Kinetic Energy vs Momentum

[Luke Skywalker]

This has been a personal brain-bug for me, and I've wasted way too much time researching it, with most results being contradictory responses from hunters of questionable expertise.

I understand the theoretical differences between KE and momentum (ie momentum is the derivative of KE with respect to velocity, momentum is a vector quantity, etc). I also understand how they relate in solid-body collisions.

What I don't understand is how they factor into calculating projectile penetration, or simply violent impacts in general. For example; all other factors (air resistance, projectile shape, etc) being equal, what is the difference between a high KE, low momentum projectile and a high momentum, low KE projectile?

If you were to shoot a bullet at a block of wood and increase its velocity while controlling for all other factors, would a graph of the penetration depth with respect to velocity be more closely linear or exponential?

Correct me if I'm wrong, but the primary factor in determining material failure is stress. So wouldn't that imply that KE indicates the distance a reaction force needs to apply on the projectile to stop it, and therefore penetration, while momentum merely determines the time the projectile will continue to move in the body? Or do I have it wrong?

[Sea Skimmer]

High momentum smashes the target, high energy pierces it. The behavior of materials changes as they are subject to higher velocity impacts, they have less time for elastic deformation, and are more likely to locally shear and be pierced instead of dented or cracked. All I have time to say right now, but that is the jist of the difference.

[madd0ct0r]

Correct me if I'm wrong, but the primary factor in determining material failure is stress. So wouldn't that imply that KE indicates the distance a reaction force needs to apply on the projectile to stop it, and therefore penetration, while momentum merely determines the time the projectile will continue to move in the body? Or do I have it wrong?

The critical stress for material failure isn't constant - it can vary depending on the time frames you're looking at, and also whether the material is elastic (steel inside 'springy' range); plastic(steel beyond 'springy' range, clay); quasi-brittle (concrete) or fully brittle (glass). In short time frames (fractions of a second) wood is quite elastic in nature. Over very long time frames wood permanently deforms even at stress less then the critical (creep).

A really extreme example would be non-newtonian fluids like cornstarch + water. You can push your hand in, but run fast enough and your feet 'bounce'.

So basically, the difference between low KE high MV and low MV high KE projectiles depends a lot on what you're shooting them at.

[PeZook]

Don't beanbag rounds go the "low energy, high momentum" route so that they don't pierce the skull but deliver one hell of a painful smack, or am I just confused?

[Sea Skimmer]

That's the jist of a beanbag yes, a typical beanbag from a 12 gauge shotgun has only about 100 joules of energy behind it and moves about 75 meters per second, which is a fraction of the power of a 9mm handgun round. They still have lead shot in them, its just slow. In contrast a high velocity 12 gauge slug is more like 3000-4000 joules. The end effect is kind of like what happens when a high velocity bullet strikes a bullet proof vest. The projectile actually does go a significant distance into the body, but then rebounds away without breaking the flesh... or at least not seriously. Bleeding wounds are common.

Even beanbags that weak though, can and sometimes still do kill people, particularly with a hit on the neck that destroy the windpipe or burst arteries, the smashing effect at work. They've also ruptured internal organs a lot more then once. Police have special targets they are supposed to train with for beanbag use, with normally green, yellow and red zones marked out. Red = this is equivalent and is treated as lethal force. yellow is just injury is damn likely, while green is ideal targets, like the legs, in which death is damn unlikely, and its a good way to knock the person down. The beanbags from 40mm grenade launchers would probably kill you if they struck your skull, but are vastly more reliable for dropping someone with one hit. They tend only to be used by military service, and some nations riot police because of that.

The end reason for penetration is energy is applied faster then it can be dissipated without damage to the target. Lower velocity inherently makes dissipation easier. When dissipation from large area deflections, and conversion as heat fails, you start to get local elastic deformation, or if the material is brittle, inelastic deformation such as cracking. Either way deformation typically turns into the projectile entering the target material, and attempting penetration. This means with high velocity impacts, local stiffness is what counts, no time exists to spread the load elsewhere, the material fails before it can do so. Elastic deformation generally uses up more energy then cracking, so good metal armor is ductile to deform under pressure, and yet as hard as possible, without cracking, to make that deformation as energy intensive as possible. It fails by actually flowing, kind of like a weld bead, and then hardens back up. Hardness also increases the chance of breaking or just deflecting the projectile which are added bonuses. Note though that this isn't how special composite armors, which are generally very hard, typically work. They are more complicated, and are a bit like microscale reactive armor. As they break they force more mass into the projectile path.

