Monday, March 27, 2006

Lies, Damned Lies, and Ballistics - Updated

NOTE: This is updated from the original post to include some new and revised numbers, and a bit more detail in some sections. Oh and I spell checked it. The damn spell check took over an hour because bloggers spell checker is incredibly stupid - which is why I never use it.

Lets talk about ballistics, and numbers. In particular I want to talk about alternative chamberings to the big 3 (9mm, .45, and .40s&w) in service and defensive auto pistols, the numbers they push out, and specifically how the external ballistics relate to terminal ballistics.

First I want to talk about a couple "super calibers", the .357SIG, the 10mm, the .45 super, the .460 Rowland and a couple of others. The commonality between them is that they are all near the limit of what an automatic pistol of reasonable size and weight can chamber.

I specifically recommended that we adopt a sidearm capable of chambering .45 super in "The Right weapon for the Job" and "Getting Down to Specifics", because a .45super chambered weapon can also chamber .45 acp, and because it offers excellent performance against soft body armor at short range with hard penetrator loads.

I have five .45's (of the six that I own) that are rated for .45 super. I don't bother with it because it doesn't seem to give me enough advantage over my +p acp in the barrel lengths I shoot, but I load high performance hollowpoints or frangibles. In a military application where we are limited to hardball, or penetrator loads; or with ANY load in a carbine or SMG; the difference in stopping power could be huge.

I figure the .45 super is the most sensible up-powered option for a .45 frame, and more importantly to my current thinking, for a pistol caliber carbine or SMG.

Just for versatility, I really like convertible firearms; in addition to my two .45 supers, up until a month ago when I sold it, I had a SIG P229 in .40. One of the best thing about that gun; a simple barrel and spring change, and its now a .357.

Of course the starting point for the .357 SIG, was the .40 s&w, and that in turn was based on the the 10mm auto. Unfortunately most people found full powered 10mm a bit unwieldy in most of the weapons chambered for it, so the .40 S&W was developed by cutting the length, reducing the charge, and lightening the bullet (basically its a compromise cartridge, with a standard .40 performing about the same as a +p 9mm, and a +p .40 performing a bit better than a standard pressure .45acp).

I really like the 10mm as a cartridge. Its a powerful and efficient design, that allows for more cartridges in the same space as the .45 acp, and can be comfortably downloaded to the same power levels as the .45, or comfortably uploaded into .41 magnum territory.

I have a 1911, and had Glock 21, either of which could be converted to 10mm with an upper change (slide, barrel, springs) and some new magazines, as the G21 and G20 use the exact same frame, and the 10mm OAL is within range to feed properly in a ramped 1911. 10mm is also very well proven in SMGs and carbines, and is in fact the most powerful chambering currently offered in the MP5 platform.

The wildcard of the group I listed above is the .460 Rowland. The only thing I know about this caliber is the numbers, which are pretty impressive, and that it will feed in a standard 1911 frame, and is a bit shorter than a .45 win mag.

Some would also include the .400 cor-bon, but honestly I just don't find it interesting. It seems to offer no advantage over the .45 super, and very little over .357 sig. 10mm and .460 on the other hand, are both serious magnum level calibers (or can be when loaded properly).

Here's some numbers from Winchester, Federal, Cor-bon, Black Hills, Triton, Doubletap, and Buffalo Bore who offer a full range from low pressure practice loads, to about the hottest factory loaded ammo you can generally buy in most places (that said, there are .41, .44, and 10mm loads at up to 50% over the standard factory energy numbers listed here - only doubletap and buffalo bore load close to maximum for those calibers).

All numbers are from either 4.5-5" automatics and 4" revolvers to be comparable. Typical magnum revolvers gain or lose appx. 50fps per inch of barrel length. Most auto pistols gain or lose somewhat less, because they are generally loaded with faster burning powders.

So here's the baseline, a +p .45 acp

45 ACP +P Factory Ballistics
165 grain at 1250fps 573ftlbs
185 grain at 1150fps 543ftlbs
200 grain at 1050fps 490ftlbs
230 grain at 950fps 461ftlbs

And a couple of magnums and common revolver chamberings etc... for comparison. The very heavy factory loads here are from Buffalobore or Doubletap, who offer the hottest ammo on the market. Most of it exceeds max SAAMI spec, but it is all definitely safe, at least in modern well constructed guns.

