Kinetic Energy, and Arrow Penetration
(And What They Mean for the Bowhunter)
Dr. Ed Ashby
To understand the relationship between
an arrow’s kinetic energy, its momentum, and their implications
towards the ability of a hunting arrow to penetrate tissues, one must rely on
the laws of physics. This discussion cannot
be made totally uncomplicated. The
following is an attempt to impart a fundamental understanding of the applicable
principles of physics, as simply as I can, and relate them to the results from
actual field data.
Before delving into the deep abyss of
the physics involved in arrow penetration, it is appropriate to first take a
few moments to discuss the field data, and the logic behind why it is collected
in the manner that it is.
Judging from questions I receive, this
appears to be a very misunderstood aspect of the study of terminal
ballistics. It is, in many aspects,
more akin to forensic medicine than to laboratory science. The aficionado of the many forensic medical
shows, now so popular on television, will recognize the methodology. One starts with a real event, something known to have occurred, and then uses
pure science to determine and explain the “how and why’ of the incident.
Penetration data collected from real
shots, into real tissues, is not a static
measurement. Outcomes differ from shot
to shot, as the uniformity of tissues encountered change. In the real world it is impossible to
control all the variables, and one does
not wish to do so. Those variables do exist. They will be encountered.
The scholar of abstract science will
cite that this testing methodology includes too many variables, but it is
precisely because of the multitude of variables that it is necessary. When dealing with infinitely complex
variables, only ‘outcome driven’ information
analysis, from a multiplicity of data, provides usable results. This is why the medical community commonly
uses ‘outcome driven’ studies.
A commonplace example of these differing
test approaches occurred with the development of automobile air bags. Engineers did enormous static testing with crash dummies, controlling all variables,
before air bags were introduced.
After the introduction of air bags into
production automobiles, outcome driven
analysis showed that significant numbers of adult humans were being
injured, and sometimes killed, by air bags during their deployment. An even larger number of children were being
injured or killed. Static testing had
indicated the deployment force would be safe. The ‘reality’ outcome
was not as the static testing had predicted.
Outcome studies of air bag performance,
in real automobile crashes, with real people on board, pinpointed the incidences where both serious and fatal
damage was caused to humans by the air bag.
It delineated the tendencies;
when the events were likely to occur.
The static test standard was a male, of
160 pounds weight, seated normally within the car. Observed injuries and deaths occurred when occupant size was
below the ‘average size’ that had been used in the static studies to determine
the safe force levels exerted upon the various parts of the body during air bag
deployment AND when the occupant was located closer to the air bag at time of
deployment than the ‘static testing standard’ (as with persons using a cushion
or pillow behind their back while driving or riding).
of occurrence of these events was tracked in the outcome studies, and found to
have a significant prevalence. Then researchers turned to the pure sciences
to find the explanations for the events, which had now been shown to occur
in the real world. Force of
impact, in relation to both occupant size and position at time of impact, was
The force of air bag deployment was
simply too violent for human tissues, under
particular sets of circumstances, which did occur in the real world application
of the air bags. The force of air bag
deployment was modified. Outcome
analysis of air bag deployment force continues today, and the regulations and
guidelines are still being modified, based upon outcome driven studies.
The above example pinpoints the major
differences in methodology between the measurements of pure laboratory science
and the outcome driven method of deriving conclusions. In
laboratory science, one starts with pure measurements and tries to predict
future events. Outcome driven studies start with events known to occur; then looks for
the scientific explanations of how and why it occurred.
Outcome driven studies factor in the probability of occurrence when a large
number of independently acting variables are randomly introduced into the
observed results. Another way of saying
this is that outcome driven studies include the Murphy Factor; to find out what
can happen; when it is likely to happen; and how often it actually
Another major difference between laboratory
science and outcome driven studies is that outcome
driven results have an ‘acceptability level’. Their validity does not have to meet any level of ‘engineering
credibility’; the ability to be repeated at will, each and every time.
For example, how many ‘unsuccessful
outcomes’, deaths or injuries, caused by
an air bag’s failure to perform as intended, are required before it is
deemed as ‘unacceptable performance’
under the real conditions of use? This
question is even more valid when the identified cause of the incidences is
The gravity of an incident; the tendency
for it to occur under particular circumstances; the frequency with which its
actual occurrence is observed; and society’s morals all determine the level of acceptability. So, one has to ask, “What
is the acceptable level of failure for a hunting arrow to perform as expected
in tissues?” As a bowhunter, I
am interested in outcome; outcome in tissues, not in a homogeneous test
medium. I think most bowhunters are!
many years I tried to find a test medium that would give results which
correlated to the observed incidents
which occurred under field conditions, as a hunting arrow penetrated real
tissues. Such a test medium would make the
investigation of terminal ballistics of hunting arrows very much simpler, and far
less time consuming and expensive.
gel, covered with a suitable elastic outer covering, gives a reasonable correlation
to tissue hits in which no hard tissues
are encountered, but I have found no combination of materials that will
correlate with the multiplicity of resistance forces encountered in penetrating
real tissues. This past year, a
European forensics team also tried to find a synthetic testing medium that
would give results comparable to that seen in real arrow wounds. They also found none.
