Momentum density & penetration

[McCall911]

Warning: Not for the math/physics phobic.

Someone else kicked off a thread in the handgun forum about sectional density, and this prompted me to start this thread, which might be of interest to some. Sectional density is commonly linked to penetration, but so far few have shown us how exactly in simple terms (assuming that simple terms could apply.)

Some time back, I came across a page on the Internet about a concept that I was unfamiliar with: Momentum density. At first glance, I thought it was junk science, just a few numbers multiplied together, so I wrote it off. Then I decided to look at it more closely, and I considered there might be more to it than first meets the eye.

The concept is mentioned on this page:
http://www.grosswildjagd.de/momentum.htm

About momentum density, the author says:

The most important feature for penetration is the momentum density. Momentum density defines the penetration potential of a projectile. Momentum density it defined as the momentum of the projectile divided by the projectile's cross sectional area.

(Of course, there are several other factors of importance that the author mentions on that page.)

The author states that momentum density is a bullet's momentum divided by the cross sectional area. But guess what? This is virtually the same thing as saying that it is the sectional density times the velocity.

Momentum density = bullet weight, lbs x velocity / area (square inches)

Since we already know that sectional density is already nothing more than a bullet's weight divided by the cross sectional area, just multiply the SD by the impact velocity.

As an example, say we have a mild .44 Magnum load with a 240 grain bullet which impacts at 1000 fps. We probably know alread that the .44 Magnum bullet is actually about .429", so the SD is 240 / 7000 / .429 / .429, or 0.186 lbs/sq inch. Multiply this by the impact velocity of 1000 fps and we get a momentum density of 186. This, according to the author of that page, is the bullet's penetration potential.

Most of us who've been studying terminal ballistics for any length of time know that there is a lot wrong with simple formulas like this. After all, what units are we talking about here? We have momentum, which is ft-lbs / sec (English system) and this divided by the area gives us nothing that makes sense. But you have to remember that momentum is also a unit of force times a unit of time. Like this:

Momentum = pounds (force) x sec

divide this by area in square inches, you get

pounds(force) x sec / sq.in.

which can be rewritten as

pounds(force) / sq. in x sec

In other words, momentum density is a unit of pressure (psi, for instance) times a unit of time. So it could be said that this simple formula may have some actual science behind it.

The problem comes when you try to relate it to practical, easy-to-work examples, such as estimating specific penetration depths, but I've found this can be done in some cases and within certain limitations.

 

[Jim Watson]

Hatcher's RSP is a measure of momentum density with a fudge factor for bullet nose profile. Internet experts say it does not consider penetration but from what your guy says, it should PREDICT penetration.

There was a penetration formula of the same era that worked pretty well on the old pine board baffle box, but I no longer have Hatcher or Josserand and cannot google the details.

 

[btg3]

To me, the pseudo-phsysics of "momentum density" is confusing and fundamentally flawed due to contorted defintions. For starters, density is classically defined as mass/volume whereas the above discussion refers to area rather than volume. Thus we have a discrepancy with units that does not seem to get resolved.

 

[Loosedhorse]

I read the article on which this one is based some time ago--African Hunter is a great magazine. That article was about bullet construction, not "momentum density." It did mention that the higher velocity of monometal bullets made up for the lower weight, so penetration (of wet phone books) was roughly the same as traditional "solids" (thick jacket over lead).

 

[jiminhobesound]

Momentum

Seems like I remember large caliber advocates, Jeff Cooper, Elmer Keith et al saying that momentum was the real factor in effectiveness. This discussion could go the inevitable route of small high velocity versus big slower velocity. I tend to believe the people that are doing autopsys with interest in the cartridge and collateral parameters can give us the most reliable data.

 

[Loosedhorse]

Jeff Cooper, Elmer Keith et al saying that momentum was the real factor in effectiveness

You are correct. They were speaking about their own experience in Africa with tough, thick-skinned game, and the experience they "borrowed" from the African professional hunters they talked to.

The "momentum" conclusion may have been an artifact of the bullets of the day: increasing the energy (lighter bullet traveling faster) was more likely to get you a bullet that breaks up early, without the penetration that such animals require. Empahsizing momentum wth slower and heavier bullets resulted in better results.

Interesting to wonder whether modern, tougher monometal bullets (which are not going to break up) might defeat this rule, but Kevin Robertson, probably today's African hunting guru for terminal ballistics, still prefers heavier (heaviest) bullets in caliber as long as they meet a certain velocity (and energy) minimum.

