Kenetic energy vs. Momentum

[Doug Treat]

I know this has been beat to death on here and I've read some peoples' comments that KE and momentum are the same thing.  You increase speed and weight and both will go up.  That's true.  But what if I have one bow and I want to know what is the best arrow I can shoot out of this bow to increase my penetration and increase the lethality of my arrows?  Let's say for hunting bigger game- elk, moose, elephant, etc.  What do I use? KE or momentum or something else? I've noticed a lot of shows on TV where whitetail deer are shot with high speed bows and the arrow waves like a flag as it runs off. Extreemly POOR penetration.  What went wrong? In my opinion, there should be no reason that a decent compound (or traditional) bow should not blow an arrow straight through a whitetail, even if a solid bone is hit. They just aren't that tough. So, what can we use as a guide?  How can we predict what our current setup will do?  Here's a little math scenario that may illustrate my point and you can draw your own conclusions.  I will exagerate my scenario to make a point:  You have 2 arrows.  Arrow #1 is the size of a .22 bullet (I know, it's not possible, but bear with me here).  It is 40 grains and we can shoot it from our bow at a blazing 1,255 fps., just like a .22 long rifle bullet.  Arrow #2 is a normal arrow that weighs 650 gr. and our bow will shoot it @ 155 fps, a normal speed for a fairly light-weight traditional bow.  How do these 2 projectiles compare in Kenetic energy and momentum?  The .22 sized arrow has 97.4 foot-pound of kenetic energy while the 650 gr. arrow has 34.6 foot pound of energy.  If I use KE as my guide then my 40 gr. arrow will penetrate 3 times better than my 650 gr. arrow (all else being equal of course).  Now, lets use momentum:  The 40 grain arrow has a momentum of 7.2 pound-feet/second of momentum and the 650 gr. arrow has 14.4 pound-feet/second of momentum.  The heavier arrow has exactly twice the momentum as the lighter.  Wait a minute here.  These 2 contradict each other.  How can one have 3 times the KE and only half the momentum.  Because KE glorifies speed as KING (speed x 2 x mass)while momentum give speed and mass equal attention. Kenetic energy, although useful when figuring knockdown power for shock in a rifle load, should not be used when trying to figure the lethality (penetration) of a certain bow/ arrow setup.  Sorry this was so long winded and I guess I really didn't let you "draw your own conclusions" like I said I would, but I still here a lot of people saying "with this setup, I'm getting great KE".  I say, "OK, but what is your momentum?"  What do you think.

[Doug Treat]

OOps, I mean, What do you think?

[katman]

I am a momemtum type of guy.

[Todd Hathaway]

You seem to be saying that your arrow will get more penetration than a .22 round. Would that be the case if the bullet did not deform on impact?

[Al Dean]

I don't see your formula as right.  (speed x 2 x weight) makes 650gr arrow @ 155 fps, 2 to 1 better than 40gr bullet @ 1255fps

[John57]

10g per LB of bow wheight,a razor sharp broadhead,an forget about all the longwinded theorys.

[Longdraw]

Sorry Doug, but you've got your physics a little scewed up.

KE = MxVxV/2

P (momentum) = MxV

Aside from using an incorrect formula your scenario fails to account for bow efficiency.  No way a very light projectile will have enough icreased velocity to maintain either the energy or momentum of a perfectly matched bow arrow combination.  The curve just is not linear.

All that being said, I tend to agree. Momentum is a much better measure of a projectiles resistance to change in motion.  In fact that's exactly what momentum measures.

- Vic

[BTH]

I agree with John. 600+ grain arrow, razorsharp two blade broadhead, and the ability to put that broadhead where it belongs. Just shoot.

[Doug Treat]

I think that, although I didn't express my formulas right, my numbers are right.  I used a chart (put out by Jennings archery) to get my KE numbers.   Todd, yes a heavy arrow will outpenetrate a .22 bullet easily, even if it's a metal-jacket (not-deforming), esp. if bone is encountered.  Yeah, I know, it's more fun to "just shoot", but I have this thing about trying to make a quick humane kill if possible.  My experiences have shown me that with elk and mule deer, heavy is better, but heavy to some guys I've talked to is 500 grains while others see heavy as 700.  There may be some out there, just getting into this, that could benefit from knowing what to start with and then see how it works.  I think most compound guys could afford to shoot much heavier arrows than they shoot, to protect their bows as well as penetrate better.  I've seen well known hunters on TV brag about the great penetration that their super fast bow got on a deer, while holding the arrow about 5 inches behind the BH.

[Doug Treat]

I guess I should say "while holding their arrow 5" behind the BH to indicate where that 'great penetration' penetrated to."

[JImmyDee]

Ignoring the gravational constant, the problem with your interpretation of the expression of kinetic energy is "speed times two" -- it should be "speed squared" and we say that "kinetic energy varies quadratically with speed."  In other words, doubling speed *quadruples* kinetic energy.  (I wouldn't say "kinetic energy glorifies speed," it's just one of many measurements.)

We should also note that the resistance encountered by the projectile upon impact also varies exponentially with speed.  You can think of it this way: the faster the arrow is moving, the faster-er it will slow down on impact.  This is why a pail of sand will stop a 30 caliber bullet when an arrow punches through; a 22LR might  fail to get through the near-side of a hog while an arrow will probably (at least) cut a lung; bullet-proof vests don't offer much protection against knife stabs.

The guys that design ships and airplanes got it figured out in their world: resistance encountered by a wing or hull or propeller moving through a fluid is a combination of friction and intertia.  The friction is simply that encountered by air or water moving over the surface and the inertia is the force necessary to move the air or water out of the way.

The same applies for us: resistance to arrow penetration is a combination of friction and inertia.  However, the body of a game animal is much different than air or water or ballistic gel: it is not at all uniform and there is no simple formula that gives the "right" answer.

I think you're on the right track, but I think you're only asking half the question.  Remembering that the momentum you calculated in yout post is the product of mass and velocity, I think you'll  realize that momentum alone won't tell the story any more than will kinetic energy alone: there is a minimum mass -- determined by the game species and shot placement -- that you must put on target for reliable kills.

The weight cited in the Dr. Ashby reports (on this site), IIRC, is about 650gr.

[Doug Treat]

Jimmy, I agree.  Momentum is not the last word in arrow penetration, but a more accurate scale to use than KE, IMO.  I also agree with the "Ashby threshold" of 650 gr. for heavy bone penetration.  I heard that he is also going to do some extensive testing in the near future with using 650+ arrows with light bows in the 45# range.  This will be very interesting to me as that seems to be about the extent that my shoulder will tolerate.

[strick9]

I also believe momentum to be a better judge for penetration . KE=velocity squared times mass/450240 not 2..got caught in the speed trap recently and finally got her quietend down,,,193 fps with 571 grn pushing to 586 grn this week  and will hopefully be 188 fps and quieter... gotta have more rubber on the pavement....

[hawgslayer]