** ARROW LETHALITY**

Part
IV: The Physics of Arrow Penetration

BY

Dr. Ed Ashby

Kinetic energy, momentum,
mechanical advantage and coefficient of resistance are a part of the basic
terminology of physics. All are used,
and often misused, in attempts to predict terminal performance of various bow,
arrow and broadhead combinations. Much
of the misuse originates from a lack of understanding of what, by definition,
these terms mean and what it is they measure.

In the terms of physics,
all broadheads are classes as "simple machines". As such, all broadheads are no more than a
series of inclined planes. The
mechanical advantage (M.A.) of a "simple machine" is the ratio of the
resistance to the effort. The
mechanical advantage of an inclined plane is equal to the length of the plane
divided by the height of the plane.

A single blade broadhead,
with a straight taper, 1" wide by 3" long can be viewed as 2 inclined
planes, each of which has a mechanical advantage of 6.0 (3" divided by
1/2"). The mechanical advantage of
the two planes combined would be 3.0 because the height would be doubled while
the length remains the same. What this
means is that with an exerted force (effort) of 1 pound, a weight of 3 pounds
can be lifted from the tip of the broadhead to the back edge of the
broadhead. The higher the M.A. the more
work a broadhead can do with the force available.

To determine the
mechanical advantage of any broadhead with a straight taper to the cutting
edge, divide the horizontal length of the cutting blade by 1/2 the width of the
broadhead (or, more precisely, the distance from the central axis of the arrow
to the highest point on the plane) multiplied by the number of blades. In an equation this would be expressed as:

M.A. = __ Length of cutting blade
__

(1/2 width of head) X (number of blades)

Example #1

As stated above, a single
blade broadhead 3" long by 1" wide has a mechanical advantage of
3.0. If that same head has three
blades, the M.A. would be 2.0, ie: (3" length/.5" lift distance X 3
blades). If it had four blades, the
M.A. would be 1.5, or one half that of the single blade.

Example #2

In a broadhead with a
cutting edge length that is 2.25" long and with each blade .75" high
(a common dimension) the M.A.'s work out as follows:

Single blade head =>
M.A. = 1.5 (Note: this is 1/2 the M.A.

Three blade head => M.A. = 1.0 of the 1" X 3" single

Four blade head => M.A. = 0.75 blade broadhead)

Five blade head => M.A. = 0.6

Six blade head => M.A. = 0.5

In example #2, a single
blade head would be able to 50% more work than a three blade broadhead with the
same applied force. It does 100% more
than the four blade, 150% more than the five blade and 200% more than the six
blade broadhead.

The mechanical advantage
equation dictates that the greater the length of a broadhead relative to the
width, and the fewer the number of blades, the more efficiently it will be able
to utilize the force applied to it.

KINETIC ENERGY vs MOMENTUM

As a base point for a
discussion of momentum and kinetic energy, one must understand that the laws of
physics dictate that energy can never be manufactured or destroyed but only
transformed or directed in its flow.
The equations for these two measurements are:

Kinetic Energy = __ Weight X Velocity Squared __

2 X Acceleration of Gravity

Momentum = __ Weight X Velocity __

Acceleration of Gravity

The kinetic energy (K.E.)
of a moving body increases as the *square* of the velocity and directly as
the mass whereas the momentum increases directly as *both* the velocity
and/or mass increases.

With the advent of compound
bows and overdraw setups, with their higher velocity capability, it has become
common to see kinetic energy figures cited as a supposed measure of the
penetration capability of a particular bow-arrow-broadhead combination. This use of kinetic energy reflects a
misunderstanding of these basic principles of physics.