An absolute ‘predictor’ of arrow penetration, on every shot, is
impossible. Outcome driven analysis from
real shots, into real tissues, does, however, give a definitive picture of any
given arrow’s incidence, tendency, and
frequency of occurrence of events during tissue penetration. Testing in a uniform medium does not. Having tried both approaches, I feel certain
that it is only through the use of outcome
driven results that reliable indicators
of an arrow’s likelihood of performance
under real hunting conditions can be developed.
Before launching into the physics of
arrow penetration, we first need some basic definitions. Those not ‘technically predisposed’ will
find the first part tedious, but it is necessary groundwork for one to
understand the propositions that follow.
It is important for one to know that the recommendations are grounded in
both the coherent logic of physics and the empirical facts;
facts confirmed through nearly a quarter century of intensively collecting and
collating detailed field measurements of the terminal performance of hunting
arrows in real animal tissues.
For the benefit of those who find the ‘highly technical’ difficult, some
of the more ‘technically precise’ clarifications and information has been set
aside in text boxes, and denoted as a “Nerd’s Note”. (Nerd: Defined as an
enthusiast whose interest is regarded by others as too technical or too scientific. Somehow, I think I resemble that
remark!). It is entirely acceptable for
those ‘mathematically challenged’ to omit reading the Nerd’s Notes! Their omission will not affect the reading
of the other text.]
Laws of Physics
FORCE: Force is defined in physics as that
which tends to change the momentum of a body containing mass. Force is proportional to the rate of
change of momentum.
Note: Force (lbf) = [mass (lbm) times
the acceleration (expressed in ft/sec²)] divided by the gravitational
constant. The gravitational constant is
32.174 lbm-ft/lbf-sec2, and is abbreviated as ‘gc’. In English units, the gc is used
anytime one goes from pounds mass
(lbm) to a force, (lbf).
MASS is a quantity of matter,
and is expressed in ‘pounds of mass’, (abbreviated as lbm). Weight is the force exerted on
an object due to the gravitational field, and expressed in pounds of force
(abbreviated as lbf). In physics,
mass (‘lbm’) is expressed as the weight of the object (in pounds force)
multiplied by the gravitational constant and divided by the force of gravity. Though the numerical value of an object’s
mass and weight can be the same, the units of measure and theory behind them
Nerd’s Note: Weight (or the force as a result of mass)
has the following equation when using English Engineering Units:
= [mass(lbm) * g (32.174ft/ sec2)/gc (32.174 lbm-ft/lbf-sec2 ), or, to
conform to the above,
Mass (lbm) = W (lbf) * gc/g
Note: The factoring in of the g
and gc does not change the resultant value; it just makes the units
consistent. This becomes a factor
anytime one talks about, or calculates, “force” and its effects, as it distinguishes clearly
between the mass of an object and the force applied by the mass.
MOMENTUM: The unit of measurement for momentum
is slug-feet per second. A slug
is a portion of the subset of coherent units known as the gravitational
foot-pound-second system. The
physical weight of one slug of mass equals 32.174 pounds. One slug of mass will acquire
an acceleration of one foot per second per second when acted on by a one
pound force (at sea level).
can also be expressed in lbf-sec, if one is not using the slug as the
unit of measure. The slug
has units of lbf-sec2/ft. It
is essentially mass (lbm) with the gc already divided into
A body of mass (M) moving at a
velocity (V) has a momentum equaling M x V. This says, “The momentum equals the mass of the
object [expressed in pounds of mass (lbm) and divided by the pull of gravity,
which will result in the mass of the object in slugs], times the velocity
[expressed in feet per second] at which the mass is moving”.
Momentum has both amplitude (an
‘amount’ value) and a direction. Because
any measurement of momentum has a specified direction it quantifies the net
force acting in that single, straight line, direction. Momentum is, therefore, known as a linear
function, and is a measurement of the force of forward movement
of an object.
Note: While there
are situations where momentum can also be angular, in dealing
with penetration the use of linear momentum is the simplest and most applicable
VELOCITY is defined as the change
in position divided by the time period during which the change
occurs. It is expressed in units of
distance per unit of time - or, for our purposes, in “feet per second”.
ACCELERATION is the rate
of change of speed, or how much the velocity of a body in motion changes
during a specified period of time.
Consequently, the acceleration of gravity is expressed in “feet
per second per second”. This quantifies
how many feet per second the velocity changes as each second
IMPULSE: Force (in our case, the momentum)
applied over a unit of time creates an impulse.
The concept of impulse
is extremely important in the study of momentum, and to the
understanding of arrow penetration.
Time passes as a force is applied to an object. When this happens we say that an impulse
is applied to the object.
When a bow launches an
arrow, an impulse is applied to the arrow. The bow applies a force on the arrow for a short time
period. According to Newton's third law
of motion, forces always come in pairs.
