Momentum vs kinetic energy for dummies:
How a traditional bow kills

 

By Francois Squirra

While reading the Frequently Asked Questions in the September 2008 issue of ABH&A, my heart stopped when I saw the table headed Generally Accepted Kinetic Energy for Hunting (see Table 1).

I hunt with a traditional bow and when I looked at the table, I realised that I was restricted to hunting animals in the 40 foot-pound group. For an aspiring bowhunter these are the most difficult to hit, if you encounter them within shooting distance, that is.

Fortunately I had seen this table before and was able to revive myself by recalling my physics classes. In recent issues of the magazine writers have attempted to describe tissue penetration and mechanical advantage and some have even conducted tests with broadheads to determine their structural integrity when shot through bone. Most of these tests were applicable to compound bows and the broadhead-arrow combination commonly used in them. Wouldn’t it be great if we could set standard criteria for hunting commonly occurring animals in South Africa with bow and arrow?

For the purposes of this discussion I would like to confine myself to animals commonly encountered by hunters in South Africa (smaller than 60 foot-pounds group), thus excluding big animals and the giraffe (those in the greater than 85 foot-pounds group), although the discussion is just as relevant for them.
Firstly, although quoted in Table 1, kinetic energy is not a very intelligent way of measuring minimum requirements for ethical hunting. Let me give you an example: I have a rifle that produces 175 foot-pounds of kinetic energy. Would you say that I could hunt a buffalo with it, according to Table 1? Yes. The only problem is that the rifle is a beautiful 1947 .22 calibre Oberdorff Mauser.

Kinetic energy can be misleading when considered on its own as a standard criterion for hunting. It should rather be used together with some other quantity (or quantities) to give a more accurate description of a projectile’s ability to penetrate an animal sufficiently and kill it ethically. The other problem I have with kinetic energy is that it dissipates too quickly. An arrow loses velocity, but seldom loses mass before hitting the target.

But wait, I’m getting ahead of myself. Let’s first have a look at how kinetic energy is defined. Kinetic energy describes the energy a body of mass possesses because of its movement. Mathematically it is defined as KEnergy = ½ m v2, where m is the mass of the body (let’s call it an arrow from now on) and v is the speed of the arrow.

You can see that as the speed of an arrow increases, so does its kinetic energy, but quadratically. For simplicity, let’s ignore the units for kinetic energy and use an arrow weighing m=2 and moving at v=2, thus producing four units of kinetic energy. By doubling the arrow’s speed to v=4, the arrow would produce 16 units of kinetic energy. Wow! No wonder kinetic energy is so popular.

Let’s consider another quantity describing an arrow’s movement – the momentum of the arrow. This is defined as the mass of an arrow multiplied by its velocity (not the speed – velocity is directional and speed is not). Try remembering your high school physics (vectors and scalars ring a bell?). That’s why I like momentum; it seems to have some purpose and direction in life. Mathematically, it’s described as M = mv, where v is the velocity of an arrow.

40 foot-pounds

50 foot-pounds

60 foot-pounds

85 to 95 foot-pounds

105 foot-pounds

Rabbit, steenbok, jackal etc.

Impala, kudu, blesbuck, warthog etc.

Gemsbuck, eland, red hartebeest, wildebeest etc.

Rhino, buffalo, giraffe

Elephant

Table 1: Generally accepted KE for hunting.

Momentum is also dependent on the velocity (speed) of an arrow but linearly, meaning as velocity increases, so does the momentum. The same applies for increasing mass. The difference between kinetic energy and momentum is that kinetic energy is almost only dependent on velocity (speed), but momentum is equally dependent on both mass and velocity. Hold this thought and let us continue.
Remember I said I was getting ahead of myself? Well, I caught up. I would now like to compare these two quantities in terms of bowhunting and arrow performance and then conclude by proving that a traditional bow, although not meeting the minimum requirements of Table 1, can ethically kill an animal when momentum is also considered.

So what would be the better way to describe an arrow’s performance, kinetic energy or momentum? (Kinetic energy is a good way of describing a bow’s efficiency. The ratio of kinetic energy to potential energy [stored energy] gives bow efficiency.)

I said that kinetic energy dissipates quickly. It’s true. The sound of the arrow in flight, spinning, friction on the shelf and a lot more, draw their “existence” from the arrow’s kinetic energy. Energy is “directionless” and likes to give itself away. By increasing an arrow’s speed it would seem that you can acquire enough energy to hunt anything (see equation 1). But be aware of penetration and here I quote Dr Ed Ashby: “As the arrow’s velocity is increased the resistance does not increase equivalently. The resistance increases exponentially. The resistance of a medium to penetration is reliant on the square of the object’s velocity (assuming objects of a given coefficient of drag; i.e., using arrows with the same external profile, material and finish). In other words, if the arrow’s impact velocity doubles, the resistance increases by a factor of four. If the impact velocity quadruples, the resistance to penetration increases 16 times.”

