Momentum vs. KE - which determines projectile penetration?

[Robert Williams]

In Traditional Archery, there is a widely held belief that momentum is the best indicator of an arrow's capacity for penetration. The most important support for this theory comes from this document: The Ashby Report<---Link to document (PDF)

This document has never been reviewed or proofed by anyone with a background in physics and is, in my opinion, so full of fundamental flaws and misapplied formulas as to render it completely useless and a second, third or 4th opinion from knowledgeable members of the Physics community would be very helpful (and very important, in my opinion).

What makes this a particularly troublesome theory is that it has convinced a huge number of archers to hunt with arrows that have high weight and low velocity out of fear that anything lighter and faster wouldn't provide adequate penetration. And this belief is rampant, causing a lot of bad shots due to inability to precisely place the looping trajectories on target.

Because the momentum formula gives equal importance to velocity and mass, archers feel their bow can generate more "penetration power" by increasing their arrow weight, thereby increasing the value of the momentum. They can't increase the kinetic energy, of course, because the bow will only generate X-amount of that.

Any help explaining whether kinetic energy or momentum is the best descriptor of the arrow "force" that generates penetration, and why it is the best descriptor, would be immensely helpful.

This topic has generated many thousands of pages of arguments and hundreds of very bad analogies like throwing ping pong balls and golf balls. Some real science and scientific explanations and a critical analysis of the Ashby report would do a tremendous amount of good and I think you in advance for any and all help in this matter.

p.s. Here are just a couple of the discussions to which I'm referring:

http://leatherwall.bowsite.com/tf/lw/thread2.cfm?forum=23&threadid=163035&category=
http://leatherwall.bowsite.com/tf/lw/thread2.cfm?forum=23&threadid=164228&messages=47&CATEGORY=5

 

[alvaros]

The key point is knowing where the energy of the arrow goes when it hits the target.

From: Biography of phisics - George Gamov :
If the velocity is of the bullet is high, most of its energy is wasted in moving away the mass of the target. In this case the penetration depends just on the length and density of the bullet, not on the velocity.
I don't know if this can be applied to archery.
( I didnt read your (large) links )

 

[phlegmy]

intuitivly
i think it depends more on KE

i'm considering 2 masses of different speeds with the same frontal area
(they could even be the same size but with different densities)[arrow heads??)

eg, a 10kg mass moving at .1m/s isn't going to do me much harm[not fatal at least]:
it has kE of .05
mom of 1

a .01 kg mass moving at 100m/s is going to hurt me more
my bodys inertia means its going to take me longer to match the speed of the projectile [which of course will be slowing down when it hits me]
untill i catch up its going to be pushing into me
has ke of 50
mom 1

and so while the momentum of both is the same
the KE of the light, fast, projectile is 1000 times higher and this is the one i fear!


i am thinking in terms of the targets {me} inertia.
if my body can match the speed of the projectile fairly quickly then a large amount of its energy appears in me as kinetic energy[of my now moving body]
if my body can't match the speed of the projectile quickly(takes me quite a while to reach 100m/s or even 50m/s) then most of the energy cannot appear in me as kinetic energy but instead will be used to tear my guts open (deformation of my body maybe a type of potential energy, are my guts elastic?)

if on the other hand, I'm somehow not free to move at all (i.e. abosrb the kinetic energy of the projectile as kinetic energy in the target) then maybe there'd be no difference.??

i'm thinking as I'm typing which is never good and this is just opinion!
but i do remember doing q's in secondry [high] school where a bullet hit a block tied to a vertical string and you'd to find the max angle the block made with the verticle, I'm pretty sure we found the velocity of the block by momentum transfer, then figured its angle from the conversion of ke to pe??

 

[phlegmy]

also for a given mass,
if you doubel the speed the momentum will doubel
but the ke will quadruple

 

[animalcroc]

also for a given mass,
if you doubel the speed the momentum will doubel
but the ke will quadruple

Interesting question that I believe is often underlooked.
I believe the answer is KE. Consider the following scenario. You have a spring fixed to a rigid wall at one end, with the other end free. The spring rests on a frictionless floor, so that it can be compressed or stretched horizontally. Now, let's say, you have a block mass of 1kg and you slide it toward the spring at 1 m/s (momentum of 1). It will hit the spring and stay in contact
until the necessary impulse of 1 is exerted on it. It will also compress the string a distance given by the formula KE=(1/2)KX^2 Now, let's say that the second time around you have a block mass of .5kg and velocity 2m/s directed toward the spring. The momentum is the same but this block has more KE. According to the formula KE=(1/2)KX^2 this block will penetrate further, namely 1.4 as much.