Thus, the arrow also puts a force on the bow, and the bow,
therefore also has an impulse applied to it.
NET FORCE is the total
amount of force exerted by a body in motion. It is the change in momentum
divided by the change in time.
When the mass of a moving object
remains constant, as with an arrow in motion, the net force
equals the mass (in slugs) times the change in velocity
divided by the time period over which the change occurs. By definition, the change in velocity
divided by the change in time gives the acceleration of a moving
body. Therefore: when the mass
of a moving body remains constant the force will equal the Mass
(in slugs) times the Acceleration. (Force equals mass times acceleration. In equation form this is expressed as: F =
Note: When using
English units, rather than slug mass, this equation would be expressed
as F = ma/gc or, if one prefers, F = (m / gc) * a. This is necessary to convert from lbm to
It is essential to understand that any
reference to the net force of a moving object is specific to the specified
time period being referenced.
In one set of circumstances, net force can equal the total
disposable force of an arrow in motion.
In another reference, net force can imply the remaining force
after deductions, as in calculating the net force remaining after an
arrow completely penetrates an animal.
When an arrow’s net force after
penetration (at the time of exit) is deducted from the (total
disposable) net force of the arrow at the time of impact
it equals the amount of the arrow’s disposable net force that was
required for the arrow to completely penetrate the animal on that particular
shot. That amount of the disposable
net force available to the arrow at impact was expended over the time
period required for the arrow to pass through the tissues.
IMPULSE: An impulse is equal to the net
force of the object times the time period over which the force
is applied. The impulse equation is
mathematically derived from the equation F = ma, which comes from Newton’s
Second Law of Motion. Study the
following. It shows the derivation of
the impulse formula.
Line 1: Force equals mass times
Line2: Substituting the definition of
acceleration for “a” in the equation.
Line 3: Algebraic rearrangement. The force multiplied by the change in time
equals the mass multiplied by the change in velocity.
line is our familiar equation F = ma.
The second line expresses
the acceleration by its basic definition, a change in velocity divided by the
change in time.
The third line is arrived at
through algebra, by multiplying each side of the equation by delta t (which is
the symbol for change in time), canceling it on the right, effectively
moving it over to the left.
Note: If working in
English units, one must not forget to factor in the gc constant to
change from pounds mass (lbm) to pounds force (lbf) in the above
equations, When doing so, the first
line of the equations above would be: F=ma/gc
The left side of the third line is
called the impulse on the object.
That is, impulse is equal to the net force times the length
of time over which that force is applied.
The right side of the third line is
called the change in momentum.
Thus, the impulse equals the change in momentum.
Impulse equals the change in momentum
An arrow in motion has a mass of M and is moving at a velocity
of V. As a result the arrow possesses a
predetermined momentum (mass times velocity) at the
instant of impact. When the arrow
strikes an animal it will decelerate (a negative acceleration value).
If the arrow stops in the animal it will have expended the entire
disposable net force available to it at the instant of impact over
the time period required for it to come to a full stop. A resistance impulse force
equaling the arrow’s disposable net force at impact will have
been applied by the tissues upon the arrow, and it will have occurred
over the exact same time period.
In this situation the arrow’s velocity change is 100%. The momentum of the
arrow at impact, divided by the time period required for the arrow to
come to a complete stop, will equal the impulse of the arrow upon the
tissues. The resistance force
of the tissues to the arrow's passage during the time required for
penetration represents the impulse of the tissues upon the arrow. The two impulses will be equal. The time factor will be equal between the
two impulses. The force
of momentum and resistance force will be equal.
If the arrow passes completely through the animal, the applied impulse
equals the arrow’s momentum at impact minus the arrow’s retained momentum
at exit, for the time period required for the arrow to pass through the
tissues. As the mass of the
arrow remains constant during the entirety of its passage through the tissues,
the arrow’s net force decreases only in proportion to the amount of velocity
loss during the course of penetration.
Given two arrows of equal momentum, but with one deriving a
greater portion of its momentum from mass than the other, the
heavier arrow will change velocity (decelerate) at a slower rate
as it passes through the tissues.
In other words, the heavier arrow will retain a higher percentage
of its impact velocity at any given time period during its
passage through the animal’s tissues, thus it also retains a higher momentum
at any given point during the time required for the arrow to penetrate.
Another way of saying this would be that, though the heavier arrow is
traveling slower, it takes a longer time to stop. The result is that the heavier arrow will have a greater impulse of force than does the light arrow.
It is momentum that gives an object in motion the tendency to
STAY in motion. The greater
the contribution of the object’s mass is to the resultant momentum
the harder it will be to stop the forward progression of a moving object. Anyone who has pushed a car in neutral and
then tried to stop it will understand this.
The more of a moving object’s momentum that is derived from its mass,
the more TIME it takes to stop it with any given resistance force.
It is common for proponents of light and fast arrows to counter that
the faster arrow will have traveled a greater distance through the tissues in
the same time period than will the heavier, and slower, arrow. This would be valid were it not for the
nature of resistance forces.
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!