.45 COLT
200 grain at 1100fps 537ftlbs (Standard pressure. There are loads with twice this energy or more)

.357 Magnum
110 grain at 1500fps 555ftlbs
125 grain at 1400fps 544ftlbs
125 grain at 1600fps 710ftlbs (max pressure load)
140 grain at 1300fps 525ftlbs
158 grain at 1400fps 688ftlbs
158 grain at 1500fps 790ftlbs (over max spec but still safe)
170 grain at 1425fps 765ftlbs
180 grain at 1400fps 785ftlbs

(switching to a 6" barreled hunting revolver will add appx. 100fps to these numbers)

.41 MAG
170 grain at 1275fps 614ftlbs
170 grain at 1650fps 1030ftlbs (Buffalobore heavy load)
230 grain at 1450fps 1075ftlbs (Buffalobore heavy load)
265 grain at 1350fps 1075ftlbs (Buffalobore heavy load)

.44 MAG
240 grain at 1260fps 848 ftlbs
240 grain at 1350fps 971 ftlbs
240 grain at 1450fps 1120ftlbs (Doubletap +p+ load)
250 grain at 1450fps 1170ftlbs (Doubletap +p+ load)
270 grain at 1450fps 1260ftlbs (Buffalobore heavy load)
300 grain at 1150fps 879 ftlbs
300 grain at 1300fps 1125ftlbs (Buffalobore heavy load)
305 grain at 1325fps 1190ftlbs (Buffalobore heavy load)
320 grain at 1250fps 1110ftlbs (Doubletap +p+ load)
340 grain at 1400fps 1480ftlbs (Buffalobore heavy +p+)

200 grain at 1450fps 934ftlbs

Finally here's the calibers we're interested in:

.357 SIG factory loads
115 grain at 1560fps 621ftlbs
115 grain at 1650fps 695ftlbs (I think this load is over max spec, but safe)
125 grain at 1350fps 506ftlbs (the default standard spec load)
125 grain at 1480fps 608ftlbs (max spec pressure load)
125 grain at 1525fps 645ftlbs
135 grain at 1320fps 522ftlbs
145 grain at 1180fps 448ftlbs
145 grain at 1250fps 503ftlbs
147 grain at 1350fps 595ftlbs
147 grain at 1050fps 360ftlbs (this is a heavy compressed tungsten subsonic frangible for use in aircraft)

10mm (in theory the full power loads, but they seem light)
135 grain at 1400fps 588ftlbs
150 grain at 1325fps 585ftlbs
165 grain at 1250fps 573ftlbs
170 grain at 1360fps 698ftlbs
180 grain at 1220fps 595ftlbs
200 grain at 1200fps 639ftlbs

.45 Super
165 grain at 1,400 fps 718ftlbs
185 grain at 1,300 fps 694ftlbs
200 grain at 1,200 fps 639ftlbs
230 grain at 1,100 fps 618ftlbs

450 SMC
165 grain at 1,450fps 770ftlbs
230 grain at 1,150fps 675ftlbs

.40 Super
135 grain at 1,800 fps 971ftlbs
165 grain at 1,600 fps 938ftlbs
200 grain at 1,300 fps 750ftlbs

460 Rowland
185 grain at 1,550fps 987ftlbs
200 grain at 1,450fps 934ftlbs
230 grain at 1,340fps 917ftlbs

.400 Cor-Bon
135 grain at 1450fps 630ftlbs
150 grain at 1350fps 607ftlbs
160 grain at 1200fps 543ftlbs
165 grain at 1300fps 619ftlbs

.38 Casull
124 grain at 1,800fps 892ftlbs
147 grain at 1,650fps 889ftlbs

One thing is certain, most of the loads listed are pretty potent, and even the "weakest" of the high performance calibers have better numbers than the strongest of the standard pressure .45 ACPs (400-450 ftlbs depending on load) and most of the +p's.

It surprised me that the 10mm and .45 super loads were so close. I thought for sure that the 10mm would be loaded much heavier, but at right around 700ftlbs, both max energy loads I listed are supposed to be max pressure loads (meaning you cant load them to be more powerful without exceeding the spec - more on this later).

The .460 Rowland, as I said before, is pretty impressive, at least according to the numbers. There's some serious energy in those cases; and from what I understand the .460 is basically a slightly shortened and downloaded .45 win mag, so that's not surprising.

It also surprised me how much energy the .38 Casull and .40 super are, in theory, delivering; but the fact is, at those 16-1800fps velocities, those lightweight bullets will probably be disintegrating on impact.

Also, it's kind of pathetic how mild the standard factory .41 and .44 magnum loads are. I've listed heavy .44 loads at over 1200ftlbs here, and .41 loads at over 1000ftlbs. These lower level loads are far below the peaks loadings that have been offered in the past, but major ammunition companies are wimping out on liability. They really just don't want someone blowing their gun up and suing them.