The key to understand is that that even though they had the same momentum the impact process was different. In the first case it was longer duration shorter penetration. The second case was shorter duration deeper penetration.

Although a spring force varies linearly with distance, the same concept should apply to any penetration

 

[animalcroc]

Heavier arrows are less prone to the effects of wind resistance.

 

[Griff38]

If the two arrows with the same momentum but different kinetic energies hit the target then since the average decelerating force acting on the arrow is equal to the rate of change of its momentum then if the heavier arrow penetrates further and takes longer to stop the force exerted on the heavier arrow must be less so therefore the force that the heavier arrow exerts on the target must be less (Newton's 3rd law), now I suppose this seems logical since if the two arrows are the same shape with the tips made of the same material hitting the same target then the resistive force would be greater acting backwards on the faster moving lighter arrow.

This is maybe something to think about:

The higher force acting against the motion of the lighter,faster moving arrow is doing negative work on it as it penetrates hence reducing its k.e and since the lighter arrow has less k.e. anyway it is not going to travel so far before the arrow comes to a stop.

The heavier arrow, traveling slower, has lower resistive forces acting on it so will definitely move further before the negative work done it by these resistive forces has reduced its higher ke to zero.

For the lighter arrow,

larger F x smaller d = k.e lost

For the heavier arrow,

smaller f x larger D = k.e lost

Since the heavier arrow has a smaller average f acting on it and more k.e. to lose the it MUST travel further. In fact even if the forces acting on both arrows was exactly the same because the heavier arrow has the most k.e. again it would travel further before it has lost all of its kinetic energy.

 

[Robert Williams]

Assuming arrows of different mass are launched with equal KE and lose none of it to drag in flight, they would arrive with the same KE. Now comes the consideration of resistance - but the resistance is not drag. Drag only occurs when an object is moving through fluid and our target is not fluid. In fact, it's actually a rather well lubricated elastic material that must be cut through.

 

[mordechai9]

Warning: I'm not really going to try and be brief, because I'd rather be thorough and careful, and it's an interesting question. I haven't read your articles you referred to, sorry, but I think this analysis may help you to get a better overall understanding.

First, let me try and clarify an assumption you have made. If I understood you correctly, you assume that the kinetic energy of the projectile is fixed and determined depending on the choice of the bow. Each bow has a fixed potential for energy transfer into whatever projectile is loaded into it. For example, a long bow will give 1000 Joules, a short bow will give 500 Joules, and that's that. Even a heavy arrow fired from the short bow will always have less KE than a light arrow fired from the long bow. This assumption seems accurate as long as the mass and geometry of projectile do not interfere with the bow's mechanism of energy transfer INTO that projectile. However, I imagine that an extremely heavy or extremely light arrow might indeed prevent an optimum transfer of energy, due to issues of imbalance and uneven distribution of weight.

Second of all, let me point out that the geometry of a projectile may be very important to penetration performance at subsonic velocities. You didn't mention this in your post, but I think you ought to keep it in mind. It is quite possible that the KE or momentum measurement is a moot point, when the geometry of the projectile may in fact be the dominant factor. So, we will also assume that the geometry is "good enough". In other words, the geometry is optimized for best performance, at all weights.

Taking these two assumptions, we have two possibilities:

1.) KE is the dominant factor in penetration performance. You observe that a more powerful bow always gives the deepest penetration. You observe that differences in arrow weight have no effect on penetration performance.

2.) Or, momentum is the dominant factor in penetration performance. You observe that a heavier arrow, even when fired from a weaker bow, outperforms a lighter arrow fired from a stronger bow.

As you can see, both of these options are unknown -- which one is correct, I don't honestly know -- you would have to figure it out based on experimental observations.

Is there a theoretical basis to say which factor is more important -- KE or momentum? Not really, because the most obvious answer is that they are BOTH important. Obviously, a heavy arrow fired very slow will not even penetrate the target, whereas an extremely light arrow with lots of speed may penetrate, only to come to a stop after very little distance. Theoretically, it is very hard to provide an answer which is little more than speculation. Penetration mechanics is a difficult and multi-disciplinary field, extremely dependent on engineering, materials science, solid mechanics.

Theoretically, I can try to give you the following basis for "intuition". Firstly, the heavier the projectile, the more "inertial force" it will possess. In other words, a very heavy object is very difficult to slow down. Secondly, the faster the projectile, the better it will be able to puncture a strong surface. This is because a faster projectile will generate a higher shear force along its sides as it attempts to "move through" the target. A slow, heavy projectile might knock the animal over like a sledgehammer; but a light, fast projectile is much more likely to shear right through the skin, albeit, perhaps not much further.