When you have high momentum without high velocity, which is generally the only way you are going to get high momentum with low energy you make it much easier for an entire structure to dissipate energy, and it is unlikely that the projectile can ever muster the required local pressure to cause penetration. But it will start damaging the entire swath of the structure by the same rule of dispersion, crushing or shearing it at its weakest points. The ideal piercing weapon of course, has high momentum, high velocity, and high energy, and delivers it all onto the smallest possible point, thus the logic of sabot ammunition, and long heavy bunker buster bombs like GBU-28.

[Sky Captain]

Is there any optimal velocity for best penetration? Suppose you had some sort of rail or coil gun that could fire a DU penetrator at variable velocity from 1 - 10 km/s. Would it be always most efficient to set it to max or some lower velocity would be preferable. I suspect at very high speed a penetrator would explode and turn into plasma on contact with target. If target has several layers of armor only first or second layer may be punctured. A slower projectile may remain more or less intact after penetrating the first layer and continue on and penetrate more layers.

[Luke Skywalker]

The critical stress for material failure isn't constant - it can vary depending on the time frames you're looking at, and also whether the material is elastic (steel inside 'springy' range); plastic(steel beyond 'springy' range, clay); quasi-brittle (concrete) or fully brittle (glass). In short time frames (fractions of a second) wood is quite elastic in nature. Over very long time frames wood permanently deforms even at stress less then the critical (creep).

Does this mean that a high momentum impactor would be more effective against a block of wood, or do I have it wrong? Which is more effective against elastic materials?

[Hawkwings]

High velocity would be better at penetrating elastic materials, as they would not have the time required to deform.

You can also relate penetration to momentum: http://en.wikipedia.org/wiki/Impact_depth

[Luke Skywalker]

High velocity would be better at penetrating elastic materials, as they would not have the time required to deform.

So would this explain why a bullet can penetrate plate mail where an arrow of greater momentum cannot?

But then is Kevlar ineffective at stopping knife attacks? Or is this for a completely different reason?

[Irbis]

But then is Kevlar ineffective at stopping knife attacks? Or is this for a completely different reason?

Compare thickness of bullet and knife, for one. Smaller energy applied to line-thin area is still greater than larger applied to large round one. Oh, and knife doesn't run out of energy after reaching first barrier.

Then, there is the fact your typical knife is sharpened, hardened steel, bullets are just lead. Shoot kevlar with actual armour piercing bullets and they will penetrate just as easily.

That's why 'teflon' bullets have 'cop killer' reputation, btw - you coat much harder core of bullet with teflon to not damage barrel, meaning you can fire something better at penetrating than lead, not because teflon traces somehow magically enhance penetration.

[Beowulf]

Is there any optimal velocity for best penetration? Suppose you had some sort of rail or coil gun that could fire a DU penetrator at variable velocity from 1 - 10 km/s. Would it be always most efficient to set it to max or some lower velocity would be preferable. I suspect at very high speed a penetrator would explode and turn into plasma on contact with target. If target has several layers of armor only first or second layer may be punctured. A slower projectile may remain more or less intact after penetrating the first layer and continue on and penetrate more layers.

Yes, there is one. That is why US APFSDS ammunition hasn't gotten appreciably faster over the last several revisions, even though propellent technology has improved, and they could toss the same weight shell faster. Instead, they made the penetrator heavier, maintaining velocity.

[Luke Skywalker]

But then is Kevlar ineffective at stopping knife attacks? Or is this for a completely different reason?

Compare thickness of bullet and knife, for one. Smaller energy applied to line-thin area is still greater than larger applied to large round one. Oh, and knife doesn't run out of energy after reaching first barrier.

Then, there is the fact your typical knife is sharpened, hardened steel, bullets are just lead. Shoot kevlar with actual armour piercing bullets and they will penetrate just as easily.

But then why do these factors give knives an edge over bullets in penetrating kevlar or (IIRC) a block of sand, but not against, say, plate mail or concrete?

[Formless]

This is my understanding, but a knife basically applies a constant pressure, whereas a bullet releases a sudden burst of energy. The bullet tries to break through every layer of kevlar at once, but because of the density of the stuff, the toughness of each strand and the flexibility of kevlar, the kevlar strands together spread the force out to a surface area much larger than the actual bullet is. Suddenly, instead of applying energy over a surface area the size of a finger, its applying it over a surface area the size of a gloved fist.

To understand why a knife doesn't have this problem, think of that old arrow breaking trick that you see in some fantasy movies, like Mulan. One arrow shaft is easy to break. Two dozen arrow shafts tied together suddenly becomes as hard to break as a two by four. This is because like a bullet against kevlar, you are trying to break every arrow shaft at once, so the actual strength is the sum of every shaft put together. However, if you instead tried to cut through the arrow shafts with a chainsaw, they don't stand a chance. A chainsaw will chew through each shaft it encounters until its finished cutting through the whole thing. Its applying constant pressure until the job is finished, so the strength of them is suddenly a matter of the individual shafts again, not the whole. Likewise, a knife can cut through each individual strand of kevlar instead of trying to cut through the whole vest at once, and it applies this pressure until its finished cutting through every strand it has encountered and can pierce flesh. Thus, where a kevlar vest may be able to keep taking bullets, the same vest will eventually fail against a knife. The only resistance a kevlar vest can provide against knives is its sheer thickness.

