate bullet type for the job.

For an example, let us consider the same rifle and cartridge that we studied with the Thorniley Stopping Power formula, two posts earlier. This is a .30-06 rifle that fires a bullet of .308 inch diameter, weighing 180 grains at approximately 2900 feet/sec. Plugging the numbers into the formula above, we get:
OGW = 29003 * 1802 * 1.5 * 10-12 = 1185.31 lbs.

This means that this formula indicates that the rifle and cartridge combination can be used to kill animals weighing up to 1185 lbs or so. Remember though that this formula is empirical and the values obtained are approximate. Also, bear in mind that we computed the velocity of 2900 feet/sec at the muzzle of the weapon and if we measured the velocity of the bullet at some distance from the rifle, it may have decreased to something like 2500 or 2600 feet/sec. Therefore the OGW value will decrease over distance.

The author included a table listing the OGW values for various common cartridges. While these numbers seem to agree with many hunters experiences, there are also some issues with this formula. Even though the author claims to have considered bullet section density, bullet diameter, bullet nose shape and bullet construction in his study, none of these appear in the formula. Therefore, according to this formula, a 150 grain bullet moving at 2800 feet/sec will behave the same, whether it is a .270 Winchester or a .30-06 bullet, whereas the performance of these two bullets are very different in real life. Also note that this formula does not care if the bullet type is a jacketed bullet, lead bullet, hollow-point, round nose etc.

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

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/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.

Saturday, December 22, 2012

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)
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.

Wednesday, December 19, 2012

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:

mbullet = Mass of the bullet in grains
vbullet = velocity of the bullet in feet per second
dbullet = 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 FactorNameMass (gr)Velocity (fps)Bullet Diameter (in)
19.6.308 Winchester16826500.308
147.50 BMG66030500.510
4.72.380 ACP959800.355
6.20.38 Special1587700.357
8.56.357 Sig12513500.355
24.9.300 Winchester Magnum18031460.308
35.5.338 Lapua Magnum25029400.338
70.3.458 Winchester Magnum50021500.458
29.8.480 Ruger32513500.475
19.9.44 Magnum24013500.429
12.3.45 ACP2308300.452
20.8.30-06 Springfield17028500.308
10.4.40 S&W16510800.400
11.3.357 Magnum15814000.357
14.9.30-30 Winchester15022500.308
7.319mm Parabellum11512500.355
6.125.56 x 45 NATO6231000.224
1.33.25 ACP507500.251

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.

Tuesday, December 18, 2012

Measuring Effectiveness of Cartridges: Momentum

In our last post, we looked at one way to study the performance of cartridges by calculating the average kinetic energy of the bullet. In this post, we will look at another method that some people use to calculate the effectiveness: momentum.

Back in physics class in school, some might have studied that momentum = mass * velocity. So all we need to do is use a scale to measure the mass of the bullet and use a chronograph to measure bullet velocity and we can calculate the momentum of a bullet.

The idea is that when a bullet hits the target, the collision will cause the momentum will be transferred to the target. Unlike the calculation for kinetic energy, which we studied in the previous post, momentum is proportional to the velocity, whereas kinetic energy is proportional to the square of the velocity.

Calculating the momentum of bullets for comparative purposes works when the bullet speeds and bullet weights are both moderate. This method completely ignores factors like bullet diameter, bullet shape, bullet material etc. Therefore some of the results could be a bit stilted. For instance, a 500 grain .458 Winchester magnum fired at 2000 feet/sec has slightly less momentum than a 3 pound bag of cotton moving at 50 feet/sec. However, no sane person will ever argue that the 3 pound bag of cotton will effectively stop a charging elephant or rhino!

On the other hand, if the bullets are made of similar material and of similar shape, then comparing their momentum values is not a bad idea to measure which one will be more effective than the other. Most of the authors that use momentum to compare bullet effectiveness tend to be fans of big-bore heavier bullets.

Monday, December 17, 2012

Measuring Effectiveness of Cartridges: Kinetic Energy

In our last post, we discussed the basics of cartridge effectiveness. In this post, we will discuss one of the methods used to measure cartridge effectiveness, by measuring the kinetic energy of the bullet.

Some of you may recall that in physics class, we were taught that kinetic energy is the energy possessed by a body in motion. Therefore, a bullet in motion has a certain amount of energy in it, given to it by the burning propellant. When the bullet hits a target, it slows down and transfers the kinetic energy onto the target.

The formula for kinetic energy that we learned in physics class is:

Ek = Kinetic energy of the bullet
m = mass of bullet
v = velocity of the bullet.

In SI units, if the mass of the bullet is in kilograms and the velocity of the bullet is in meters/sec, then we have the energy of the bullet in joules. Similarly, in imperial units, if we have the mass in pounds (lb. or slugs), and velocity in feet/sec, then we have the kinetic energy in foot-pound force (ft-lbf).

In some American firearms related articles, the kinetic energy is also defined as:
Ek = m * v2 / 450240
m = weight of bullet in grains
Ek = kinetic energy of the bullet in foot-pounds force

So why is this formula different and where did this 450240 come from. Actually it is simply a reworking of the previous formula.

Recall that 1 grain = 1/7000 pound force (lbf)
Also, acceleration due to gravity is roughly 32.16 feet/sec2. Therefore, 1 pound force (lbf) = 1/32.16 pounds mass (lb.)
Therefore, if we measure our weight in grains, we need to convert it into lb. first, which works out to:
Mass in pounds = weight in grains / (7000 * 32.16) = weight in grains / 225120
Now applying this to the formula Ek = 1/2 * m * v2, we can write this as:
Ek = 1/2 * weight in grains/225120 * v2

which can be further simplified into:
Ek = weight in grains/450240 * v2

The listing below shows the kinetic energies for some common pistol cartridges in both imperial and SI units.
Average kinetic energies for common cartridges
CartridgeKinetic energy
.380 ACP199270
.38 Special310420
9 mm Luger350470
.45 ACP400540
.40 S&W425576
.357 Mag550750
10mm Auto650880
.44 Mag1,0001,400
.50 AE1,5002,000
Calculations of the above were taken from wikipedia.