Based on this intuition, it seems that you need some kind of combination of KE and momentum to do the job. The KE must be sufficient to easily puncture the animal, and the momentum much me sufficient to make the projectile "go the distance". Ultimately, it's more of an engineering problem I think, and unfortunately, it's not as easy as saying "This one is more important".

 

[Robert Williams]

Actually, the assumption is that a GIVEN bow will have X-amount of potential, which will wind up as KE in the arrow being launched. How much KE will depend on the mass and velocity, of course, but it's typically a variation of less than 10% with a bow's efficiency being best at heavier arrow weights (because the bow's energy has to propel both it's limbs and the arrow, the heavier the arrow, the greater percentage of the bow's energy moves the arrow).

We are assuming, however, that projectiles of different mass have the same KE with velocity and mass being the changing variables. Or we can assume that the projectiles have the same MO with velocity and mass being the changing variables. Using both models, which would show the most difference between depth and capacity for penetration and percentage of increase or decrease of the respective value (KE or MO).

My suspicion, yet to be confirmed, is that Momentum plays little, if any role and that because (at least for one thing) an arrow is designed to cut through it's target; not pound or wedge. It requires energy to do the work... The arrow comes to a stop once the energy has been expended through work (or vibration or chemical energy). Momentum and energy are different values and it's the energy that's used for the work.

Momentum will tell a lot about the direction and momentum of both objects after a collision. Momentum is always conserved. It tells us more about how much of a change in direction and velocity of the target in an elastic collision than it does about how deeply the projectile will imbed itself in the target.

Intuitively, the lighter the target, the more of a hindrance a high momentum formula would be, allowing more loss of energy due to momentum transfer. Where a slow but heavy projectile will knock a tin can off a stump, a lighter, high velocity projectile can punch a hole all the way through it without even moving it.

This is not simple science, I'm afraid. If it was, there would be an expert answer immediately available, but instead it is something debated heavily.

I can tell you for certain that a 1730 grain fishing arrow traveling at 102 fps has the same momentum (0.78 slug feet) as a 500 grain arrow at a velocity of 350 fps. The latter arrow has an enormous amount of KE (over 100 foot pounds) and it's capacity for penetration is proportionally greater than that of the slower, heavier fishing arrow even though they BOTH have the SAME momentum. This should tell us that a momentum based formula for penetration would be fundamentally flawed to the point of being totally useless while a KE model would be useful but not necessarily linear in it's relationship due to other forces at work that change with velocity and mass.

 

[beprepn]

It is fairy easy to answer this question by integrating Newton's second law.

One finds that if a frictional force dominates, as one has when shooting a target where the resistance is caused by the target material squeezing the arrow shaft, that KE is correct.

If the resistance to penetration is proportional to the velocity, then penetration distance is proportional to the momentum.

And if the resistance is proportional to the velocity squared, then the arrow never stops and penetration is infinite.

This meets the common sense test as when the resistance increases with velocity, higher velocity arrows penetrate less.

The real world is not pure, of course, and the resistance to arrow penetration will be a mixture of forces. Personally, I like momentum for deer and KE for targets...or maybe 75% momentum and 25% KE?

At any rate, heavier arrows also have higher KE out of a specific bow as they couple to the bow better leading to a slightly higher initial KE and they shed velocity more slowly flying through the air (they penetrate the air better!) leading to higher KE's down range.

So heavier arrows are always better for penetration, all else equal, as long as one can hit what they are aiming at with them!

 

[nanoWatt]

On Mythbusters, they showed that a thatched reed going around 300mph can stick an inch or so into the trunk of a palm tree (which is much harder than an animal). If these straw-material which weigh only a few grams can do that because of their stiffness, you'd probably want a good balance of speed and mass.

I thought the KE formula was 1/2 mv^2. This means velocity is much more important to the overall energy (it has a square relation while mass is linear).

So you'd want do do some formulas to maximize the 1/2 m v^2 formula. However, you have to account for air resistance and such.

They could do this archery test with ballistics gel, and see how far the arrows penetrate.

 

[Robert Williams]

Interesting

There is a general assumption that increase in velocity means increase in resistance, which is true in projectiles moving through fluid (drag equation) but not for projectiles moving through solid or semi-solid objects. Animals such as deer and other vertebrates consist of muscle and bone, both of which are elastic even though muscle is much more elastic in nature. In neither case can the drag formula be applied because neither is a fluid (which has no form of it's own). In the case of elastic materials, a combination of higher velocity and lower mass is more efficient than higher mass and lower velocity. Penetration requires work and work is performed by KE. The problem with lower velocity projectiles and penetration of elastic materials is that elastic materials give and stretch and the slower the attempted penetration, the more stretch and give there will be before the mechanical process of severing the fibers occurs. The stretching and pushing expend energy that cannot be recovered and is not used for cutting or severing the chemical bonds of the target.