Now, steel armor like plate or chain resists blades because its steel on steel. The blade isn't sharp enough to cut through the armor, and the two materials are approximately similar hardness. Further, the metal flexes instead of breaking, especially if we're talking about chainmail. A blade can cut or stab through plate mail, but now it needs to act more like a bullet and simply overwhelm the material with one sudden impact at the tip. With chainmail, a powerful enough blow with a sword can break individual links in the chain, but it dissipates all its energy in the process thus saving the wearer's life. Stabbing chainmail means that the blade has to force aside the links in the chain, so a thinner tip will go through easier; however, the chain can trap the blade after it goes through and make it difficult to pull back out. Again, this can save the life of the victim because the blade itself keeps blood from escaping the wound.

Now, sandbags are a different story altogether. You are looking at a granular impact, where the energy of the bullet spreads out very quickly into every single grain of sand in its path. As the Box of truth discovered, this will actually stop a high velocity bullet from a rifle faster than it stops a low velocity pistol bullet. This is because the rifle bullet's own energy is being used against it: it is releasing energy so quickly into the sand that it overwhelms the strength of its own materials. The jacket shreds and the lead core disintegrates into a hundred tiny shards. Those shards now have an exponentially lower sectional density than the bullet before it shatters, and the whole thing stops after only an inch or two. A pistol bullet on the other hand has low enough energy that it remains intact (intact enough they even speculated they could reload the recovered bullet). However, it still dissipates its energy in six inches of sand or less.

[Sea Skimmer]

Is there any optimal velocity for best penetration?

Depends on the target and the projectile. Fire a projectile too fast against too strong a target and it will breakup with sub optimal results. About nothing we can make today can avoid breakup when you start flinging it at much over 2km/s against modern armor; but this does not mean penetration becomes impossible. It means your going to hit diminishing returns and need vastly more energy for ever smaller results. After a point breakup turns into vaporization, which can destroy anything, but only for a huge price. Stuff vaporizes into vapor, this may or more likely may not involve any plasma generation. Depends on the materials.

But then is Kevlar ineffective at stopping knife attacks? Or is this for a completely different reason?

Kevlar fibers are not inherently ineffective at knife attacks, rather kevlar made into bullet proof vests are ineffective against knife attacks. This is because the weave and thread choice is optimized to catch bullets in a highly elastic manner that involves some fibers being torn, this is why the vests have 20-30 layers of fabric. With a knife that loose elastic design just makes it easier to cut through the fibers in turn and the point of the knife may easily go inbetween the threads. Many stab vest are in fact made of kevlar, and use larger threads which are far more tightly woven together. You could still saw into one with a knife given time, but they are very effective against sudden stabs which are likely to cause deep fatal wounding, the normal test is with an icepick. A slashing attack with a knife blade is not really effective against any kind of kevlar vest, but the main reason for stab vests is prison weapons which are almost all some kind of icepick like stabbing device with a very small point. Some companies do make armor piercing handgun bullets with very small needle like steel cores, which exploit the same weakness of kevlar.

Stab vests are much more rigid and the result is less, but not total loss of, protection against bullets. Protection against fragmentation actually very hard to do with kevlar, because explosive driven fragments can have sharp narrow edges, and yet far more velocity then bullets. Worst of both worlds. So this has meant that the vests and suits worn by bomb squad/EOD guys are yet different in turn, rated for neither bullets or knifes, but rather very specific fragmentation threats. I couldn't tell you offhand exactly just have they differ, but they tend to be rigid leaning.

Other kinds of kevlar armor exist too, though this should really be spoken of in the context of aramids, as kevlar is only a trademarked name of one specific type, the original formula of which is now outdated. The stuff used for vehicle armor for example is a lot different then any vest. The layers of fabric are bonded together rigidly with reinforcing resin. This among other things, reduces the ability of fragmentation to damage the kevlar fiber itself, and makes much larger areas of the armor bend under load, absorbing more energy. You can also cut the stuff with a power saw without ruining to make the pieces fit inside of doors or ship bulkheads or whatever.