Measuring the kinetic energy of bullets tends to favor bullets of higher velocity and lower mass, because the kinetic energy increases as a factor of the square of the velocity. Measuring the cartridge effectiveness by calculating the kinetic energy does not take into consideration factors such as diameter of the bullet, shape of the bullet, physical characteristics of the bullet (solid vs. hollow point, round nosed vs. flat nosed etc.). Some manufacturers tend to favor this method as it has a basis in science, as well as the fact that it is easy to measure the velocity of the bullet and its mass.

Measuring Effectiveness of Cartridges: An Overview

There are several formulae to determine the effectiveness of various types of ammunition. Some of these methods are based on some scientific principles, others are just empirical formulae that produce a number (i.e. a figure of merit) which can be used for comparative purposes against other ammunition types. We will study some of these formulae in the following posts.

One of the terms that is often mentioned in discussions of this sort is "stopping power". It is defined as the ability of the cartridge to cause enough ballistic injury to incapacitate a target where it stands. The physical characteristics of the bullet, the type of target and the shot placement has a large effect on stopping power. For example, a bullet capable of stopping a human will not stop a charging cape buffalo, a grazing shot will not have as much effect as a hit to center of mass etc.

Some of the methods used to measure cartridge performance include:

  1. Kinetic energy - This is a scientific method and is often used by cartridge manufacturers to tout the superiority of their products.
  2. Momentum - Another scientific method that is sometimes used by cartridge manufacturers
  3. Taylor KO Factor - This is an empirical formula devised by John (Pondoro) Taylor, a 20th century big game hunter and poacher in Africa.
  4. Thorniley Stopping Power Formula - Another empirical formula, devised by Peter Thorniley, who is another big game hunter in North America and Africa.
  5. Hatcher Formula - Developed by Major General Julian Hatcher of the US Army in the 1930s, to determine the effectiveness of various types of pistol ammunition.
  6. Optimum Game Weight Formula - Developed by Edward A. Matunas and first appeared in the April 1992 issue of Guns magazine.
Some of these methods (such as kinetic energy and momentum) ignore the diameter of the bullet and some of the others formulae take the size of the bullet into consideration. Only one of these methods (Hatcher formula) considers the construction of the bullet (e.g. an expanding bullet may be more effective than a non-expanding one) and the shape of the bullet (e.g. a solid wide nose bullet vs. a round nosed one) to be factors in performance, the others don't consider these factors as important at all. However, they all do help in simplifying some of the details of bullet effectiveness.

We will study these methods in detail in the following days.

Wednesday, December 12, 2012

What's Wrong With This Picture?

A couple of months ago, we discussed a hilarious video about how not to shoot a firearm. Now here's a screen capture from the popular TV series The Walking Dead.

Scene from The Walking Dead. What's wrong with the above picture? Read on to find out.

So what's wrong with the above picture? First, notice the character's off hand. He's holding a pretty sharp knife in it, while attempting to support his rifle at the same time. Not a very good idea.

Next, notice how he's intently peering through the sights. Wait, did I say sights? I should have said "sight". If you look closely at the picture, notice that the rifle is missing the rear sight! This person must be one heck of an expert shot to not need a rear sight!

Tuesday, December 4, 2012

Always Make Sure of Your Ammunition Type

In the world of firearms, ammunition comes in several calibers (e.g. .22, .357, 7.62 mm, 9 mm, .45 etc.). However, when purchasing ammunition, one must be careful to specify the exact type of ammunition. We will see the reason why in this post.

Back when we studied rimfire cartridges, we noted that there are several cartridges in .22 caliber, such as .22 Short, .22 Long and .22 Long Rifle (i.e. .22LR). Besides these three, there are other cartridges too, such as .22 Remington Jet, .22 Reed Express, .22 CB etc. Most of these, except the last one have .223 inch diameter bullets, but the length of the cartridges and the bullets differ. Therefore, if a firearm takes (say) .22 Long cartridges, the user will never be able to fit a .22 LR cartridge into the chamber.

The same thing is true with other calibers as well. For example, we have .380 vs. .38 S&W vs. .38 Special vs. .38 Short Colt vs. .38 Long Colt. In mathematics, we are taught that 0.380 = 0.38, but when it comes to cartridge sizes, they are two completely different things, as the image below shows:

.38 Special (left) vs. .380 (right)

Not only are the lengths and cartridge profiles dramatically different, the two bullets are also slightly different diameters as well: .38 special has a .357 inch diameter bullet, whereas .380 has a .355 inch diameter bullet.

Similarly, when referring to .45 caliber ammunition, it should be specified if the user wants .45 ACP, .45 GAP, .45 Webley etc. As before, all of these have dramatically different cartridge shapes and bullet weights.

.45 GAP (left) vs. .45 ACP (right)

As before, the reader may observe the difference in the sizes and shapes of the cartridges. Incidentally, ACP stands for Automatic Colt Pistol and GAP stands for Glock Automatic Pistol, the names of the manufacturers whose products these cartridges were originally designed for.

Finally, we have many cartridges in 7.62 mm: 7.62x25 mm. Tokarev, 7.62x51 mm. NATO, 7.62x39 mm. Soviet, 7.62x54 mmR etc. The two most famous ones are the 7.62 NATO (i.e. 7.62x51 mm. used by FN-FAL, M14, Heckler & Koch G3 etc.) and the 7.62 Soviet (i.e. 7.62x39 mm. used by the AK-47, AKM and Type 56 rifles).

