Question about sectional density and penetration

[roscoe]

So - I think sectional density is pretty straightforward except for one thing: the influence of velocity. Is there some formula for calculating some sort of penetration quotient?

Obviously, given two projectiles with the same mass and cross-section, the faster one will penetrate more.

But, what about an object of lesser sectional density moving faster? How does one calculate the overall interaction of sectional density and velocity?

When discussing our fabled interaction with grizzly bear we often talk about how heavier loads penetrate better, but what about much faster loads with the same cross section? There must be some point at which speed overcomes weight (assuming the bullet stays intact) and provides better penetration.

Is there a name we can put to this and how do we calculate it?

 

[Yooper]

Try the following webpage for a penetration calculator using the Poncelet equation.

http://home.snafu.de/l.moeller/Poncelet/Poncelet_english.htm

 

[saspic]

There must be some point at which speed overcomes weight (assuming the bullet stays intact) and provides better penetration.



I'm by no means an expert, but could that point effectively be where a pistol ends and a rifle begins?

 

[RyanM]

The pistol and rifle thing seems to be at 2000 to 2500 fps. That's the velocity at which fragmentation and temporary cavitation can cause very large amounts of tissue damage through tearing.

 

[junyo]

Sectional density is a diameter to mass ratio. Remove sectional density from the equation and I think the main factor is kinetic energy, which really is just mass and velocity. Therefore a 115 grain bullet (0.00745187465 kilograms) exiting the muzzle at 1200 fps (365.76 meters/second) will have about 500 joules of kinetic energy, whereas a 147 grain bullet (0.00952543977 kilograms) would have around 638. Now, zip the 115 grain bullet up to 1358 fps (414 meters/second) and the kinetic energy figure about matches the slower 147 grain bullet. Assuming that cross section stayed the same, and not accounting for expansion, it should penetrate the same amount.

 

[roscoe]

Remove sectional density from the equation and I think the main factor is kinetic energy

True, and that is easier to figure out if you have two objects with the same diameter, but let's say you have a 180 grain .357 round and a 350 grain .44 round. Although the ballistic coefficient on the .44 is better, what happens if you can get the .357 going faster. At some point it will offer better penetration. Now, we are not discussing the advantages of a bigger hole, etc., but it would seem like there must be some way to quantify the penetration offered by the combination of velocity and sectional density.

(I am still working on that Poncelet thing with excel, but it requires input of a particular density of target and I am looking for a more universal number.)

 

[RyanM]

No, energy is actually the opposite. Energy increases exponentially as velocity increases (double vel = quadruple energy). Penetration increases logarithmically as velocity increases (at most firearms velocities it's 1.2x to 1.5x gain in penetration for a doubling in velocity).

Think of it this way. If you throw a balloon gently, then throw another one real hard, twice as fast as the first, the one you threw really hard isn't going to go 4 times as far as the other, more like 1.5 times as far. That's because drag increases as velocity increases.

 

[Yooper]

It is really an optimization problem. There is a point of diminishing return in penetration for each sectional density with respect to velocity. The relationship is curvilinear and for each sectional density there is an optimum velocity, beyond which increases in velocity yield proportionately smaller increases in penetration. Sectional density is the primary determinant and the relationship seems to be independent of absolute mass. As the sectional density increases, the optimum velocity also increases and with the heaviest projectiles for a given caliber, optimum velocity can exceed pressure/recoil limits.

 

[roscoe]

Follow-up to the Poncelot equation in Excel: (assuming Poncelot works, and choosing a target material at random and holding it constant), you can get a pretty good rough idea of the penetration by simply multiplying the natural log of the sectional density by the kinetic energy. It does not produce the exact result, but it correlates pretty highly (.90, significant at .05). Why, I am not sure, because the Poncelot equation uses the log of the energy, but using that predicted Poncelot less well.

Anyway, for me the surprising result is the effectiveness of the .44 magnum out of the rifle. Assuming this is all accurate, a heavy .44 magnum round is not far behind the 45-70 except in the most powerful loadings. Because of this, I can justify (at least to myself) a .44 Trapper!

 

[Werewolf]

Interesting link. I played with it a bit.

It seems that (after doing the conversions to metric) a 230 gr (14.7 grams) .45 ACP (11.4mm) at 300 m/s gets 23 cm of penetration in flesh and that a 158 gr (10.1 grams) .357 (9mm) Mag at 450 m/s gets 31 cm.

That's a lot of penetration on the .357.

Now the next question is which one does more damage?

Not being a hunter and a paper puncher only I've never really given much thought to that.

In flesh - penetration is about 9" for the .45 and 12" for the .357. Both would go clean thru most folks on a front to back shot. It would seem to me that with both rounds fully penetrating that even though the .357 is faster that the .45 would do more damage as it would make a bigger hole?

Am I missing something here - intuitively it seems that the .45 is a better choice for non-expanding and probably a better choice even if one uses expanding bullets. Wouldn't bigger be better?

But then one must consider kinetic energy. IIRC (and I may not be) the formula for that is e=m*(v*v) i.e. energy=mass times velocity squared. Dividing the results by 1000 for ease of use the numbers come out to 1325 for the .45 and 2045 for the .357 at point blank range. So given those results it would seem that the .357 has more energy and thus could break any bones it hit easier than the .45 but then isn't that dependent on the mass of the object being hit :banghead: :banghead: ohhhhhhh - my head is starting to hurt.

NOTE: Standard ball .45 runs around 830 FPS which is less than 300 m/s. I chose 300 m/s for the .45 ACP because that's pretty close to the velocity that I load my 230 gr .45 ACP to (around 912 FPS).

 

[RyanM]

The general rule is ignore energy! Energy is almost totally irrelevant to wounding effect.

http://www.thehighroad.org/showthread.php?t=132353

Mass, velocity, expanded diameter, fragmentation, etc., must all be taken into account seperately. Increasing mass increases penetration proportionally (double mass = double penetration). Decreasing frontal area increases penetration inversely proportionally (halving area = double penetration). So sectional density correlates directly with penetration depth (double SD = double penetration). Increasing velocity offers diminishing returns after a certain point, however. You can test that out for yourself in the calculator and see.

 

[Control Group]

IIRC (and I may not be) the formula for that is e=m*(v*v) i.e. energy=mass times velocity squared

Close: Esubk = (1/2)mv^2

 

[roscoe]

Increasing velocity offers diminishing returns

Yes, that seems to be correct: (assuming non-deforming solids) a 115 grain 9mm at 1100 fps appears to penetrate a bit better than a 55 grain .223 at 3200 fps, which is counterintuitive to me. Yet, that is what folks say about the 9mm and the .223 when worrying about shooting indoors through interior walls.

I guess that is why the .38 is so effective, despite the fact that is moves pretty slow. The 150 grain .38 at 845 fps appears to penetrate almost as well as a 147 grain 9mm at 1225 fps. The .380 comes out as a pretty poor comparison to the .38, at least in penetration.