The effect of exponentially increasing resistance is easy
to experience. Try holding a hand out
the window of the car, while the car is going at a velocity of 30 miles per
hour (which is only 44 feet per second), and feel the air’s resistance
against your hand. The resistance
is very slight. Now accelerate
to 60 miles per hour (a mere 88 feet per second). The velocity has only gone up by a factor of two, but the
air’s resistance to your hand passing through it is now four times
Now imagine the effect on an arrow passing through tissues. Tissues are more solid than air. They have a greater density. Their resistance to an object’s
passage is higher. Visualize the effect
as an arrow’s velocity increases from 150 feet per second (a fairly
typical velocity from a mid-draw weight traditional bow) to 300 feet per second
(as from a top line compound bow).
Let us now assume an arrow weighing 700 grains for the slower bow (150
fps is easily achievable with that weight arrow and a ‘traditional’ bow) and a
390 grain arrow for the faster bow (the advertised velocity
rating for one of the newest compound bows on the market, using that weight
arrow). The slower arrow has 0.466 slug
feet per second of disposable net force. The faster arrow has 0.519 slug feet per second.
Lets also assume these two arrows are of same materials, have equal
physical external dimensions (easily achievable), and both have perfect flight
characteristics. The tissue’s resistance
increase is totally dependant upon the velocity of the
The lighter arrow has 10.22 percent more disposable net force (and
123.2 percent more kinetic energy) than the heavier arrow but, because
of its higher velocity, it is met by four times the resistance to
penetration. Which arrow will penetrate
further in real tissues? Empirical
evidence from the outcome studies
provides an overwhelmingly definitive
answer. Both the frequency and degree to which
the heavier, slower, arrow out-penetrates the lighter one is of such a
magnitude that it must be viewed as the norm.
ALL MOMENTUM IS NOT THE SAME
Given two arrows, identical in shaft and broadhead
materials and profile, and having EQUAL momentum, but possessing UNEQUAL
mass, the arrow deriving the greater portion of its momentum from
its mass will penetrate better.
Laws of Physics requires this to be true, and ALL of my field
test data validates this to be the case.
To say this in another way, arrow momentum derived through
increasing arrow mass results in a greater gain in penetration than does
momentum gained by increasing an arrow’s velocity. This is true because the tissue’s resistance
is increased by the square of the velocity.
Let’s look at two arrows of equal momentum, but unequal mass,
both of which expend all their available net force in the tissues. If the momentum is equal
between two arrows at impact, the one with the greater mass has to be
traveling at a slower velocity.
As shown above, the slower arrow will be met by a lower resistance
force than the faster arrow.
With the momentum of the two arrows equal at impact, their disposable
net force will be equal, but the resistance force will be greater upon
the faster arrow. Because of the higher
resistance force, the faster/lighter arrow will lose velocity
more rapidly, and its momentum will diminish at a faster rate than that
of the heavier arrow. It will stop in a
shorter period of time, thus it will have a lower impulse of force than
the heavier arrow.
To quantify the potential for penetration we must
first quantify ALL the directional FORCES involved.
When an object is in motion, it has kinetic energy. Kinetic energy is defined as the total
energy of a body in motion. Kinetic
energy is scalar, or non-directional, in nature - it is the
TOTAL energy, of all types, in all directions. That is: kinetic energy has magnitude,
but it does not have direction.
(Note that kinetic energy is defined as ENERGY, not as FORCE.)
Kinetic energy includes all the types of energy
of a body in motion, and is very dependent on the object’s velocity. When a moving object with mass
strikes something, the kinetic energy is transferred, as one or another
form of energy.
An arrow’s kinetic energy at impact is the basic ‘potency’ of
the collision - how hard the arrow strikes the target. Kinetic energy is measured in
"foot pounds". A 'foot pound'
is the amount of energy needed to exert a one pound force
for a distance of one foot.
(Note that foot pounds is a measure of the energy
required, not a measure of the force itself). Force is a portion of the
arrow’s total energy.
The formula for kinetic energy is: Kinetic energy equals
one half the mass (lbm) times the velocity squared and divided by
the gravitational constant (gc).
Kinetic energy is often cited by the advocates of light
weight, high velocity, arrows as the standard for predicting an arrow’s
ability to penetrate. But consider a
A baseball weighs 5.12 ounces (that’s 2240 grains) and can be thrown in
excess of 95 mph (which is 139.33 feet per second). It has 96.5 foot pounds of kinetic energy. It actually strikes much harder than a heavy
hunting arrow at ‘traditional bow’ velocities, but I can't really see hunting
buffalo with a fast ball! Kinetic
energy determines how hard the baseball strikes; it has no direct
bearing on how well it penetrates.
As with the baseball, a tuning fork, once struck, has high kinetic
energy (it can shatter a crystal wine glass), but has almost no momentum. It would make a darn poor weapon against an
animal of even modest size!
The kinetic energy of a moving arrow includes ALL the energy,
of all types, inherent to the arrow.