NATO 7.62x51 mm. (top) vs. Soviet 7.62x39 mm. (bottom)

As the reader may note, it is pretty easy to tell that the two cartridges are drastically different.

The same thing applies to many other calibers as well. So, the buyer must note the exact cartridge type when purchasing new ammunition. As amazing as it may seem, quite a few people are not aware of the differences between the cartridges or that the other cartridges exist. There have been several instances where a buyer has walked into a local firearm store and asked for .38 cartridges and ended up walking out with either .38 S&W or .38 Special, when they really wanted .380 cartridges, or asked for .22 Long when they really wanted .22 Long Rifle etc. It is a source of frustration to both the buyer as well as the owner of the firearms shop.

Therefore, it is very important to note down the exact type of cartridge that a firearm accepts.

Saturday, December 1, 2012

Measuring a Barrel's Twist Uniformity

In our last post, we discovered how to measure the twist rate of a barrel. Measuring the twist rate of the barrel is one thing, but how do we ensure that the rifling inside the barrel is uniform? We will study that process in this post.

Back when we studied different methods of rifling barrels, one of these methods was called Button Rifling, where a tool is pulled or pushed through a barrel to create the rifling grooves. One of the problems we noted at the bottom of that post was that if the button slips inside the barrel while it is being pulled or pushed through it, the grooves may be non-uniform. So even if the twist rate is (say) 1 turn in 10 inches, the rate of twist may not be uniform throughout the 10 inches of length. So how do we verify the uniformity of the twist throughout the barrel.

The solution was an invention called the Twist Deviation Machine invented by Mr. Manley Oakley of Seattle, WA.

Mr. Manley Oakley, inventor of the Twist Deviation Machine

Mr. Oakley was a world-class benchrest shooter and had won the National Bench Rest Shooters Association (NBRSA) trophy several times. He was a well known figure in the Northwest chapter of NBRSA and they now award an annual trophy in his name.

Mr. Oakley's theory was that if a barrel had a uniform twist rate, or the twist rate slightly increased as it approached the muzzle, all other features being normal, the barrel would be an accurate one. On the other hand, if the barrel's twist rate changed non-uniformly (e.g. speeded up and then slowed down, or vice-versa) or if the twist rate decreased towards the muzzle, the barrel would be less accurate. He verified this theory by testing barrels that were known to be accurate against other less accurate barrels of similar physical characteristics. In order to determine the uniformity of a barrel's twist, he invented the Twist Deviation Machine to measure it.

It consists of a hollow steel tube with a thinner steel rod passing inside it. The steel rod is free to rotate inside the steel tube. To the steel rod is attached a plastic washer and about three inches away from this washer is a second plastic washer attached to the steel tube. This apparatus is then pushed through the barrel. As the washers are being pushed through the barrel, they engage the rifling and start to rotate as they are pushed through, which results in the steel rod and the steel tube rotating as well. Now if the rifling is uniform throughout the barrel, the rod and tube will both rotate at the same rate. However, if the rifling is not uniform, then one of them will rotate faster or slower than the other and this can be easily observed by looking at the parts of the rod and tube that are sticking out of the barrel. This allows the user to measure if the twist rate is increasing or decreasing through the whole length of the barrel.

Monday, November 26, 2012

Measuring a Barrel's Twist Rate

When we studied the basics of rifling a while back in this blog, there was mention of a term called "twist rate" for a barrel. The twist rate is defined as the length of the barrel required for a bullet to make one complete turn of the barrel. For instance, a standard M16A2 barrel makes 1 turn for every 177.8 mm. of barrel length. It is very important to match the twist rate based on the weight, diameter and length of the bullet to ensure accuracy. We studied a couple of methods on how to calculate the twist rate for a given bullet using the Greenhill formula and the Miller Twist Rate formula. All this is good in theory, but how do we actually measure the twist rate of the barrel? This post discusses a simple way to do so.

Recall that about 16 months ago, we discussed various tools used for cleaning firearms. Well, people can use some of those very same tools to measure the twist rate of a barrel. All that is needed is a standard cleaning rod with a rotating handle, a jag and a cotton cleaning patch. The user simply attaches the cleaning patch to the jag end of the cleaning rod and then pushes it into the barrel until the cleaning patch engages the rifling of the barrel. The user then takes a piece of sticky tape and attaches it around the back of the cleaning rod near the handle like a tiny flag (or uses a marker and marks a spot on the cleaning rod). The user then measures how much of the cleaning rod is sticking out of the barrel (a ruler measuring the distance from the barrel's base to the start of the flag or mark point should do it). Then the user pushes the cleaning rod into the barrel. Because the cleaning patch has engaged the rifling, the rod rotates as it is pushed into the barrel. When the flag or mark point has made one complete rotation, the user then measures how much of the cleaning rod is sticking out of the barrel again. The difference between the two gives the twist rate. For instance, if the rod initially has 22 inches sticking out the barrel when the first measurement is made and 12 inches sticking out of the barrel when the second measurement is made, this means that the twist rate is 1 in 10 inches.

For those of you who would like to see a video of the process:

Happy viewing!

Sunday, November 25, 2012

Firearm Myths - 4 (More Movie Madness)

Folks, it's time for another edition of firearm myths. We already dealt with this topic previously here, here and here. We will look into some more firearm myths which are prevalent in movies today.

1. If someone gets hit by a bullet, they get lifted off the ground and fly backwards.

In several movies, we have a scene where someone (villain, sidekick, red-shirt guy etc.) gets hit by a bullet shot from a handgun and is lifted clean off their feet and through a plate glass window, all by the force of the bullet hitting the person. Can this actually happen in real life? Well, let's recall some basic physics from middle school here. Newton's third law of motion states that, "For every action, there is an equal and opposite reaction".