So, what would you rather increase? Mass. It takes force to get mass moving, but it also takes force to stop it. Think of a car accelerating to 100 kilometres per hour. Direct the car at a brick wall that is say, five metres thick. Now accelerate a train to speed so that the momentum of the car and the train is the same. Hit a similar brick wall. What do you think will happen? What will penetrate deeper into the wall?

As I mentioned previously, an arrow does not lose mass. For the sake of penetration, if two arrows have the same momentum but different mass, the arrow with the bigger mass will penetrate better. That’s the thing about momentum; a heavy object moving slowly is much harder to stop than a light object moving fast.

This is where momentum gains my favour over kinetic energy. Momentum is equally dependent on mass and velocity as mentioned before. Yes, momentum is also increased by increasing the velocity of an arrow, but as I stated above, rather gain momentum by increasing mass. Velocity hampers penetration. It is by far better to use an arrow’s momentum to determine ethical minimum criteria for hunting than the kinetic energy. Like all things, there is some kind of optimum weight velocity ratio to be used. We will leave this for another article.

To achieve the second objective of this article, I’ve rewritten Table 1 into a format using momentum rather than kinetic energy. I’ve obtained the required momentums for hunting from previous issues of ABH&A. Where Cleve Cheney quoted momentum I’ve used it and in some cases adopted it for other animals for which I could not find minimum criteria. I used a similar required momentum for all animals with similar body mass and structure, like red hartebeest and black wildebeest, and I’ve put the kudu with gemsbuck and blue wildebeest. The eland, although a female kudu, in my opinion should rather fall into a lower category.

In Table 2 the minimum momentum required to hunt the “common” species in South Africa is listed. This table is not foolproof and much research still has to done. For instance, I have not considered the tissue penetration index (TPI), where the shape of a broadhead makes quite a difference when penetration is considered.

Species

Minimum momentum (in slugs)

Eland (?), gemsbuck, blue wildebeest, kudu

0,4

Red hartebeest, black wildebeest, tsessebe, nyala

0,38

Impala, blesbuck, bushbuck, reedbuck

0,35

Springbuck, mountain reedbuck and all small game

0,3

Table 2: Minimum momentum required for hunting the common species in SA.

Arrow weight (grains)

Velocity

(feet per second)

Kinetic energy (foot-pounds)

Momentum

(slugs)

511

167

31,6

0,38

810

130

30,4

0,47

490

179

34,9

0,39

511

180

36,8

0,41

810

140

35,2

0,50

466

160

26,5

0,33

Table 3: Kinetic energy and momentum from three traditional bows.

On this, I would like to make a proposal: the momentum describing the penetration capabilities of an arrow should be replaced or used in unison with the TPI of an arrow-broadhead combination. The broadhead used will, together with the momentum of an arrow, set a very reliable benchmark for establishing minimum hunting criteria. I would like to set up such a database, but this would require data from hunters (see notes).

Back to my traditional bows. However much I hate sending an arrow through a chronograph, I had to, just to prove that a traditional bow can, in modern times, still kill an animal ethically. It has been doing so for a few thousand years already. Here is a table containing the kinetic energy and momentum of two medium-weight traditional bows, a 53 pound longbow and a 54 pound recurve. To make it interesting, I have added the statistics of my wife’s 40 pound recurve (in italics). Table 3 contains the data.

To conclude, it seems, as mentioned above, that momentum is a different and perhaps more accurate way of describing an arrow’s performance when penetration and killing capabilities are considered. Using a very sharp broadhead and placing the correct arrow, momentum-wise, in the correct spot on an animal will result in success.

Furthermore, even a slow bow can effectively kill an animal if the correct weight of arrow is used, together with a good cut-on-impact broadhead.

Notes:
In the above discussion I’ve left out quite a few technical discussions, as well as some of the not too difficult physics. I would like to refer those interested in more technical information to the writings of Dr Ed Ashby. See acknowledgements.

Formula for calculating kinetic energy:
Kenergy = ½ Mass (in grain).velocity2 (in feet per second)/225218 foot-pounds

Formula for calculating momentum:
Momentum = Mass (in grains).velocity (in feet per second)/225218 slugs

I would appreciate it if readers could send me info on animals hunted. Arrow weight, broadhead (if you can calculate mechanical advantage it would help), penetration depth, arrow speed and shot placement.

Acknowledgements:
• Dr Ed Ashby. Google him with the keywords: TPI, momentum, bow hunting or e-mail me for some of his articles at fphuntingsaf@mweb.co.za.

• Africa’s Bowhunter & Archer magazine for previously published articles.

Updated: Wednesday, October 22, 2008 10:53 AM