M vs. KE, why carbon better for target penetration

[beprepn]

With apologies, I model physical systems for my job. What we do is take measurements and do the math and use the math to understand the measurements.  And I know that most posts that start out this way are BS. And I don't think the following is a particularly good way to think about penetration, but it does explain the carbon arrow youtube comparisons that make light arrows look better than they are.

I have integrated Newtons second law from the surface of an animal to where it reaches zero velocity - the penetration distance - for different forms of the resisting force.

The results are different for friction - such as slowing down an arrow by squeezing on the shaft - and for "plowing" resistance - such as a boat moving through water.

This is friction in the physics sense - friction forces oppose the motion and are independent of the velocity. For frictional forces the correct variable to calculate penetration is KE.

For plowing forces, forces which increase in strength with velocity, the correct variable is momentum.

So if I shoot a light carbon arrow and a heavier aluminum arrow into the same target, since friction is the primary force opposing penetration, the carbon arrow is not penalized for having less momentum. But it does benefit from having a smaller shaft such that the target squeezes on it less hard and the frictional force is much less. It penetrates much farther and a new industry is spawned and a million deer are lost due to lack of penetration.

For an arrow penetrating a deer, we do everything we can to minimize friction. I believe that the other forces that oppose the arrow's movement are proportional to the velocity of the arrow. For example, the arrow must move the flesh out of the way of the arrow - the faster the arrow is moving the more violently the flesh is thrown aside and the higher the force to do this. The calculus says that for forces proportional to the velocity the correct variable for predicting penetration is momentum.

Thinking about it, the light arrow is moving faster so the force opposing it is higher (since the assumed force is proportional to velocity) - the light arrow is penalized for its higher speed. Consequently the light arrow loses more energy per inch of penetration than does the heavier arrow.

Now things get weird. If the forces slowing down the arrow were all proportional to the velocity the arrow would never stop! As the velocity went to zero, the forces would go to zero, and the arrow would continue to move forever, although too slowly to see after a while. Of course there is always some friction so the arrow does stop - at least in the dirt.

And, for the sake of full disclosure, the fact that "plowing" forces are proportional to the velocity is from a web site that talks about modeling boats plowing through water (for game simulations. This kind of explains the last paragraph - a big boat really does take "forever" to slow to a halt.

F = ma = change in momentum per unit time

impulse = F times delta time = change in momentum

It is almost dawn.

[LittleBen]

Although I think you've lost your mind and should sleep more (same for all engineers, myself included) I think you have some interesting points.

The thing I take from this is that if the "plowing resistance" is the primary resistance to an arrow penetrating flesh, then the sharpness of the broadhead should be just as important as momentum.

[beendare]

I've been a heavy arrow guy for a long time now- hard to argue when you see them on heavy game like water buff.

I think your plowing theory makes some sense. We are dealing with a varying density target in animals so that may explain your discrepancies.

[Wallydog]

Its my understanding that resistance is squared in direct relationship to velocity. In the real world heavy slower beats lighter faster everytime when penetration is the only consideration. I rarely take more than a 20 yd shot bowhunting. And my 630 gr arrow will pass thru a large deer after hitting bone everytime unless the braodhead integrity wont tolerate the momentum generated and sunsequent impact of the shot. This may be a clue as to why light fast arrows with mechanical heads dont penetrate heavier game. If the resistance is squared to velocity then the  deployment violence of the initial imapact of a fast mechanical head on heavier game should certainly be questioned.

[Mike89]

The other thing to keep in mind is that these equations are just approximations.  I imagine as the arrow comes to a stop, different parts of the shaft and head will sort of "catch" flesh, stretch it to a point and then release as it cuts or slips past-

I guess my point is that is that real world is more complex than any equations we can come up with, so you may have to add something to your model to stop it below a certain speed to make it behave appropriately.

[amazonjim]

I plowed with one mule, but with two the plow seemed to go through the dirt easier.

Interesting, above my head but interesting.

[halfseminole]

The embodied energy of the heavier arrow is higher.  We know there is a difference between a wad of paper thrown at us and a baseball thrown at us, even if the person exerts the same amount of effort.  Air resistance affects the lighter one more, as there's not enough mass to counteract its effects-so while the carbon arrow might be faster, external forces are already working harder against it in flight.  Without the additional mass behind the head, it doesn't matter how fast it is or how sharp it is-it's not going to penetrate as well because of lower embodied energy.

I heavily prefer bamboo shafts with tanged heads-quite a lot of weight for an arrow.  The embodied energy however is very high-all that weight behind the head gives it quite an impact.  We have to realize that the head is important for cutting, but the mass behind it plays an important role.  The more overall weight, the better the overall penetration as the arrow is not as susceptible to loss of momentum upon impact.

There's a lot of fun formulas to calculate that too, but it's easier to just work off the rule of thumb that the harder you want to hit it, the heavier the arrow ought to be.  I did physics for a living for a while too, but all those fun formulas really point back to the same thing here.

[nleroux2]

Thanks a lot! Now I won't get any sleep tonight. I'll be trying to determine the skin friction of an aluminum arrow vs. a carbon arrow.

[Marshallrobinson]

The broadhead ferrule size, broadhead cut area/ratio and shaft diameter of the arrow can really upset the assumption that it is simply heavy verses light.
A skinny carbon such as an old AFC with an outsert, combined with a broadhead ferrule that exceeds the diameter of the outsert is going to be a far better penetrater than a common shaft of the same weight for which the components are matched to form a perfect surface, broadhead design being the same.
Especially when a non-hollow bone (such as the scapula)is the subject of penetration.

With this being a fact, it is only a matter of finding just how much lower the thinner shaft can go in weight, before things become equal again.

[315stickbow]

scroll down to the bottom. should give you some good reading =]  http://www.alaskabowhunting.com/Dr.-Ed-Ashby-W26.aspx

[315stickbow]

particularly  http://www.alaskabowhunting.com/Ashby-On-Momentum-Kinetic-Energy-Arrow-Penetration-W19.aspx