More on that: It seems that the 10mm loads listed are a bit watered down, and there are hotter commercial loads available, but not from the majors because they are all apparently exceeding, or at least bumping real hard on the max SAAMI spec.

Winchester offers a 175gr load at 1290 fps and about 650ft/lbf, and that's about the highest energy full weight load (there are some ultra lightweight 10mm's loading 125gr and 135gr bullets which are loaded to insane velocities) that the major manufacturers are pushing out.

Small companies like Doubletap Ammo, and Buffalo Bore are going a little hotter, and apparently exceeding SAAMI specs. That doesn't mean they aren't safe, (SAAMI is full of pansie asses) just that the SAAMI spec is low. The same could be said of the .45 super (the only one of the other uprated calibers that is a SAAMI standardized offering); and of the .357SIG. The manufacturers of .357 SIG pistols are already going to SAAMI with a much higher pressure spec than initially adopted because the pistols and the cartridge can be uprated by as much as 20% over the current "max" spec loads without any excessive pressure issues.

Hunting around online, I found a chart listing these 10mm loadings. According to the site, all are over SAAMI max pressure specs, but still appeared to be safe in a G20.

165 grains at 1400 fps 718 ftlbs
150 grains at 1476 fps 726 ftlbs
220 grains at 1335 fps 792 ftlbs (yipes)

Here's the loadings Doubletap is offering.

135grains at 1600 fps 767 ftlbs
155grains at 1475 fps 750 ftlbs
165grains at 1425 fps 744 ftlbs
180grains at 1330 fps 707 ftlbs
200grains at 1270 fps 715 ftlbs

On the extreme end, some lunatic worked up the following:

13.4 gr blue dot, 155gr Hornady XTP at 1618 fps 900ftlbs

which is heading into .44 magnum territory, but which apparently produces no signs of excessive pressure from a Glock 20 (which because it is the most common 10mm pistol in public hands, has become the standard testing pistol for the caliber).

The original "proof" load for the 10mm (which is supposed to be the maximum load used to test the gun to see if it will blow up) was 170gr at 1400fps and 740 ftlbs (though it never actually hit 1400 from a real gun, that number was from a pressure barrel ), which was the load that SAAMI set their original max by backing down 10% from.

That's plenty stout for any human target, through a car door and heavy clothing, which is why the FBI loved it (by the numbers anyway).

Personally would think 13.4gr of blue dot behind a 155gr from an unsupported, and relatively loose chamber (in a G20) would be pretty close to suicidal, but the guy says that after testing to max pressure, it's what he backed off to, to the point where there were no signs of excess pressure. Also he was using a 24lb recoil spring, a tungsten guide rod, and a titanium firing pin.

In that same realm of slightly insane loads, here's a fun load out from a 5" .45 super

255 grains at 1180 fps 788 ftlbs

That is by far the heaviest load I've seen for .45 super, and I don't think I'd want to push that far, but there are a bunch of loads right around 700ftlbs that I would be comfortable with.

What's real fun though; is when you scale up from a 5" barreled pistol, into a 10-12" barreled SMG, or a 16" barreled carbine. From a 16" barrel, that same 255gr load is pushing 1650 fps and 1540ftlbs. The 185gr max velocity load listed above, will clock in at 1940fps and 1545 ftlbs.

Okay so I've been throwing around a lot of numbers, but what exactly are they, and what do they mean?

First, they were derived from the standard energy calculations:

Energy = Mass in grains * (velocity in fps)e2/450400

Here's an online calculator

Unfortunately, while muzzle energy is easy to calculate (as are retained energy, and momentum), they don't necessarily mean very much. They are just numbers.

You would think that given the amount of raw data available, someone would be able to write a computer simulation of terminal ballistics that had relevance to killing power, and more importantly to stopping power, but we still haven't figured that one out.

An aside: Not only is muzzle energy not a useful measure of stopping power, but muzzle energy isn't even a reliable indicator of recoil, because there are quite a few factors determining how much recoil is felt. Just off the top of my head, the weight of the bullet, the composition of the powder, the hold (or mount), the grip or stock shape and material, the total energy, the speed of the recoil impulse, the length of the barrel and the weight of the gun all are major factors in felt recoil.

As an example of how these factors interact, the 500 s&w is the heaviest recoiling standard pistol cartridge out there, even at the almost 5 lb weight of the gun. Out of a 7 lb carbine, the same cartridges recoil is comparatively mild.