Therefore, if there's enough energy in a bullet to send a person flying back 10 feet in the air, there should be an equal amount of recoil energy acting on the firearm, which would send the shooter of the firearm flying backwards 10 feet as well! (assuming that they are both the same weight, of course).

There are firearms that can deliver enough energy to knock a human off his feet (such as a very large caliber bullet or cannon shell), but these firearms are mounted on a weapons platform for a reason and are not carried in someone's pocket.

2. Cocking the pistol/shotgun/revolver to show that someone means business.

This scene is shown in several movies. Hero has a bad guy covered by his pistol and the bad guy is refusing to answer questions. The hero then dramatically pulls back on the hammer to cock the gun (or racks the shotgun's slide) and show the bad guy that he means business and the bad guy immediately starts talking.

In other scenes, the dramatic clicking sound happens when a character walks into a room and announces his/her presence and indicates that he or she is in control of the situation now.

In real life, no one ever walks into a dangerous area without their firearm already cocked and loaded and ready to go at a moment's notice. This is called Condition 0, which we studied a while ago when we were reading about carrying conditions.

Even more amusing is to hear the hammer cocking sound, when the actor is holding a hammerless action pistol, such as a Glock.

3. Firearms exhibiting features that shouldn't happen in real life.

As mentioned above in the previous myth, we often hear cocking noises out of firearms that don't typically make such a noise in real life. One example is the sound of a hammer being cocked back, when the character in question is holding a hammerless pistol, such as a Glock. Another example that is seen in several movies is the sound of a racking slide of a pump-action shotgun, when the character is holding a double-barreled model!

Then there's the scene where someone shoots 15 times without reloading, while holding a 6-shot revolver.

4. It's only a flesh wound.

In the movies, there seems to be a general idea that if someone gets shot in the arm, shoulder or the leg, the person will generally hobble around a bit, but will survive in the end. Then there is the scene where a person will shoot someone in the leg to prevent them from running away and the person getting shot doesn't die. Persons getting shot in the shoulder make a full recovery in the hospital after a couple of days.

In reality, there is no safe place where a person can be shot and be assured of not dying. There are some large arteries in the arms, shoulders and legs that can cause a person to die from blood loss within minutes if these are punctured. The shoulder joint is very complex and is hard to put back together if shattered by a bullet.

Police and soldiers also don't aim to intentionally wing someone. This is because there is a chance that the shot could miss and hit some other innocent person. Therefore, they are trained to aim at the center of mass, so that there is a greater chance of hitting the target. No one in their right mind ever shoots to wing someone, they always shoot to kill.

5. Put down your gun, officer, and step away, or else the hostage dies.

This is the classic scene in several movies. The hero or heroes barge into a room with their guns out, only to find the villain holding a gun pointed at a hostage's head. The villain then tells everyone to throw down their weapons and kick them towards him, or else he'll shoot the hostage. The heroes will do that and then the dramatic music starts.

No one ever wonders why the villain doesn't now just shoot the hero and the hostage and then make his escape. This is why, in real life, police officers and military forces are trained never to put down their guns in a hostage situation.

Tuesday, November 20, 2012

What is a MOA?

When referring to firearm accuracy, several texts use the term MOA a lot. So what is this MOA business anyhow? We aim to study that topic in this post.

MOA stands for Minute Of Arc (though some in the firearm industry call it Minute of Angle). Let's go back to your middle-school geometry classes and recall that a circle is divided into 360 degrees. If we want to measure angles smaller than a degree, we divide 1 degree into 60 minutes (and if we want to go even smaller, we divide 1 minute into 60 seconds). Therefore a circle is 360 degrees or 360*60 = 21,600 Minutes of Arc (MOA). So what does this have to do with firearms accuracy, you ask?

Well, if you have a circle that is 100 yards in radius, the length of one minute of arc at this distance works out to approximately 1 inch (to be precise, it is closer to 1.047 inches). How do we get this value, you ask? Here's the arithmetic behind it:

Radius of circle = r = 100 yards.
We know that the circumference of a circle = 2 * pi * r.
Taking pi = 3.1415927 approximately, we have circumference = 2 * 3.1415927 * 100 = 628.31854 yards
Now we know that 1 yard = 3 feet and 1 foot = 12 inches.
Therefore, circumference of the circle in inches = 628.31854 * 3 * 12 = 22619.46744 inches
Now, we know that a circle has 360 degrees or 21600 minutes of arc.
Therefore, length of 1 minute of arc = circumference / 21600 = 22619.46744 / 21600 = 1.04719756667 inches.

Since target ranges are usually set in multiples of 100 yards, this makes the measurement rather convenient for shooters. For example, if the firearm is shooting about 3 inches to the right of dead center at 100 yards, then we simply need to adjust the sights 3 MOA to the left to make it hit dead center. For greater ranges, we simply scale up the measurements as required: e.g. for 200 yard range, 1 MOA = 2 inches approximately, for 300 yard range, 1 MOA = 3 inches approximately and so on. Most modern telescopic sights are set to be adjustable in 1/2, 1/4 or 1/8th MOA per click, so it makes zeroing the sights very easy, since we know that 1 MOA = approximately 1 inch for 100 yards distance. Quite a few telescopic sights come with an MOA scale printed around the adjustment knobs.

Telescopic sight which measures in 1/4 MOA increments per click of the adjustment knobs.

As we mentioned above, strictly speaking, 1 MOA is about 1.047 inches for 100 yards. Therefore, some people define a separate term called SMOA (Shooters Minute of Arc) which is defined as exactly 1 inch for 100 yards. Some scopes come with an SMOA scale rather than an MOA scale. The difference between true MOA and SMOA is pretty small: for 1 MOA at 1000 yards range, true MOA works out to be 10.47 inches and SMOA works out to be 10 inches, therefore the difference between the two is less than 1/2 inch for 1000 yards distance. However, if one was to make adjustments of say 20 MOA, then at 1000 yards, the difference between the two would be around 9.4 inches! So it is good to know which unit the scope is calibrated to.