Of course, here in the US, we also have momentum fans for pistol bullets for SD.

 

[Pigoutultra]

This could be another excuse to do a bullet penetration test. Perhaps the best way would be to use bullets of the same caliber and same nose profile, maybe a 90% meplat flat point. Then see how the sectional density effects penetration in relation to velocity and energy. Energy I would assume would only be comparable with two bullets of the same caliber as a 5" bullet with the same energy as a .452 bullet isn't going to penetrate at all. These test would be nearly impossible to use to make an equation that accurately predicts penetration based on sectional density and velocity alone. Definitely impossible to use with expanding bullets as you can't predict the expansion rate with a precise enough accuracy to use a formula.

 

[Owen Sparks]

I saw an momentum demo once where a man shot a gallon milk jug filled with sand with a .45 auto and the 230 grain round nose bullet did not exit. He then shot another sand filled gallon jug with a heavy arrow from a 35 pound recurve at fairly moderate speed and the arrow passed right through the jug.

 

[Pigoutultra]

Bullets and arrows aren't comparable at all when talking about penetration. A bullet crushes, an arrow slices.

 

[harrygunner]

Imagine two rods of the same mass, with different cross sectional radii, shot length-wise into a material.

Model the material's resistance to penetration as pressure. Then, the force the material imposes on each rod is that pressure times the cross section of each rod.

The rod with the lower cross section will experience a smaller decelerating force. And, will therefore travel further inside the material, until its initial momentum is reduced to zero.

P.S. The concept of density is not limited to 'stuff per unit volume'.

 

[McCall911]

I think the term "momentum density" is just a convention, like the term "sectional density" which is used to describe the ratio of a bullet's weight to its cross sectional area. Momentum density is used in particle physics in a completely different context and doesn't really refer to a volume either. But when you factor everything out, you can reduce "momentum density" to pressure x time, which makes more sense to me.

 

[RTR_RTR]

I think the term "momentum density" is just a convention, like the term "sectional density" which is used to describe the ratio of a bullet's weight to its cross sectional area. Momentum density is used in particle physics in a completely different context and doesn't really refer to a volume either. But when you factor everything out, you can reduce "momentum density" to pressure x time, which makes more sense to me.

::thumbsup::

 

[McCall911]

I've noticed a couple of shortcomings when it comes to momentum density, if you define it as the velocity times the sectional density. I'll have to illustrate one of them.

Say you have that .44 Magnum bullet in my example above. SD of 0.186 and impact velocity of 1000 fps. We are to perform two tests. One is a conventional .44 Magnum cartridge and firearm, while the other is a specially-designed cartridge and firearm which will shoot the .44 bullet sideways. (And by some magic the bullet is able to remain stable in flight turned sideways.)

Which bullet is going to penetrate deeper and why?

(Remember we haven't changed the static sectional density at all.)

 

[Pigoutultra]

(Remember we haven't changed the static sectional density at all.)

Actually you just did. Sectional density is all about the the diameter of the object in relation to its orientation, by shooting a bullet sideways you have completely changed the orientation thus changing the sectional density. In your example the sectional density would decrease as the cross-sectional area was increased while the mass has stayed the same.

To answer your question, the properly stabilized and oriented bullet will penetrate deeper.

 

[McCall911]

Actually you just did. Sectional density is all about the the diameter of the object in relation to its orientation, by shooting a bullet sideways you have completely changed the orientation thus changing the sectional density. In your example the sectional density would decrease as the cross-sectional area was increased while the mass has stayed the same.

To answer your question, the properly stabilized and oriented bullet will penetrate deeper.

That's what I figured. The surface area which is presented to the target is significantly larger, therefore it penetrates less. So for momentum density to work properly, I think you'd have to use the surface area which makes contact with the target. In the case of a sideways bullet, it would be roughly half the area of a side of a cylinder.

That's what I noticed when it comes to flat nosed bullets in particular. You have to use the cross section of the flat part of the bullet, or the meplat, instead of the cross section of the caliber.

I'll go back to the case of the .44 Magnum bullet at 1000 feet per second. It is a flat-point bullet with a 0.300" meplat. So to get the effective sectional density, you would use 0.3 instead of 0.429, the caliber:

240 / 7000 / .3 / .3 = 0.381

So the momentum density is 381.