Historically the .357 magnum is the best manstopper out of a carry pistol, but its energy levels are a lot lower than some other calibers with worse stopping records. Penetration, bullet construction, expansion etc... are all important considerations.

These things are just too complicated for anything other than real world experience, and maybe high end computer modeling. Of course the problem with that is, you have to have good data, and know how to weight each factor etc... and we just don't know how to weight things.

The best numbers we have are from Marshall and Sanow, and while their methods were sound by actuarial standards, they certainly weren't scientifically sound.

My personal favorite numbers only solution is that preferred by the mad ogre (a reader of this site)...

Defensive Power Factor

Bullet Weight in Grains, Times Caliber, Times Velocity, Divided by 1000 = Defensive Power Factor or DPF for short.

For 9mm and such you of course use its actual measured Caliber .355 or what ever your bullet is actually sized at. 10mm is .40 cal etc.

A 230 grain .45 load: 230*.45= 103.5 *900/1000 = DPF 93.15

It has no real scientific basis, but it generally ranks loads known to perform well, in about the same the same order as the real world data.

Actually, I really badly need to qualify that statement. I said it would rank loads in about the same order as real world data; this is only true when the velocities are close to each other.

Let me calc out some examples to show what I mean:

Calculating the DPF of the superhot 10mm load listed above comes out like this:

200gr * .40 caliber * 1778fps /1000 = DPF 142.4, 1404ftlbs

The .30 carbine come out like this:

110gr * .30 caliber * 1990fps /1000 = DPF 65.67, 967 ftlbs

Okay that looks like it's probably pretty close to the real world effectiveness of the two by comparison.

Where things get interesting is when velocity differences are more than a few hundred FPS.

MadOgre suggests scaling rifles by 100 instead of 1000, but that's not really all that useful. There's no scientific justification for the arbitrary scaling factor, and even from rifle to rifle its not that useful. The problem is that energy is an exponential function of velocity. When there are relatively small differences in velocity the disparity is relatively small, but when you are talking about 1000+ fps differences in velocity, the energy differences are pretty huge.

Lets take the standard M193 5.56 NATO, and M80 7.62 loads as an example:

55gr * .223 caliber * 3250fps/1000 = DPF 39.8, 1289 ftlbs

147gr * .30 caliber * 2700fps/1000 = DPF 119.07, 2380ftlbs

I don't think anyone would argue that the .30 carbine was 50% more effective than the 5.56 NATO, as the DPF would indicate, nor would I argue that the 5.56 was 50% more effective than the .30 carbine.

I also wouldn't try to argue the 7.62 NATO was 3 times as effective as the 5.56. I might, say twice as effective, but not 3 times, and no-one on the planet would say that a 10mm carbine would be 20% more effective than a 7.62 NATO as the DPF would indicate.

Oh and you might note that 9mm pistol rounds at 147 grains are a little over 1/3 the velocity of the 7.62 nato, at about 1/6th the energy; and I'd still rather be shot in a vital area with 3 or even 6 147gr 9mm pistol bullets than one single 7.62 rifle bullet.

Sooooo, how do we deal with the disparity?

Okay here goes.

Neither muzzle energy or the DPF listed above have any useful proportionality across great velocity differences. They either weight caliber too high, or they rate velocity too high.

So let's factor both energy AND caliber. Since energy already takes into account mass and velocity, adding caliber covers all the ballistically significant factors (other than bullet construction).

We'll call it the WAG Power Factor, or WPF for short.

The loads for comparison are as follows:

9mm: 147gr .355 caliber at 1050fps = DPF 54.8, ftlbs 360, WPF 127.8
9mm+p: 124gr .355 caliber at 1250fps = DPF 55.02, ftlbs 430.17 WPF 152.7
.45ACP: 230gr .45 caliber at 800fps = DPF 82.8, ftlbs 338, WPF 152.1
.45+p: 165gr .45 caliber at 1250fps = DPF 92.81, ftlbs 573, WPF 257.85
10mm: 170gr .40 caliber at 1350fps = DPF 91.8, ftlbs 698, WPF 279.2

Jump to carbine and rifles here

.30cb: 110gr .30 caliber at 1990fps = DPF 65.67, ftlbs 967, WPF 290.1
10mm: 200gr .40 caliber at 1778fps = DPF 142.4, ftlbs 1404, WPF 561.6
5.56n: 55gr .223 caliber at 3250fps = DPF 39.8, ftlbs 1289, WPF 386.7
7.62n: 147gr .30 caliber at 2700fps = DPF 119.07, ftlbs 2380, WPF 714

Hmm, looking at those numbers, I get the feeling that it might actually be semi-useful. The proportionality might not be quite right, but based on my own experience the order of effectiveness comes out the same as the WPF lists.