So when someone mentions that his/her firearm shoots 1 MOA, that means that under ideal conditions (no wind blowing, match-grade ammunition used, firearm mounted on a bench rest, barrel and chamber are clean etc.) the firearm will shoot groups of bullets inside a 1 inch circle on average at 100 yards distance. Most quality rifle manufacturers will guarantee that their rifles shoot sub-MOA groups with specific ammunition brands. A sub-MOA means that the rifle will shoot groups of bullets in a circle smaller than 1 inch at 100 yards range. With really high end rifles with match grade barrels and quality ammunition, 0.2-0.5 MOA or better is easily achievable. US Army sniper rifle standards from 1988 require the rifles to shoot a 5 shot group with 0.605 MOA accuracy over 300 yards distance when using M118 special ball cartridges and a government approved bench rest. This works out to shooting 5 shots inside a 1.9 inch circle at 300 yards. Any rifles that fail to meet this standard are returned back to the manufacturer.

To give the reader some idea of accuracy of various firearms, a typical assault rifle shoots about 3-6 MOA, a typical sniper rifle (depending on whether it is used by police or military) shoots about 0.25-2 MOA, but a real accurate competition rifle may easily shoot 0.15-0.3 MOA groups. With new advances in metallurgy and machining techniques, several manufacturers are now offering civilian rifles that are guaranteed to shoot 1 MOA or better out of the box, and cost less than $1000 too. Just a few years ago, such accuracy at such a low price would have been unthinkable. People used to pay their gunsmiths hundreds of dollars to bed their stocks and fit precisely machined custom barrels, all in order to get 1 MOA accuracy from their rifles. Now, due to advances in technology, they can now achieve the same accuracy or better from off-the-shelf guns and ammunition.

Monday, November 19, 2012

What is a Boolit?

The term "boolit" may be seen on some forums on the internet these days. Well, what is it, the reader wonders? Well, wonder no more.

The word "boolit" is a made-up word and is not part of the English language. It is a deliberate misspelling of the word "bullet". The origin of this term seems to have come from a forum called and spread from there on to many other shooting forums (mostly those with a lot of American members). As per the forum, a boolit is a projectile that is hand cast by a person for use by an individual, whereas a bullet is a machine-made projectile made by a commercial company for mass consumption. Another common difference per the forum members is that a "boolit" is cast from a mold, whereas a "bullet" is a jacketed projectile.

In the good old days, many fire arms came with their own bullet molds.

Some enthusiasts like to make their own bullets, just the same way that their forefathers used to. And quite a few of them refer to their own custom made products as "boolits".

Monday, November 12, 2012

Point Blank Range

When referring to firearms, we sometimes hear the term "point blank range". So what is the meaning of this term and how did it come about.

The popular usage of the term "point blank" usually means shooting a target very close to the muzzle of the firearm. However, in the field of ballistics, the definition is a bit different: "point blank range" is the range between which the user can hit a target without adjusting the elevation of the weapon. This varies depending on the size of the target and what range the firearm was zeroed at.

In our discussion about zeroing a sight many months ago, we mentioned that the sights are calibrated for a known distance (e.g. for the M16, the US military recommends zeroing the sights at 300 meters). Remember that bullets travel in a parabolic path, such as the diagram below:

Public domain image from Wikipedia.

As can be seen from the figure above, when the bullet is fired, it comes out of the barrel below the line of sight, then quickly rises above the line of sight and then starts to fall again after it has traveled some more distance. If the rifle is sighted to say, 300 meters, the bullet strikes a point that coincides with the line of sight when the target is at 300 meters distance. If the target is closer than 300 meters, the bullet will fly above the line of sight to the target. If the target is farther than 300 meters, the bullet will strike a point below the line of sight of the target. Say the target is something like 5 inches (12.7 cm.) tall, the rifle is zeroed at 300 meters and when the user shoots at it, the bullet never rises 2.5 inches above line of sight at distances over (say) 275 meters, nor does it fall 2.5 inches below the line of sight for distances below (say) 325 meters. Therefore when we shoot this target by aiming for the center of the target using line of sight, we can't miss it between these two distances. Thus we can say for a 5 inch target, the zero range is 275 meters and the maximum point blank range is 325 meters.

There are multiple reasons given for the origin of the term "point blank". One reason given is that the traditional center of targets is a white circle and the French word for "white" is "blanc".

Another charming reason given for the origin of this name has to do with the famous Italian mathematician, engineer, designer of fortifications and bookkeeper, Niccolo Fontana, known to the world as "Tartaglia". The reason for his nickname has to do with an incident where he was badly wounded in the face by an invading French soldier when he was just a boy. The scars from those wounds affected his ability to speak normally and he picked up the nickname "Tartaglia" ("The Stammerer") as a result. As an adult in the 1500s, Tartaglia proved to be a skillful mathematician and was the first to translate Euclid's Elements from Latin into a popular European language (Italian), so it could be more readily understood by common people (only well educated people spoke Latin). During the 1530s, he started to develop an interest in military fortifications and gunnery. In 1537, he published a study on the science of gunnery and over the next 10 years, he invented an instrument known as a "Gunner's Quadrant" to help gunners aim their artillery properly. Incidentally, he was also the first to demonstrate the path of a bullet is always parabolic.

A Gunner's Quadrant.