This includes such things as the flexional energy; vibrational energy
(some of which is transformed into the sonic, or sound, energy); all of
the rotational energies; gravitational energy; potential energy;
and the heat (frictional) energy generated by its passage.
An arrow’s momentum is also a part of
the arrow’s kinetic energy - the only part that relates to its ability to
penetrate. Some of an arrow’s kinetic
energy is dissipated as other forms of energy during flight and on
impact. Even the ‘sound’ of a hit is
derived from the arrow’s kinetic energy.
As shown above, the Laws of Physics dictates that
momentum, and not kinetic energy, is the correct unit of measure
to quantify the linear (straight line) "potential disposable net
force" that is available to an arrow. Momentum determines THE AMOUNT OF FORCE
which an arrow has available to it for penetration.
(Perhaps this is a good point at which to digress for a moment. Kinetic energy is frequently used as
a guide to the potential lethality of a high speed bullet. This is because a bullet can cause tissue
damage in ways an arrow can not.
Bullets carry massive amounts of kinetic energy, relative to an
arrow. Much of a bullet’s kinetic
energy is transferred through the tissues as a ‘shock wave’, caused by the
rapid compression of tissue fluids.
As the bullet strikes, a ‘hydraulic force’ is transferred,
through the tissue fluids, over a wide area.
This causes histologic tissue shock, disrupting tissue
functions. It is this hydraulically
induced ‘shock wave’ that causes the ‘bruising’, or ‘blood-shot’
tissues surrounding a bullet induced wound channel.
If one researches the literature of terminal ballistics and killing
power of firearms, they will find that, even there, the use of kinetic
energy as an indicator of bullet lethality falters badly as the size of the
animal increases. Its usefulness also
diminishes with firearms producing low (by firearms standards) kinetic
energy, as with handguns. This is
the reason that such other ‘indicators’ of bullet lethality as “Taylor’s
Knock-Out Value”, the “Optimum Game Weight” and the “Power Factor” find their
way into firearms literature, all of which place more emphasis on the bullet’s momentum
and/or impulse of force.
Studies conducted by
the U. S. Army’s Ballistics Research Facility indicate that tissue shock
from hydraulic compression becomes a significant “wound factor” only at impact
velocities around 2500 feet per second, or greater. Creating ‘hydraulic shock’ is not an
option with an arrow).
Kinetic energy is NOT the
correct unit of measure for calculating ANY of the forces relevant to
penetration. It is applicable for
calculating neither the force of a moving object; the disposable net
force at impact; the net force at exit; net force consumed
during penetration; the applied impulse; nor the resistance
impulse force affecting penetration.
With a given arrow,
if its kinetic energy is increased, there will be a measurable increase
in its penetration, but only because the velocity increase necessary to
achieve more kinetic energy has also increased the arrow’s momentum. The increase in penetration will not
be proportional to the increase in kinetic energy. It will be proportional only
to the resultant increase in the arrow’s momentum (with
the increased resistance created by the higher velocity also
Kinetic energy IS applicable for
calculating the mechanical efficiency of one’s bow.
Efficiency is defined as the ratio
of the amount of energy (Ah, now we get to use ENERGY) used by a machine
to the amount of useful work done by it.
A “machine” is defined as
a device with moving parts used to perform a task. Work is defined as the
transfer of energy, measured as the product of the force
applied to a body multiplied by the distance moved by that
body in the direction of the force. Work is force
times a distance. Work
can also be defined as being equal to the change in kinetic energy.
For a bow and arrow system, the bow’s efficiency
is defined as the proportion (percentage) of the bow’s stored energy
that is transmitted to the arrow when it is fired. The more efficient a bow is the higher will be the amount
of its stored energy (i.e., the potential energy that is stored
in the limbs of the drawn bow) which is transferred to the arrow when
the bow is fired.
The arrow’s kinetic energy is derived
directly from the ‘output kinetic energy’ of the bow, and represents the
useful work performed by the bow.
The arrow’s momentum will be a function of the
bow’s output kinetic energy and the arrow’s mass, but it is not
the product of them. (In
mathematics a “function” is a quantity whose value depends upon the
varying values of other quantities, while the “product” is the result of
the multiplication of two or more quantities.)
When one looses and arrow, a portion
of the bow’s stored potential energy is used to apply a force
upon the arrow. The applied force
acts upon the arrow over the time period during which the arrow remains
on the string.
This force, applied over this time
period, will be the impulse of the bow upon the arrow. It is this applied impulse which
causes the movement of the arrow’s mass. In other words, it changes the velocity of the arrow, and
the arrow’s mass times its launch velocity determines the arrow’s
momentum at the instant it departs from the bowstring.
A bow’s output kinetic energy
allows one to estimate the bow’s ability to cast an arrow. The greater a bow’s output kinetic
energy, the more capable it is of casting a heavy arrow with acceptable
levels of velocity and trajectory for ethical hunting ranges.