*By definition*, kinetic energy is the capacity to do work. It is the **TOTAL ENERGY** of a body in
motion. ** K.E. is scalar, or
nondirectional, in nature**. As
applied to an arrow in motion, K.E. includes such things as: radial energy due
to arrow flexion; rotational energy due to arrow spin; sonic energy due to
vibration; heat energy due to friction; and potential energy (all other
remaining energy). Simple use of K.E.
alone as a predictor of penetration capability also fails to take into
consideration the mechanical advantage of the broadhead and the coefficient of
resistance. The kinetic energy of an
arrow,

Momentum is the measure
used in physics to quantify the "impulse"; the force exerted over a
period of time **IN ONE SPECIFIC DIRECTION**. ** Momentum is a unidirectional force vector**. Another
of those basic laws of physics states that "in cases of collision, whether
the bodies are elastic or inelastic, the momentum before collision is equal to
the momentum after impact". This
means that momentum is the measure of how much energy, due solely to the weight
and velocity of an arrow, must be transferred to whatever it impacts before the
arrow comes to rest. (Again, momentum
alone will not fully predict the penetration capability of an arrow, and the
mechanical advantage of the broadhead and the coefficient of resistance must
also be considered.)

Assuming there is no
bending of broadhead or arrow shaft, how far into the target an arrow will go
before all available energy is lost (the amount of penetration) depends on four
MAIN factors: the resistance of the object impacted (target), the momentum of
the arrow, the efficiency with which the arrow (broadhead) utilizes the force
available to it and the resistance of the arrow (often expressed as the
frictional or drag coefficient). The
resistance of the target we have little control over. Arrow and broadhead selection we do have control over.

Use of a broadhead with a
high MA results in maximum utilization of whatever momentum is available. High MA broadheads also offer lower tissue
resistance.

Heavy arrow mass helps
maintain high momentum throughout the arrow's full flight. Velocity diminishes very rapidly but arrow
mass remains constant throughout the full flight of the arrow (which is why bow
draw-weight is less important than arrow mass).

Shafts with diameters
equal to, or smaller than, the ferrule reduces the drag factor of the shaft on
tissues as the arrow penetrates.
Broadheads with smooth contours, having no abrupt angles or juncture
points, also reduces the drag factor.
Combined, these factors help maximize the penetration capability of
hunting arrows, regardless of what target resistance is encountered.

The following charts give
the calculated momentum of one of the better performing combination tested in
the Natal Study (adequate for all shots on game up through zebra size when used
with a ** good** broadhead of at least 3.0 MA) and compares it with
the momentum of common rifle and handgun loads. They also demonstrate the measured effect on the momentum of
increased arrow mass. While arrows
certainly penetrate tissue with much lest effort than a bullet, an arrow used
on big game must still maximize the use of its very limited energy. There is no excess to spare!

Graph XIV is intended to
allow the reader to calculate various arrow weight and velocity combinations
that will give a momentum (.57 Pound-Seconds) equal to that 'adequate' basic
big-game arrow used in the example (assuming the same 'good' broadhead is
used).

It will be noted that,
with current equipment, it is impossible to generate this amount of momentum
with light weight arrows. Even at an
arrow mass of 520 grains, the velocity needs to be near 250 feet per second,
yet a heavy arrow of 740 grains need only be traveling a little over 160 feet
per second to reach this level of momentum (a velocity easily within the
capability of most conventional and compound bows of only modest draw weight) .

In Part V: Predicting
Arrow Penetration on Real Animals we will look at the * potential*
practical application of the information presented in the previous articles,
and a first field test of its capability.

** MOMENTUM AND MASS**

MOMENTUM = __MASS IN LBS.
X VELOCITY IN FT./SEC__ => POUND SECONDS

32 FT./SEC./SEC

2219 shaft w/190 gr.
Grizzly (710 gr.) at 180.5 Ft./Sec. => .57 Lb-Sec

22 Hornet 45 Grain at 2690
Ft./Sec. => .54 Pound-Seconds

.38 Special 158 Grain at
755 Ft./Sec. => .53 Pound-Seconds

.357 Magnum 158 Grain at
1250 Ft./Sec. => .88 Pound-Seconds

**TO ACHIEVE .57
POUND-SECONDS OF MOMENTUM: (The average
momentum of the "Adequate" performing arrow/broadhead combinations in
Natal Study, all of which had broadheads of 3.0 mechanical advantage and
shafts equal to, or smaller than, the broadhead ferrule in daimeter):**

A 900 Grain Arrow must
reach a velocity of 142 Ft./Sec.

A 740 Grain Arrow must reach
a velocity of 161 Ft./Sec.

A 550 Grain Arrow must
reach a velocity of 234 Ft./Sec.

A 450 Grain Arrow must
reach a velocity of 285 Ft./Sec.