This is actually a very complex problem and one that is absolutely not "solved" with a simple and erroneous conclusion that with any given amount of KE in a projectile, the projectile with the highest momentum factor will penetrate best. That solution doesn't work and a logical test would be to use a pin to pop a moderately inflated balloon. An attempt to do it slowly but with great momentum will only push and/or stretch the balloon but with sufficient velocity, it will not move or stretch but be punctured/penetrated, instead.

 

[hArryMarx]

Poncelet developed equations to explain this. I played around with it and found some interesting insight. Ek is mv^2, while momentum is mv^1. Poncelet penetration in bone is proportional to mv^2, while in flesh mv^1.2 ! This is due to the the respective friction experienced, that the importance of mass vs. velocity alters. "beprepn" hinted at this in his post. Ek and momentum represents the extremes of the answer and therefore opinions fluctuate between the two. In reality, the target defines the relative inportance, and of course, targets are generally not homogenous.

 

[amppatel]

well personally i think momentum has nothing to do with it, at all, i am mp wrong, but the initial energy of the bullet ie the k.e energy is more important, as the how far the arrow goes inside is Resistive Force X Distance which will equal the K.e before so if you increase K.e then you have to increase the distance as the resistance is constant. i think, lol

 

[Robert Williams]

I agree, amppatel. The mistake in assuming that momentum is the primary predictive indicator in penetration is that it is based on an errant postulate that penetration of animal tissues is modeled most effectively as a fluid and that drag forces are the primary resistive force encountered.

As a point of fact, animal tissue is solid with both elastic and plastic properties. Resistance to shearing does not increase with a projectile's velocity and actually decreases in elastic mediums. For a given amount of kinetic energy, a projectile can have an infinite variation of velocity and mass configurations to equal that amount of kinetic energy but it is the energy, itself that determines the capacity for work and not the momentum.

While it may be difficult to comprehend that 125 grains traveling at a velocity of 400 feet per second can have the same potential for damage as 600 grains traveling at only 180 feet per second, the fact that they both have approximately 43 foot pounds of KE tells us that despite a very large difference in momentum values, the actual capacity for penetration will be nearly identical, although some variation should be expected due to the complexity of the problem. In order to penetrate, both projectiles must cleave/shear a path for themselves and this is work. The amount of work to be done determines the energy requirements and this requirement does not increase due to increased velocity. It will, if anything, be reduced, depending on exactly how elastic the material to be sheared will be and how much mass the object to be sheared will have. The slower the velocity of the projectile and the lower the mass of the target object, the more energy will be expended moving the object rather than shearing the object.

 

[Proeliator]

An old topic but I thought I'd add to it abit.

First of all comes the question: What determines how hard a projectile hits its' target?

The answer is Kinetic Energy (KE), and just like drag, KE increases with the square of the speed; i.e. an object doubling its speed has four times as much kinetic energy as before.

The kinetic energy of an object is also directly related to its momentum using the formula:

p = Momentum
m= mass

And as already pointed out you could ask yourself the following; What is the more dangerous to stand in the way of:

A 20 ton truck moving towards you at 0.1 m/s? (Momentum= 2000, KE= 1 Joules)

or

A 20 kg 1 sq/m metal plate moving towards you at 100 m/s ? ( Momentum= 2000, KE= 1,000,000 Joules)

Now the truck you can sort of prepare to come into contact with, bracing yourself by acquiring a good stance, and it really won't hurt you much; Ofcourse providing the engine isn't running, as then it would naturally just run you over ;) Whilst nomatter how much you brace yourself a 20 kg 1 sq/m metal moving towards you at 100 m/s is going to slam you into pulp.

Next comes the question of penetration of solid objects;

If hardness & density are equal, and a higher penetration is desired, then Kinetic Energy spread over as small a surface area as possible is what you need to be looking at.

It is for this very reason that the main armour piercing projectiles fired by todays tanks are long, slender [subcaliber] arrow like tungsten projectiles moving at extremely high velocities, instead of massive full bore projectiles moving at much slower velocities.

As for drag, it is true that it affects lighter objects more than heavier ones of the same size & shape, and that is why mass is important for projectiles, cause the more of it you have the better the energy retention over any given distance will be; again assuming size, shape & surface is the same ofcourse.