Thusly, the output kinetic energy
OF A BOW is a useful INDICATOR of how much arrow momentum it can
Impulse is the FORCE
applied by a body in motion, over a period of time, upon
the object it hits. Momentum
has FORCE. Kinetic energy has
ENERGY. An arrow’s net
disposable force equals its momentum at the instant of impact, and
must be met by an equal resistance force, acting over the time period
of the impulse, for the arrow to come to rest.
Kinetic energy does not enter directly
into any of the calculations relating to penetration. THE KINETIC ENERGY CARRIED BY AN ARROW AT
IMPACT HAS NO DIRECT BEARING ON ITS ABILITY TO PENETRATE.
If one fills a 5 gallon plastic pail with sand and fires both a .357
magnum and a heavy hunting arrow at it, the bullet will be stopped by the sand,
while the arrow will penetrate the pail completely. The .357 magnum handgun has a 158 grain bullet traveling at 1250
fps, for a momentum of 0.83 slug-feet per second, and a kinetic energy
of 520 foot-pounds. A 710 grain arrow
at 183 fps has only 0.57 slug-feet per second of momentum, and a mere 52
foot-pounds of kinetic energy.
These are actual combinations I have used to demonstrate the
penetration power of a heavy hunting arrow.
Our baseball, with 96.5 foot pounds of kinetic energy, and 1.39
slug-feet per second of momentum, will simply bounce off. What makes the difference?
A major factor
between the bullet and the arrow is the increased resistance force met
by the higher velocity bullet.
While the bullet has ten times more kinetic energy, and
37.5% more momentum, than the arrow, its almost seven times higher velocity
causes the bullet to be met by nearly fifty times as great a resistance
force as that encountered by the arrow!
Another major factor
between the handgun’s bullet and the arrow (yes, we will get to the baseball
shortly) is the longer time period of the arrow’s impulse; which
results from its higher mass.
Though the arrow is traveling much slower than the bullet, and has less momentum
than the bullet, it derives a greater percentage of the momentum it does
possess from its mass. It is
The heavier (and lower
velocity) arrow “decelerates” more slowly than the bullet or, if one
prefers, it has a longer time period over which the force
acts. Remember? Force multiplied by the time
it acts equals the impulse. The
heavier arrow retains a higher percentage of its force for a
longer period of time than does the bullet. The bullet’s total net disposable force, though very high
relative to the arrow, is entirely dissipated in milliseconds.
Now, to our
baseball. Our pale of sand also has a
differing resistance to the passage of projectiles having differing
cross sectional areas and profiles.
The baseball has a
much larger surface area presented to the bucket, in relation to its mass,
than does the bullet. The bullet
presents a larger surface area per unit of mass than does the
arrow. In physics this difference in
the ‘penetration ability’ is defined by the sectional density of the
DENSITY of an object of round (cross sectional) profile is defined as
the mass of the object divided by the
square of its diameter. The heavier the
object is in relation to its cross sectional area, the higher its sectional
density. The higher the sectional
density, the less the amount of frontal surface area (per unit of its mass)
that is presented to the target, and the less of the target’s ‘matter’
(relative to the penetrating object’s mass) that will be displaced by
the passage of the object through the target.
This translates into a lower level of resistance on the
frontal area of the projectile.
If the mass of
an arrow is increased without changing its external dimensions, it will weigh
more per unit of cross sectional area.
Its sectional density will be increased, and it will penetrate
farther with any given applied force.
Note that the sectional
density refers only to the resistance on the penetrating object’s
frontal area and the amount of ‘matter’ displaced in relation to its mass. In tissues, an arrow’s “shaft drag” is
also an important feature influencing penetration. Shaft drag results from the frictional forces
between the arrow shaft’s surface and the substance being penetrated.
Shaft drag is one
major reason that arrow penetration test into artificial test media often
differs from actual results derived from testing on real animal tissues. Most ‘target materials’ rely heavily on shaft drag to
stop the arrow. They are made from
materials specifically chosen and designed to ‘close down’ around the shaft,
exerting the maximum possible shaft drag.
Muscle fibers, on the other hand, tend to retract, actually spreading
apart, when cut by a sharp broadhead.
When cut, muscle
tissues also release blood, which lubricates the shaft, reducing the coefficient
of friction between the arrow shaft and the tissues. This reduces the drag on the shaft. These biologic reactions are a major reason
why accurate and reliable measurements of hunting arrow penetration can only be
achieved through testing conducted on live (as when actually hunted) animals,
or VERY freshly killed animals.
Even when testing on
freshly killed animals, physiological tissue changes occur rapidly, and testing
must be done within minutes of death.
If the time lag is longer, results become erroneous, due to changes in
tissue resistance forces encountered.
difference in the ability of hunting arrows to penetrate tissues, as opposed to
bullets, is that they are tipped with a broadhead. Yes, the broadhead slices through tissues,
rather than having to ‘push’ through them, but there is more.
A broadhead is a “simple machine”, a series of inclined planes. These inclined planes allow the arrow to
accomplish more work with any given applied amount of force. The profile of the broadhead offers a mechanical
Mechanical advantage is defined as the improvement gained by use of a mechanism
(machine) in transmitting force (There’s that word
again!). Specifically, it is the ratio
of the force that performs the useful work
of the machine to the force that is applied to the
machine. In other words, broadhead
design can multiply the force of the arrow, increasing its
ability to do work.