I was observing a number of soaked newspaper (wetpack) tests recently and was able to come up with a couple of constants which gave an expected range of penetration, when you use the momentum density. The constants are 20, for a high value, and 24 for a low value. So using this, I get:

381 / 20 = 19"

381 / 24 = 15.9"

Since the resisting force on the bullet goes up with the square of the velocity, I would expect that the penetration depth of this mild .44 Magnum load would be in the neighborhood of 19" in wetpack.

 

[Pigoutultra]

No, sectional density isn't about the meplat, its about the total frontal area. A .308 flat point and a .308 spitzer point have the same cross-sectional area. For a .44 that would be .429, it does not matter what the meplat is for sectional density. For penetration, the design of the nose is very important though, so sectional density doesn't give the whole story.

 

[wally]

What he is calling momentum density is sectional density times velocity. Penetration is proportional to this to within a constant dependent on the bullet nose profile (for non-expanding bullets). Other complications, like yaw and cavitation make simple generalizations futile.

The key is when your 9mm JHP expands to .69" the sectional density is greatly reduced which greatly reduces penetration, but when your 230 gr .45ACP expands to .69" penetration is not reduced nearly as much.

 

[McCall911]

No, sectional density isn't about the meplat, its about the total frontal area. A .308 flat point and a .308 spitzer point have the same cross-sectional area. For a .44 that would be .429, it does not matter what the meplat is for sectional density. For penetration, the design of the nose is very important though, so sectional density doesn't give the whole story.

I'm basing the meplat idea on the work of the author of this page
http://www.rathcoombe.net/sci-tech/ballistics/methods.html#flat-nosed
where he pointed out:

The following method is adapted from the work of Veral Smith of Lead Bullet Technology (LBT), as described in his book, Jacketed Performance with Cast Bullets.
He observed that the wound (or channel in test media) caused by hardcast flat-nosed lead bullets was proportional to velocity and to the diameter of the flat portion of the nose. I am impressed by the tests which he conducted, because he found by blacking the noses and shoulders of the bullet types he tested that the notion long held that a Keith-style semi-wadcutter cut a hole on the basis of its shoulder diameter (as opposed to its nose) was incorrect; the blacking on the shoulders was intact, only the nose was clean (this phenomenon may differ at much lower velocities or different nose shapes). On the basis of his testing, he developed cast bullet designs which are nearly cylindrical and feature a very wide flat point.

So his (or Veral Smith's) findings were that only the meplat is presented to the target in penetration and that the shoulder isn't involved.

I'm not saying that I was necessarily right when I did this, but in the testing that I've been observing the method I mentioned seemed to work out fairly well for the type of flatnosed bullets that were being evaluated.

(For those interested, the tests are found beginning on this page:
http://forums.accuratereloading.com/eve/forums/a/tpc/f/4711043/m/2861098911/p/1
By now, it's on the 150th page, although not all of it relates to terminal ballistics. But hey! It's the Internet! )

What he is calling momentum density is sectional density times velocity. Penetration is proportional to this to within a constant dependent on the bullet nose profile (for non-expanding bullets). Other complications, like yaw and cavitation make simple generalizations futile.

The key is when your 9mm JHP expands to .69" the sectional density is greatly reduced which greatly reduces penetration, but when your 230 gr .45ACP expands to .69" penetration is not reduced nearly as much.

One thing I have noticed about terminal ballistics is that no matter how much you think you might have something nailed down, something will pop up to make a fool out of you. It's a humbling field of study for a dummy like me!

 

[jiminhobesound]

Momentum etc.

Great discussion, I am impressed by the knowledge exhibited by forum members; a lot of physics concepts, a lot of ballistics and what was proven? This is a great techncial discussion of performance and it further enforces my belief that the coroners, and aids, given enough parameters, are the only folks that can make some practical conclusions.

 

[McCall911]

Inasmuch as medical personnel have to "clean up the mess", I agree. They may not be able to predict what a given projectile might do, but they're the ones who see the effects firsthand.

 

[Manco]

Seems like I remember large caliber advocates, Jeff Cooper, Elmer Keith et al saying that momentum was the real factor in effectiveness.

It depends on the numerous and varied parameters of the shooting, including the type of target, of course. If we view living creatures in terms of fluid media, and the intended targets are large relative to the characteristics of the caliber and firearm, then momentum may well be a superior indicator of effectiveness than kinetic energy because penetration against the force of drag is the key in such cases.