It consists of two wooden arms joined together at 90 degree angles to each other, similar to a carpenter's square tool. At the joint between the two arms is attached a thread with a weight at the end, called a "plumb bob". Between the two arms is an arc marked off into 12 divisions, called points. The cannon would then be fired at various angles of elevation and the ranges measured for each angle and noted down in a book called a "gunner's table". To use the device, the gunner would insert the long arm into the cannon barrel and then tilt the barrel up or down until the plumb bob would intersect the point that his gunner's table showed for the distance to the target. When the barrel was near vertical, the plumb bob would cross Point 12 and when it was near horizontal, the plumb bob would cross Point 0.

Using a gunner's quadrant with a cannon. Public domain image.

Remember though that this was 16th century Europe and most people were still using Roman numerals. Therefore the points on the arc were marked using Roman numerals I, II, III, IV .... XI, XII, since most people were used to seeing numbers written like this anyway. The concept of "zero" doesn't exist in the roman numbering system and therefore point 0 was marked with a blank and was called/translated as "Point Blank"!

How to Calculate the Twist Rate - III

In the last couple of posts, we saw the Greenhill formula and the Miller twist rate formula as two methods to calculate the twist rate of barrels. There are also a few computer programs that were written to help calculate the barrel twist rate.

A well known program called McGyro was written by the late Robert L. McCoy in the 1980s. Bob McCoy was a distinguished scientist employed by the Ballistic Research Lab (BRL), now called the Army Research Lab (ARL) at Aberdeen Proving Grounds in Maryland. For nearly 30 years, it was his job to conduct both theoretical and experimental ballistic research on everything from .22 caliber to large cannon shells. During this time, he received 3 of the highest civilian awards from the US government for his work and even wrote a book on the subject called Modern Exterior Ballistics. So this is certainly a person who knew what he was talking about. In the 1980s, he wrote a series of small programs in the BASIC programming language to estimate the twist rate. The most famous of these are McGyro and IntLift. As an item of historical interest, his original programs are available here.

Bob McCoy's programs were later improved by William Davis Jr., who was also another well known ballistic engineer. Bill Davis qualified as an expert shooter in rifle, carbine and pistol when he was enlisted in the US Army during WW-II. After the war, he worked at the above mentioned Ballistic Research Lab (BRL) for a few years and later at Frankford Arsenal and Rock Island Arsenal. He was involved in the development of the 5.56x45 mm. cartridge and the M16 rifle as well (He held the ad-hoc title of 'AR-15 Project Directory' during its development). It was Davis' team that identified many of the early ammunition problems with the M16, including primer sensitivity and cyclic rate issues due to high port pressures; they also diagnosed and offered solutions to the ill-fated change to ball-type propellant. He later went on to found Tioga Engineering in 1980 and was ballistics editor for American Rifleman magazine for several years until his death in 2010. He wrote the NRA's book on handloading ammunition and contributed to the Encyclopedia Britannica for the Ammunition section.

The original programs that these gentlemen developed were programmed in the BASIC programming language, mainly because it was available on practically every home computer model sold in the 1980s and therefore could be run by a wide audience. Unfortunately a BASIC interpreter doesn't come distributed with computers these days, but many free versions can easily be found by using google and searching for "BASIC interpreter". The programs have also been translated into more modern programming languages such as JavaScript and may be found online at several places on the web.

There are also other translations, such as WinGyro, which is McGyro modified to run under a Microsoft Windows environment.

Sunday, November 11, 2012

How to Calculate the Twist Rate - II

In our last post, we looked at the venerable Greenhill formula, first developed in 1879 and refined over the years. In 2005, Don Miller suggested an improved formula to calculate the twist rate, in the March 2005 issue of Precision Shooting magazine. We will look at that formula in this post.

Some of the problems with the Greenhill formula was that it was more suited to bullets from another era, where the bullet shapes were more oblong shaped (like an American football) and largely made of lead alone. Modern bullets are longer (e.g. spitzer or boat-tail shape) and made of multiple materials (such as copper and brass jackets, steel core etc.). The corrected Greenhill formula does work better than expected for modern bullet velocities of 2800 feet/sec, but it doesn't work so well for black powder velocities.

What we studied in the previous post about the Greenhill formula was actually its simplified form. The original Greenhill formula was much more complicated and involved calculating such esoteric items as polar moment of inertia, transverse moment of inertia, pitching moment coefficient, angle of attack, air density etc. To work out the original Greenhill formula, one would need a degree in physics and mathematics to make sense of all these terms, as well as access to some high quality scientific instruments to make the measurements needed to calculate all these items. What Miller did was start with the original Greenhill formula and used some empirical data to simplify the calculations so that one didn't need an advanced degree to do the whole calculation and could use basic instruments to do the measurements. The Miller formula is:

     T = Twist rate in inches per turn
     m = Weight of the bullet in grains
     s = Gyroscopic stabilization factor (see below for how this is evaluated)
     d = Bullet diameter in inches.
      l = Bullet length in calibers, which is calculated as L/d, where L = length of bullet in inches.

With the Miller formula, all the information needed for the calculation (length, diameter and weight of the bullet) can be easily obtained from the manufacturer, or can be easily measured by anyone with access to a vernier caliper and a small weighing scale.

Miller notes that his constant 30 in the equation above was taken assuming a standard temperature of 59 degrees fahrenheit, standard pressure of 750 mm. of mercury at 78% humidity, velocity of bullet at 2800 feet/sec and altitude at sea level. He also notes that under the standard conditions, the gyroscopic stabilization factor s in the equation above runs from 1.3 to 2.0 (for military, it runs from 1.5 to 2.0) and he says 1.75 is a good starting value for it. He also states that cold temperatures significantly affect air density and therefore s as well. Hence, he recommends assuming s = 2.0 to account for usage in low temperature environments for preliminary calculations of twist.

With all this in mind, let us perform a sample calculation for a Sierra bullet with the following specifications:
    m = 180 grains
    L = 1.180 inches
    d = 0.308 inches

First we calculate l = L/d, which gives us l = 1.18 / 0.308 = 3.83117

Then we assume s = 2.0 to account for low temperature conditions and calculate the twist rate T, which works out to about T = 12.081. This means that a twist rate of 1 turn in 12 inches ought to work well for this bullet.