Not all broadheads
offer an equal mechanical advantage.
As with any inclined plane, the longer the slope of the plane in
relation to the rise of the plane, the higher will be the mechanical
A long and narrow
single blade (2 cutting edges) broadhead will have a higher mechanical
advantage than one of equal length and width, but having more blades. Also, as the profile of a broadhead’s
blade(s) becomes shorter and/or wider the mechanical advantage becomes
lower. Having either a convex or
concave cutting edge profile, rather than a straight taper, also lowers a
broadhead’s mechanical advantage.
Any abrupt rise in
the contour of a broadhead results in a profile which lowers the broadhead's mechanical
advantage. This is why a very
smooth and gradual fade-in of the broadhead’s ferrule into the blade is
important in broadhead design. It
detracts less from a broadhead’s mechanical advantage.
In trying to maximize
arrow penetration, there is also the efficiency of the bow/arrow system
to consider. Up to the limits of the
bow’s ability to move the arrow, bows become more efficient as the mass
of the arrow increases.
A heavier arrow
causes a bow to shoot more quietly than with a lighter arrow. This is because of the increased efficiency. More of the bow’s stored energy is
transmitted to the arrow and less is ‘wasted’ in the form of bow vibration,
which causes increased hand-shock and noise.
Increasing bow efficiency through the use of greater arrow mass
results in both a quieter shooting bow and one which imparts more force
to the arrow. A win-win situation for
For almost a quarter
century I have been actively collecting terminal arrow performance data from
shots into real animal tissues, and have the world’s most extensive ‘real
tissue’ arrow wound database from which to extract comparative outcome information. All empirical data supports the conclusion that
the above laws of physics apply to hunting arrow penetration in tissues.
In real tissues, it
is easy to get a very light, very fast, arrow combination, generating high
amounts of kinetic energy, which averages significantly less penetration
than an appreciably heavier arrow producing only one third as much kinetic
energy. A high frequency of this outcome
is demonstrable; with both arrows having identical broadheads and the same
shaft materials and dimensions.
What does all this mean for the bowhunter?
Let’s try to put everything into context. Relative to virtually all big game hunting
weapons, hunting arrows have a very low amount of force available with
which to do their job - penetrating animal tissues.
Lack of penetration
is the number one cause of a hit being non-lethal. The terminal arrow performance data from
each and every one of my studies overwhelmingly verifies
that fact (and the data is of sufficient magnitude that it must viewed as fact,
at least until data of an equally substantive nature, derived from outcome testing on real animal tissues,
demonstrates any reason to believe otherwise).
If one wishes to
maximize the hunting arrow’s ability to penetrate then consider the following.
(1) Maximize the bow’s efficiency. That means shooting the heaviest arrow one
can while still maintaining a trajectory that is adequate for ethical
Most bows show a
rapid increase in efficiency with increasing arrow mass up to the
point of approximately 12 to 14 grains of arrow mass per pound of bow
draw weight. (The exact point where the
rate of efficiency increase begins to decline varies from bow to bow and
shooting style to shooting style. There
are many variables, and the value of a chronograph to the shooter should not be
underestimated.) Beyond this point of
arrow mass per pound of bow draw weight a bow’s efficiency will
still increase as the arrow gets heavier, but the rate of efficiency
increase slows down.
(2) Use broadheads of high mechanical
advantage. This becomes
increasingly important as the bow’s draw weight becomes lighter, or the size of
the animal being hunted becomes larger.
Use of a high mechanical
advantage broadhead also becomes increasingly important as the power
stroke (the distance the arrow travels before it leaves the bow string)
becomes shorter. A shorter draw length
gives a shorter power stroke, which also means that, regardless of the amount
of force stored in the bow’s drawn limbs, that force will be
exerted upon the arrow for a shorter period of time.
For any given amount
of applied bow force, the longer one’s draw length, the more time
the bow has to exert its force upon the arrow; i.e.; the bow’s
impulse upon the arrow will be greater, and the
bow’s efficiency increases. (Force
applied over time equals the impulse.)
(3) Use broadheads with a cut-on-impact
tip. Broadheads of a
cut-on-impact tip design penetrate soft tissues with less resistance
that other broadhead tip designs. The
various tip designs, and their effects on penetration in bone, are still under
investigation in the current study.
(4) Accept nothing less than perfect arrow
flight in your hunting arrows.
It minimizes energy loss during the arrow’s flight, and reduces resistance
forces on entry (due to less shaft flexion), which results in the arrow
retaining more force to apply directly to penetration.
Achieve perfect arrow
flight through wise selection of arrow shafting materials and spine, perfect
broadhead-to-shaft alignment, careful bow tuning and the use of sufficient
fletching to stabilize the arrow in flight.