The proponents of kinetic energy will always have a point that wounding is a form of work, and that work (in physics) is represented by energy as opposed to momentum. While there is no arguing over the fact that greater kinetic energy represents greater wounding potential, in practice everything depends on how that energy is used. To take an extreme example, blowing a wide but shallow crater in a critter does a lot of damage but may not have the intended effect. In defensive scenarios against humans, the service calibers could theoretically waste much of their energy in minor temporary stretch cavities that usually do little or no damage in elastic tissue (eventually being converted to a piddling amount of heat).

That said, if we change the parameters of the shooting, such as reducing the size of the target, changing the type of target (e.g. solid instead of fluid), greatly increasing the energy of the bullet, or simply desiring a different effect, then kinetic energy and/or velocity can play the major role in determining effectiveness. Otherwise, momentum--in conjunction with sectional density and other factors--is the prime determinant of effectiveness in most of the cases that we, as a group, consider.

This discussion could go the inevitable route of small high velocity versus big slower velocity.

Naturally, since such trade-offs are necessary in order to keep firearms within the range of shootability for people. I doubt that anybody would argue against the idea of large AND fast being more effective, but it's not all that interesting.

I tend to believe the people that are doing autopsys with interest in the cartridge and collateral parameters can give us the most reliable data.

Reliable empirical data, yes, but there are so many random and often unknown variables involved that it is difficult to draw any useful conclusions, hence all of the ballistic testing that is done under controlled conditions.

The "momentum" conclusion may have been an artifact of the bullets of the day: increasing the energy (lighter bullet traveling faster) was more likely to get you a bullet that breaks up early, without the penetration that such animals require. Empahsizing momentum wth slower and heavier bullets resulted in better results.

Interesting to wonder whether modern, tougher monometal bullets (which are not going to break up) might defeat this rule

That is one factor, definitely, but even if it is overcome I think that we also need to consider any differences between how the medium reacts to mass versus velocity, even though physics treats them as equivalent in terms of momentum. For one thing, drag in a fluid medium is proportional to the square of velocity, which means that even though lighter bullets have greater velocity to start with, they slow down much faster, even relative to their velocity, theoretically resulting in shallower penetration. Note that this assumes equal momentum with heavier bullets, and that assuming equal kinetic energy would result in even worse relative performance for lighter bullets. As for mass, a fluid medium doesn't react to it in any particular way, except perhaps that greater inertia makes it harder to slow down.

In terms of kinetic energy, heavier bullets may have somewhat less of it than lighter bullets that have the same momentum, but they use more of their energy to penetrate rather than imparting it to the medium in other forms. In other words, if penetration is what you want, which is usually the case with hunting huge animals, for example, then relatively heavy bullets are generally more efficient.

Note that all of this is in line and works in conjunction with ballistic coefficients, which are enhanced by mass.

Kevin Robertson, probably today's African hunting guru for terminal ballistics, still prefers heavier (heaviest) bullets in caliber as long as they meet a certain velocity (and energy) minimum.

(italics changed to bold so that your emphasis shows up in quotes)

That's a good point--everything we've discussed is only valid within a range that most of us would consider reasonable for firearms use. Too heavy a bullet, for example, would not only require an impractical amount of compensation while aiming, but in the most extreme cases would not even penetrate at all, and simply push the target (i.e. transfer its momentum). Within the practical range, however, heavier bullets tend to penetrate fluid media better, in my opinion. Change anything and all bets are off, though--such is the nature of what we're dealing with here.

Of course, here in the US, we also have momentum fans for pistol bullets for SD.

I think it makes sense for those who give priority to penetration and think little of temporary cavitation, at least with the service calibers and smaller calibers that many people carry for concealability (because they're weak relative to the size of the human body). Furthermore, even when given equal momentum, there are those who favor heavier bullets (within the practical range) for various other reasons (e.g. less easily deflected, less blast & flash, more momentum and penetration retained out of short barrels, etc.).

 

[Loosedhorse]

For one thing, drag in a fluid medium is proportional to the square of velocity

True, of course, but that may be less important than it "sounds."

From my perspective (see below), impact, which can produce not just deceleration but also deformation and fragmentation, affects bullet penetration more than drag. Impact force (as the bullet goes from air to skin/bone/flesh) is a different deceleration than drag, but is also greatly affected by speed and bullet profile (as anyone familiar with "belly-flop" diving can attest!). The effects of impact are also influenced greatly by bullet construction.