[Sky Captain]

Now, sandbags are a different story altogether. You are looking at a granular impact, where the energy of the bullet spreads out very quickly into every single grain of sand in its path. As the Box of truth discovered, this will actually stop a high velocity bullet from a rifle faster than it stops a low velocity pistol bullet. This is because the rifle bullet's own energy is being used against it: it is releasing energy so quickly into the sand that it overwhelms the strength of its own materials. The jacket shreds and the lead core disintegrates into a hundred tiny shards. Those shards now have an exponentially lower sectional density than the bullet before it shatters, and the whole thing stops after only an inch or two. A pistol bullet on the other hand has low enough energy that it remains intact (intact enough they even speculated they could reload the recovered bullet). However, it still dissipates its energy in six inches of sand or less.

IIRC Mythbusters once tested bullet penetration through water and also find out that slower pistol and shotgun bullets could penetrate deeper while high velocity rifle bullets shattered on impact. So it seems that if the goal is to penetrate as deep as possible through soft materials more efficient is to use heavy but relatively slow bullets.

[Sea Skimmer]

The main reason pistol bullets will go further in water is because they normally have blunt noses which are enormously superior for traveling through water. A sharp nosed bullet wants to tumble and travel backwards, which may or may not cause breakup depending on the strength of the round. This is why torpedoes have such blunt noses among other things, and why the Japanese created special shells for underwater trajectories which had breakaway noses. They could travel several hundred feet underwater after impacting at rifle bullet like velocities. Blunt rifle ammo would have gone much deeper into the water then a blunt pistol round. The Soviets actually designed ammo like this to use against enemy frogmen and for certain underwater weapons.

In fact in the Mythbusters pool test, a blackpowder rifle firing a miniball was more effective then an AR-15, side effect of its much blunter projectile. But they were just testing the required depths for safety, not why it worked out the way it did.

[mrtspence]

I'm just a hobby physicist, so take what I have to say with salt!

My understanding is that, as kinetic energy increases relative to momentum (which it does quite rapidly as the projectile gets higher velocities) the strengths of the materials involved begins to become irrelevant. The materials start to behave more like liquids. This means that, as velocity increases, the projectile is more likely to break up on impact. At hyper velocities, you get explosions from the contacting materials turning into plasma--but that is a different story. Under 3km/s, metals still mostly behave like metals.

This means that small objects at very high velocities (their kinetic energy is very high while their momentum is much much lower) will be less likely to go very deeply into an object as they will instantly fragment on contact. This is why whipple shields (think tin foil with space between it and the regular armour) work so well on satellites--little tiny asteroids blow up on contact with it while the space prevents the shockwave from damaging the hull. If the whipple shield wasn't there, a 10g asteroid going 30km/s could inflict serious, serious damage.

More massive objects with less velocity are more likely to behave like Newton predicted. Someone already linked to that wikipedia page. Instead of breaking up on impact, they are more likely to retain their integrity and continue pushing through something until they are stopped. An arrow would sail clean through a whipple shield. However, this retention of integrity means that the target gets that same benefit.

This effect kind of scales down. This is why when mythbuster's shot fast rounds into water, they didn't penetrate--the rounds broke up as they had a lot of kinetic energy. Rounds with more momentum and less velocity (less relative kinetic energy), such as the big slow miniball, are not going to break up as quickly and will instead push through the target.

As far as an optimal velocity? It depends on what you are shooting. Slower rounds will pierce soft things better than extremely fast things, maybe. They might get more depth of penetration, but a hypervelocity round is going to take a huge spherical chunk out of something unless specifically designed not to (like the current KE rounds tanks use).

Modern composite and reactive tank armours could stop basically any non-hypervelocity round. This is because they can retain their mechanical strength, which is immense, and use special mechanisms to further protect from penetration. It doesn't matter that your ballista bolt is 200kg, if it is only going a few hundred meters per second, it isn't getting through that Abrams as it cannot defeat the internal mechanical strength of that armour. A round going fast enough is going to make part of itself and whatever it hits into plasma--doesn't matter what it is. Above 3km/s, your super-modern composite armour might as well be bronze, the part that gets hit is going to behave like a liquid anyways. A pop can going 30km/s is going to do far, far more damage to that tank than the 200kg ballista bolt.

Plus, when things go fast enough, there are some really really complicated plasma science magic that kicks in. I actually don't know how it works and I don't think many people do. It's possible that the whole projectile will become a plasma (think big spherical explosion, not a lot of depth of penetration). There is other talk that, depending on how the projectile is shaped, the first bit might become plasma while the rest pushes the superhot/superdense plasma through the target material (think insane, ridiculous depth of penetration). I think the second mechanism best describes how tank-fired anti-tank KE shells work. The long thin rod shape pushes through the tank armour and, once through, the plasma rapidly expands (boom). This would at least superficially explain why the armour plates pierced by the round have a thin little hole in them while the inside of the tank explodes!