Guess it's not a totally Wild Ass Guess, just mostly.

An online associate of mine know as Toad (don't ask) responded to this idea with this:

Hmm, my $0.01 on this is that the caliber should have a non-linear effect, since the frontal area goes up by the square as does the energy. Simplifying I get this for approximate areas. A = Pi (D/2)squared

5.6mm - 24.3 square mm (.22)
7.6mm - 45.4 square mm (.30)
9.0mm - 63.6 square mm (.356)
10.0mm - 78.5 square mm (.40)
11.4mm - 102.1 square mm (.45)
12.7mm - 126.7 square mm (.50)

Between 5.6 and 7.6 the area increases by about 87%

Between 9mm and .45 the area increases by about 64%

That would dramatically increase the weight of caliber in the equation.

Lets look at the example of the standard .45. 10mm, 5.56, 10mm carbine and 7.62 NATO:

.45acp WPF 152.1
10mm WPF 279.2
10mm carbine WPF 561.6
5.56 NATO WPF 386.7
7.62 NATO WPF 714

If we change our criteria from caliber to surface area we get the following:

.45acp 34510
10mm 54793
10mm carbine 110214
5.56 NATO 31323
7.62 NATO 108052

It would seem the surface area numbers weight caliber too highly, putting the standard .45acp as more powerful than the 5.56, and the 10mm carbine load as more powerful than the 7.62 NATO.

By these same numbers a .454 Casull would be more powerful than the 300 win mag.

Toad came back with...
Yes it would, but anecdotal evidence seems to weigh bullet diameter a little higher that the straight diameter formula. To a certain extent a larger bullet of the same weight and velocity is going to transfer energy and make a bigger hole than smaller one. Of course a .22 bullet that weighed 230 grains would probably tumble through like a buzz saw. If not an increase by area for a factor then perhaps an a linear add to the bullet diameter. Say just Pi times diameter?
(Notice folks I'm trying to get Chris to do all the hard work on this)

Hell I don't mind. I do numbers like this in my head for fun, and it's nice to exercise the engineering and math degrees every once in a while.

There's a reason why I'm an insomniac.

There's a huge logical hole in Toads suggestion there. All the results would still be proportional.

Okay lets run the numbers again as an example.

45 acp would go from 34510 to 478, which is 3.14 times my original WPF number of 152.1

7.62 NATO would go from 108052 to 2242 which is 3.14 times my original WPF number of 714

Clearly the size hole made is important, and yes there are substantial differences in area vs caliber proportionality.

Some maxims:
  • Killing power increases as the velocity of bullets of the same caliber and mass increases.
  • Killing power increases as the mass of bullets increases at the same caliber and velocity.
  • Killing power increases as diameter increases at the same mass and velocity.
So there are three independent factors which will increase killing power.

Leaving out bullet construction and shot placement here, and assuming a bullet doesn't fail catastrophically, how do we figure out what is the most important. How do we weight them?

We know that bullet energy is a real measurement of power derived from mass and velocity, but that it is not directly indicative of killing power.

A higher velocity load at the same mass and caliber will have significantly higher in killing power (I.E. .38spl vs .357 magnum).

A higher mass load at the same velocity and caliber will increase killing power, but not all that much unless the animal is extremely heavily structured. The heavier the animal, and the more penetration required, the more important mass becomes. Lets assume we are talking about a human.

A higher caliber load, at the same mass and velocity (thus the same energy), will have more killing power than the higher mass load, but its non deterministic in relation to the higher velocity load.

So mass would seem to be the least important factor, but how do you rank the others?

Well if we take a look at the .454 and the .500 S&W, and we load both to the same mass and velocity, the .454 would have more killing power due to it's higher sectional density, rather than the .500 with its higher caliber, though the differences would be pretty small.

That would seem to indicate velocity is more important than mass, however it requires a larger proportional change in velocity vs caliber to produce a proportional change in killing power I.E. a 25% increase in caliber at the same mass and velocity will produce more killing power than a 25% increase in velocity in the same caliber and mass.

Oh and to physics folks, yes I know that momentum and impulse are more relevant, but most gunnies aren't used to talking in those terms, besides which momentum is a derived dimesnion from velocity and mass anyway.

That means the curves cross, possibly at multiple points, and cant be figured with simple arithmetic, though we may be able to estimate it.

I'm just not sure how to do that, and neither is anyone else, though it will never keep us from trying (and then from writing, or bitching about it at the range, and in gun shops and hunting camps).