In his paper, Miller says that he used experimental data from the US Army's Ballistic Research Lab (BRL) to verify the validity of his formula.

Friday, November 9, 2012

How to Calculate the Twist Rate - I

Many moons ago, when we first studied the basics of rifling on this blog, there was mention of the term twist rate of a barrel. The twist rate is defined as the length of the barrel required for the bullet to make one complete turn in the barrel. For instance, a standard M16A2 barrel makes 1 turn for every 177.8 mm. length of barrel.

As we studied earlier in this blog, the whole purpose of rifling is to spin the bullet as it comes out of the barrel, therefore stabilizing the bullet as it flies through the air. This gives the firearm its accuracy. Different bullet shapes will need different twist rates. For instance, large diameter bullets have more inherent stability because the large diameter has gyroscopic inertia, whereas long thin bullets need to be spun faster. Therefore, a rifle firing a round lead ball can get away with 1 turn in 60 inches rifling, whereas a rifle firing a longer conical bullet (such as the M16) needs a rifling rate of something like 1 turn in 7 inches (177.8 mm.). On the other hand, having too high of a twist rate increases barrel wear and can tear the bullet's jacket apart while flying through the air. Therefore, one must pick a barrel with a twist rate that keeps the bullet stable in the air, without causing performance issues for the bullet or barrel.

To determine the optimum twist rate of a barrel, several people have developed various rule-of-thumb equations over the years. The first equation we will study is the Greenhill formula, which was developed by Sir George Greenhill, professor of mathematics at the Royal Military Academy in London, way back in 1879. This formula is completely empirical (i.e. the professor made several observations and came up with an equation that approximately fit the observations, rather than having a complete scientific explanation of how the formula was derived). The formula was originally developed for rifled artillery, but was found to work just fine for small-arms bullets as well. The original Greenhill formula was:
Twist = C * D * D / L

     D = Diameter of the bullet in inches
     L = Length of the bullet in inches
     C = A constant (defined to be 150 in the original equation, but read on for modern modifications)

Of course, this equation only worked for lead-core bullets, so it was modified a little to account for bullets made of other materials as well. The new reworked formula is:
Twist = C * D * D / L * sqrt(SG / 10.9)

    sqrt = Square root
    SG = Specific gravity of the bullet material. For a lead core bullet, SG = 10.9, therefore the right part of the equation evaluates to 1. For other materials, the value of SG is as follows: Copper = 8.5, Brass = 8.9, Steel = 7.8.

Say we have a .308 Winchester cartridge (the commercial version of the NATO 7.62x51 mm. cartridge) with a 190 grain Sierra Matchking bullet (manufacturing number #2210). This bullet has a length of 1.375 inches and a diameter of .308 inches. Assuming SG = 10.9 for lead core bullet, we have:
Twist = 150 * 0.308 * 0.308 / 1.375 * sqrt(10.9 / 10.9)

Doing the math with a simple calculator gives us the result of 10.3488 inches, which means a rifling rate of about 1 turn in 10 inches should work out for us. This is why many rifles that use .308 Winchester caliber come with barrels with a twist rate of 1 turn in 10 inches.

While the value of C = 150 worked for bullets in Greenhill's day, it didn't work so well with modern bullets that travel at much faster speeds. In fact, as far back as 1929, a book called The Textbook of Small Arms which was published in England, stated that: "In actual practice Greenhill's figure of 150 can be increased safely to 200 and still control the bullet". A more modern rule of thumb is to assume C = 150 for bullets that travel at speeds less than 2800 feet/sec and for bullets that travel faster than 2800 feet/sec, C = 180 can be assumed. Another way to compute C, which was  proposed by Les Bowman in Gun Digest, 1962, is to use the following equation for C:
                                       C = 3.5 * sqrt(V)
     V = velocity of the bullet in feet/sec.

Note that in Greenhill's time, the average velocity of a bullet was around 1800-1850 feet/sec, which would give us C = 150 approximately, if using Bowman's method to calculate C. Similarly, using a more modern velocity of 2800 feet/sec would give us C = 185 per Bowman's calculation.

With Bowman's alteration to the formula, the new Greenhill formula can be written as:
Twist = 3.5 * sqrt(V) * D * D / L * sqrt(SG / 10.9)

Let us work out the twist rate needed to fire a cartridge that shoots the same bullet as above at a velocity of 2500 feet/sec. In this case:
Twist = 3.5 * sqrt(2500) * 0.308 * 0.308 / 1.375 * sqrt(10.9 / 10.9)

Again, doing the math with our trusty calculator, we get about 12.0736 inches or about 1 turn in 12 inches. 

In the next article, we will study some more rule-of-thumb equations to calculate barrel twist rates.

Thursday, November 8, 2012

The Effect of Temperature on Ammunition Performance

Today's topic of study is going to be an interesting one: the temperature of the ammunition at can affect how it performs. We will study the correlation between these two factors in this post.

It was known for a while that cold ammunition does not perform as well as ammunition at a warmer temperature. The higher the temperature of the ammunition, the higher the velocity of the bullet. Of course, the increase in velocity is also accompanied by an increase in the chamber pressure. This is caused because the powder in the ammunition burns at a faster rate when it is warm and at a slower rate when it is cold.

This phenomenon affects how ammunition should be stored. For instance, in a National match in 1930, it was found that ammunition that was exposed to direct sun rays for several hours before the match, was generating excessive pressure, causing some firearms to malfunction. The problem got so bad that the organizers had to stop the match and replace the ammunition with others stored at cooler temperatures.