A 350 Grain Arrow must
reach a velocity of 367 Ft./Sec.

**EFFECT OF INCREASED
ARROW MASS:**

With 94# longbow:

A. Arrow of 650
Grains has 184.5 FPS velocity and Momentum = .54 Pound- Seconds.

B. When arrow
mass is increased to 785 Grains, velocity is 175 FPS velocity and Momentum =
.61 Pound-Seconds.

C. When arrow
mass is increased to 1286 Grains velocity is 154 FPS and Momentum = .88
Pound-Seconds.

In these
examples an increase in arrow mass of 21% results in a velocity decrease of
only 5.1% but increases momentum by 12.9%
(Arrow 'A' compared to Arrow 'B').
A mass increase of 98% results in a velocity decrease of only 16.6% and
a momentum increase of 63% (Arrow 'A' compared to Arrow 'C').

Of historical note, Art
Young and Saxon Pope used 75# self-wood longbows and 3/8" birch shafts
with broadheads 1" wide by 3" long (arrow mass of *approximately*
800 grains at *approximately* 155 FPS for a Momentum of .55). With these they were able to completely
penetrate (with arrow exit) Alaska Brown Bears, and Young successfully took
many of the larger Africa species, including several lion and cape buffalo,
with the same equipment. He was unable
to achieve adequate penetration on black rhino with this combination.

** POSTULATES BASED ON ARROW
PENETRATION TEST**

** OF GAME ANIMALS**

I have formulated the
following postulates which were originally developed from data secured during
the Natal Study. All subsequent testing
I have conducted to date have supported these original postulates.

1. Many
broadheads are too fragile, bending or breaking on impact, thus limiting
penetration.

2. Broadheads * must*
remain undamaged. A broadhead that
becomes bent or broken on impact will severely limit penetration.

3. There is a * direct
correlation* between the depth of arrow penetration and percentage of
killing hits. The deeper an arrow
penetrates, the more likely it is to kill.

4. There is *no
correlation* between broadhead size, increased number of blades or 'cut
area' and the percentage of killing hits.

5. Given a well
placed hit with a sharp broadhead, broadhead failure and inadequate penetration
are the only two things which cause failure to kill.

6. Rigid single
blade broadheads are the least prone to damage on impact.

7. The most
lethal shot angle is with the animal quartering away from the archer.

8. The least
lethal shot angle is with the animal quartering towards the archer and the shot
hitting in the neck-shoulder junction area.

9. All
multiblade broadheads offer insufficient penetration when heavy bone is
encountered.

10. Single blade
broadheads penetrate significantly better than multiblade broadheads in both
soft and hard animal tissue.

11. Four and
five blade heads penetrate bone better than three blade heads.

12. When a rib
is hit on entrance, a single blade broadhead is almost twice as likely to be
lethal as 4, 5, and 6 blade heads and three times as likely to be lethal as
three blade heads.

13. When heavy
bone is encountered, a total arrow mass of at least 650 grains, as well as a
tough single blade broadhead, is required to achieve adequate penetration.

14. A single
blade broadhead is more than twice as likely to produce an exit wound as a
multiblade broadhead.

15. The degree
of blood trail is dependent on the location of the hit and the presence/absence
of an exit wound, not the number of blades on the broadhead.

16. When all
shots are considered, the degree of wound inflicted (depth of wound channel
times the blade cut) by single blade broadheads is equal to or greater than
that inflicted by multiblade broadheads.

Postulates (Cont.)

17. No
multiblade broadhead can reasonably be expected to penetrate even a deer size
animal when the hit is from the forward quartering angle and in the area of the
neck-shoulder junction.

18. The most
important factor in achieving adequate penetration is a well constructed single
blade broadhead.

19. The second
most important factor in achieving adequate penetration is adequate arrow mass
(a minimum mass of 650 grains is recommended for 'standard' big game animals
and 900 grains for 'super size' game).

20. The third
most important factor in achieving adequate penetration is having a shaft
diameter no larger than, and preferably smaller than, the broadhead's ferrule
in diameter.

21. Game animals
have reflexes faster than even the very fastest of arrows. No archer can guarantee where his arrow will
strike an animal.

Copyright 1996

Dr. Ed Ashby