You cannot however compare air [a fluid medium) resistance [drag], with the resistance to penetration of solid objects such as flesh, bone and steel. And this because of effects such as plastic, elastic and elasto-plastic deformation, which do not occur when objects slip through the air. The distance at which a projectile will travel through a solid object depends on many of the same factors however, such as weight, shape, size & speed, albeit with different emphasiz put on each factor and now with the added importance of both projectile & target hardness.

 

[mrspeedybob]

It seems like this would be a very easy question to get a definitive answer to.

Take a deer carcass, shoot it with several different weight arrows from the same bow. Post the results and end the discussion.

 

[pervect]

I'm not sure about arrows, but I recall looking up some information about armour penetration by anti-tank rounds way back when. The title I was referred to, which had some actual experimental information was something like _the design of armoured fighting vehciles_ , but I wasn't able to remember or google for the exact title.

The conclusion was that it requires a certain amount of energy to displace the armour enough to create a certain size hole - so for penetrating armour, the energy is what's needed.

Thus fast, long, and dense rod penetrators are the historical projectiles of choice for this purpose.

Wiki has some info, see

The title of the article gives away the conclusion - "kinetic energy penetrators".

http://en.wikipedia.org/w/index.php?title=Kinetic_energy_penetrator&oldid=395763942

 

[DHag]

This question really cannot be answered completely by physics alone. The affect an arrow has on its target has far too many variables involved. Based on what I learned in getting my Chemistry/Physics degree, I suspect that if two arrows, identical in every way except for mass, are shot from the same bow under controlled conditions, the penetration would be pretty much identical shot after shot. The heavier arrow would likely "win" at long ranges, but bowhunters generally take 99% of their shots at ranges of 30 yards or less. Also, the heavier arrow would show less drift in a cross-wind, but that is not part of the penetration question.

Perhaps now the question that is more important is whether today's equipment makes the penetration issue moot.

I took my first buck over 40 years ago, at the ripe old age of 13, shooting a recurve by instinct that drew 50 pounds at 31" of draw length. It's velocity was perhaps 120 fps, shooting 3/8" diameter cedar arrows with glue-on Bear Razorhead broadheads. With my 20-yard shot, the arrow penetrated about 8 inches, but the broadhead did its job and left a blood trail that was easily followed.

My biggest problem was that deer would often "jump" the string, and I would watch my arrow pass harmlessly behind the deer's butt. Penetration was rarely an issue to me. ;-)

From that bow, I moved up to an early Pearson compound with round wheels. Speed with just-as-fat-but-lighter aluminum arrows jumped to about 160 fps. Penetration was better, but mostly from better broadheads, not from great oodles of speed. And deer still jumped the string. I was using my first sight that had a pin for every 5 yards of range. I could actually shoot a consistent 6" group at 30 yards!

The I got a twin-cam (cams!) Browning, almost 200 fps. The greatest improvement in penetration came from the introduction of 125 grain Muzzy broadheads with "Trocar" tips and replaceable blades. I would occasionally have an arrow penetrate so its head was showing out the other side. I had one pin for each 10 yards of range on my sight. Deer still jumped the string, but not as badly with the addition of big fuzzy balls of fluff on the string. Down to a 4" group.

Then came a single-cam PSE Whitetail Extreme, making 235 fps with early-style carbon fiber arrows, and 250 when I could afford the new "skinny" carbons, at only 15 grains per inch. Having a deer jump the string finally was only an occasional occurrence with "Limb Savers" and "String Leeches." Penetration was now 90% clean pass-throughs. Adding a fall-away rest meant that two-inch groups at 25 yards became my norm.

So here I am today, now with a new PSE Stinger that (with my long draw length) clocks at 302 fps, using carbon arrows at 10.9 grains per inch and 100 grain Muzzy broadheads. Traditional fletching has been replaced by FOBs. Arrow weight is almost down to the test standard of 5 grains per pound of peak draw weight. I have one pin on my peepless sight for shots out to 30 yards. Two more pins extend the range to 45 and 60 yards each. At 45 yards I have to shoot at separate spots on the target, because I now shoot 1-inch groups at that range. Even though the arrows are lighter than ever, I don't have to compensate for wind drift. The arrows are so fast, and the ventilated broadheads and FOBs catch so little wind, that even a 30-mph cross-wind isn't a a factor anymore.

I still take 95% of my shots on deer inside of 25 yards. Half of them inside 15 yards. Every shot is a clean pass-through, even through a shoulder blade. The Muzzies are simply devastating. With the silence of parallel limbs and a string stop, plus raw speed, deer don't jump the string. Ever.

I think the whole question of arrow weight vs. arrow speed is nice for discussion, but absolutely unimportant with today's equipment. Nowadays the determining factor in getting a good, clean kill has everything to do with hunting skill and shot placement.