Robertson, for example, used to recommend "solid" bullets on Cape buffalo, with the sole exception of a dead-profile first shot. He now recommends HP monometals (like the Barnes TSX) for all reasonable first shots on buffalo. That has to do with the fact that such bullets don't lose mass, and so penetrate great even when they expand and their BC changes drastically.

Robertson reports instances of a few ultra-high velocity (typically above 2500 fps) soft-points breaking up on impact with buffalo (and even lion!), so that penetration did not even get past the ribs--fluid drag was not the chief component of such failures, IMHO. It is because of these spectacular failures that I wonder if monometal non-HP bullets might do well with high-velocity (and high-energy) loadings, given their ability to resist deformation and fragmentation on impact.

Perhaps all of this matters little. In hunting and SD, we all agree that penetration is important--we sometimes disagree on how much penetration is "optimal." And it is that disagreement that usually drives the selection of less penetrative (HP, SP, light, soft lead or partially jacketed) bullets that deform a lot on high-speed impact, versus more penetrative ("solid"-nosed, hardcast, FMJ or monometal, heavier) bullets.

It is perhaps noteworthy that TSXs are generally lighter that corresponding traditional expanding hunting bullets, and are often driven faster in commercial loadings--yet they still penetrate deeper.

 

[Manco]

True, of course, but that may be less important than it "sounds."

Right, I was just trying to give an idea of the factors these bullets have to overcome in terms of basic physics. For the reasons you've given, they may indeed be able to overcome their theoretical deficiencies. The subject is sufficiently complex that the only way to know for sure is empirical testing and actual usage. I think in general bullet design potentially has a greater effect--even with pistol calibers--than relatively minor differences in external ballistics that people usually focus on. "All else being equal" is often assumed in discussions but rarely achieved in practice, and bullets can differ quite a lot, obviously.

 

[Zoogster]

One consideration I see left out of most formulas is projectile hardness.

While determining penetration in gel is fairly straightforward, it does not tell the entire story in shooting things that are not of uniform density and hardness like ballistic gel, paper, water, or many other mediums people use.



For example one small round that theoretically just barely penetrates far enough using the usual math in gel may be deflected or lose significant energy and even change shape entirely after hitting a bone.
While another round that theoretically penetrates the same depth using the same math can in reality penetrate much further, losing a lower percentage of energy when it impacts and/or deforming less.


Also different mediums are penetrated differently by different bullet profiles and projectile hardness. What may penetrate say metal really well can have mediocre penetration in tissue, while something that penetrates in tissue to massive depths can be stopped by the same metal.
It is not simply X momentum and Y sectional density penetrates Z depth.
Then there is variations in between, where some rounds that still penetrate say the hard object lose significantly more of their energy in doing so, and so their performance after is very different. A round that normally penetrates tissue really deep can penetrate tissue minimally after penetrating a hard object, while another round that only penetrates bare tissue moderately can still retain most of that penetration after penetrating a hard object and greatly exceed the penetration of the other round in soft tissue even though it penetrates soft tissue less without that hard object in the way.

The formulas are typically kept simple and assume a uniform medium like gel to keep comparisons practical for likely uses. Yet they certainly do not tell the entire story.

 

[Zoogster]

However I should add that sectional density and momentum are probably the best gauges of most handgun performance, and I have argued that momentum is far more important than energy before. (Rifles around or over 3,000 FPS add in additional factors that are not adequately explained by just momentum and sectional density.)


While many people are touting energy figures for handguns it is a poor way to determine many things. Momentum is a much better determiner of soft tissue penetration. And as you stumbled upon how much that momentum can accomplish is determined by the sectional density of the round. A .50 takes more than a .45 to do the same thing, .45 takes more momentum to penetrate than a .40, and a .40 more than a .355, etc
However even that is simplified, because it is only precise if the round remains the same shape and diameter the entire time, and it remains oriented forward which some rounds won't even do after losing spin stabilization.

While many handguns bullets do stay oriented forward, simplifying that concern, many also expand. So the diameter and shape, and as a result the sectional density changes during the course of penetration modifying what the momentum can accomplish before it is expended. Changing the end result. So most formulations will be a close approximation, because to be precise they would have know the exact shape and remaining momentum at every moment of the penetration.
A round may rapidly expand to one diameter, then remain about that diameter for a significant amount of time, then change diameter yet again to something more or less based on bullet construction.
Each change modifying the sectional density.
A round also expends a certain amount of its energy and momentum just in deforming the metal during expansion or in overcoming bone etc, that does not get transferred towards the penetration, missing momentum otherwise not accounted for if you will.