However, ammunition stored at colder temperatures will ignite slower and not generate as much pressure and velocity, which could cause the bullet to reduce its range. Therefore, when people publish data about velocities and pressures generated by different ammunition types, this data is always measured at a certain standard temperature (in the United States, this is generally 59 degrees fahrenheit)

Modern ammunition is not as susceptible to the effect of temperature swings, but it still happens. So what is the amount of loss or gain due to variations in temperature, the reader asks?

Several different authorities attempted to answer this question in the 1930s. According to studies conducted by the Frankford Arsenal (located in Northeast Philadelphia, Pennsylvania) and Burnside Laboratory (located in Carney's Point, New Jersey), the ammunition they tested increased in velocity on average by about 1.7 feet per second for every one degree fahrenheit rise in temperature for ammunition loaded with IMR type smokeless powders. What this meant is that ammunition heated from (say) 70 degrees fahrenheit to (say) 130 degrees fahrenheit would experience an increase in velocity of about 102 feet/sec.

Correlation between ammunition temperatures and velocities for three different types of ammunition. Click on image to enlarge
This test was conducted by the US Ordinance Department and the image is now in the public domain.

Of course, the composition of the ammunition plays a large part in this and different ammunition types react to temperature in different ways. The US Ordinance Department also conducted its own tests in the 1930s and the results are shown in the graph above. As you can see from the above graph, Dupont's 1489 powder generates about 51000 PSI (pounds per square-inch) of pressure at 70 degrees fahrenheit, but generates about 56500 PSI of pressure at 140 degrees fahrenheit. This is an increase of about 5500 PSI (or 11% increase) and some firearms may not be able to handle the excess pressure generated.

In case you're wondering why 70 degrees fahrenheit appears in the graphs and notes above, it is because that was the "standard temperature" that measurements were taken against in the 1930s. These days, the standard temperature in the US is considered to be 59 degrees fahrenheit

The following table is based on U.S. Army tests on .308 Winchester ammunition, which is the commercial variant of the 7.62x51 mm NATO standard ammunition.

Bullet Drop at 600 Yards
(200-Yard Zero)
-10 2400 feet/sec -109 inches
+25 2500 feet/sec -100 inches
+59 2600 feet/sec -91 inches
+100 2700 feet/sec -84 inches

Bear in mind that air is a good insulator, whereas brass is a good conductor of heat. Therefore, ammunition that is exposed to direct sunlight may reach higher temperatures than the surrounding air and 130-140 degrees fahrenheit is not an unusual temperature for ammunition to reach. The excessive pressures may cause some firearms to malfunction and can also affect the accuracy of the weapon.

Consider that a hunting rifle has been zeroed in at a temperature of about 80 degrees fahrenheit (which is the average temperature on a typical day where the author lives in Southern California). Now let's say the person goes hunting with that rifle in Alaska, where the average temperature in early morning is typically about 0 degrees fahrenheit. That is a 80 degree variation in temperature, which can cause the velocity of the bullets to drop by a significant amount. This means the sights will need to be readjusted because of the drop in velocity and chamber pressures.

In the late 1800s, the British author and firearms manufacturer, W.W. Greener, came up with a pretty convenient approximate formula to account for changes in temperature. His formula is based on a standard temperature of 60 degrees fahrenheit (rather than the modern ballistic standard of 59 degrees F) and is as follows:

Range adjustment for temperature = (degrees +/- from 60 degrees fahrenheit) * (target distance in hundreds of yards) / 10

So if the temperature is above 60 degrees, we should subtract this distance from the actual range and if the temperature is below 60 degrees, we should add this distance to the actual range. For example, say the temperature is 100 degrees fahrenheit and the actual distance to the target is 800 yards. Therefore, we calculate:
Range adjustment for temperature = 40 * 8 / 10
which works out to 32 yards. What this means is that though the actual distance to the target is 800 yards, we should treat it as being at (800 - 32) = 768 yards and adjust the sights accordingly.

Sunday, October 28, 2012

Self Lubricating Bullets

Your humble editor recently came across an interesting book called "Modern American Pistols and Revolvers". The book dates from 1896 and describes the state of firearms that was considered "modern" in that era. Among its many pages was a mention of something called a self-lubricating bullet.

Self-lubricating bullets seem to have been invented by Daniel B. Wesson, one of the founders of Smith & Wesson. He received a patent for his design in 1893. During this era, fouling in firearms was a big problem because smokeless powders hadn't been invented yet and black powder left a lot residue behind in the barrel. Excessive fouling would cause the finest revolvers in the world to become inaccurate, especially when they were shot rapidly in dry conditions.

Smith & Wesson self lubricating bullet. Click on image to enlarge. Public domain image

In the above image, we see a Smith & Wesson self-lubricating bullet. The bullet has a hollow core in its base, about 1/8th inch in diameter. Inside this core, a copper plug is inserted and the core is filled with lubricant A. The base of the core has a brass plug B. Four tiny passages C are drilled along the side of the bullet and these passages are also filled with lubricant.

Self-lubricating bullet when fired. Click on image to enlarge. Public domain image.

When the bullet is fired, the brass plug B is pushed inside the bullet by the expanding gases, thereby forcing out the lubricant out of the passages C in the conical front of the bullet. The theory was that the lubricant would distribute itself to the walls of the barrel and keep it moist. The black powder residue, which is very hot, would cool down more rapidly because of the lubricant and adhere only loosely to the barrel and could be cleaned easily with only a wire brush.

The book goes on to say that this cartridge was tested by the US Government and seemed to show more accuracy than using regular ammunition, while being slightly more expensive. Smith & Wesson used this idea for several of their cartridges, .38 S&W, .38 Special, .32 S&W, .32 S&W Long, .44 S&W Russian etc.

With the invention of more modern and cleaner burning powders, this type of bullet seems to have lost its popularity.