Start with a really
good broadhead and then set your hunting arrows, and your bow, up around the
broadhead. In testing I have used a
couple of hundred different types and designs of broadheads. As long as the broadhead is aligned so that
it spins in precise balance, on a straight shaft, I have yet to meet ANY
broadhead that I cannot get to fly perfectly.
This applies even to stone points!
The ‘balance’ of the broadhead does not have to be perfect. The ‘balance’ of the arrow system does!
[Tip: If the broadhead spins true, and the shaft is
correctly spined to the bow (for that weight broadhead), and it is straight, yet
the arrow still ‘wind planes’, there is not enough fletching to overcome the wind
shear effect created by the broadhead’s blades as they rotate through the
air. To stabilize the arrow in flight,
use more fletching surface area. This
is especially important when the broadhead itself is not well balanced;
presenting surfaces with varying shear angles to the air, such as with a stone
Once you have your
hunting arrow flying perfect, make your practice arrows (be they for target,
field, small game, roving or stump shooting) shoot just like your hunting
not the other way around!
It is foolish to sacrifice good broadhead construction, profile and mechanical advantage just to get one’s
hunting arrows to ‘shoot just like a target arrow’.
[Tip: A well tuned bow/arrow combination
will shoot ALL equal weight broadhead/field tip/target points into the same
group at any range. If the point of
impact is different between field tips and broadheads of matching weight, there
is a ‘tuning’ problem.]
The hunting arrow is
the single most important piece of equipment that the bowhunter carries
afield. The broadhead chosen is the
most important part of the hunting arrow.
A hunting bow merely
launches the hunting arrow. The arrow
delivers the broadhead. When the
broadhead hits it must perform, without failure, each and every
time. To do otherwise risks a wounded
animal and failure of the entire hunt.
A perfectly placed hit can frequently be non-lethal
when there is a failure of the broadhead tipped hunting arrow to perform its
task; penetrating and disrupting the body’s life support functions.
(5) Mechanical Broadheads. Mechanical broadheads have become very
popular in recent years. Mainly this
has occurred because it is extremely easy to get them to shoot much like a
target or field point of equal weight, even when the arrow’s fletching area is
insufficient to stabilize a fixed blade broadhead. In flight, mechanical broadheads present less surface area to the
air. They have a lower wind shear
Mechanical broadheads do, however, encounter
significant resistance upon opening in tissues. Outcome studies show that they require a
substantially higher level of impact momentum to achieve the same amount
of penetration as a broadhead of a more ‘traditional’ design.
This needless loss of disposable net force
reduces penetration. Remember? Outcome studies show that lack of penetration is the number one cause of
a hit being non-lethal and, in
all testing to date, mechanical broadheads average less penetration, on an
arrow of a given mass and momentum,
than does either a replaceable blade broadhead or a more 'traditional'
broadhead of comparable mechanical advantage.
In addition to their needless loss of disposable
net force during blade deployment, mechanical broadheads pose some other
penetration problems. All of the many
mechanical broadheads thus far examined in field testing have a low mechanical
advantage. As the field data shows,
this further inhibits penetration capability when tested on real animal
In all testing to date, mechanical broadheads
have also suffered by far the highest damage
rate of all categories of broadheads
tested. The outcome data manifestly
shows that a broadhead which becomes damaged during the course of penetrating
an animal causes a dramatic increase in resistance, and
penetration is severely decreased.
It is highly likely that the high
damage rate to the blades of
mechanical broadheads results from the abrupt increase in resistance
encountered at the time of blade deployment.
Though the total amount of resistance
force encountered by the blades may not be any greater than that
encountered by a fixed blade broadhead, a major portion of the resistance force is encountered over a
very short time period; abruptly upon deployment. This ‘spike’ in resistance force must be met by utilization of a higher proportion
of the arrow’s disposable net force;
reducing the arrow’s retained disposable
net force, which, in turn, lowers the arrow’s overall impulse of force upon the tissues.
Fixed blade broadheads enter the tissues with
blades fully deployed. They can utilize
any mechanical advantage they do have from the instant of impact, i.e.:
the mechanical advantage is available to them in penetrating the very elastic
skin. Mechanical broadheads cannot use
the mechanical advantage of their blades until after the blades are
The skin’s property of elasticity
imparts a ‘give’ to them as the arrow hits.
This can drain off substantial amounts of an arrow’s disposable net
force. This ‘give’, when an arrow
impacts, is why a loosely hung carpet makes a pretty fair arrow backstop. More ‘work’ is required of the arrow to
penetrate the carpet. Remember? Work is force times distance. The resistance force has to be moved
over a greater distance by the arrow’s impact force before the
It is because less work is required for them to penetrate the skin (and the other soft
tissues) that broadheads with a cut-on-impact tip penetrate better in soft
tissues than do broadheads having other tip configurations. The bevel of the tip’s cutting edge is also
an inclined plane - a simple machine. It, too, offers a mechanical advantage.
The longer the bevel (the lower the
sharpening angle), the higher the broadhead tip’s mechanical advantage will be.
But there is a lower limit. The
tip MUST be strong enough to resist damage upon impact with hard tissues (bone). A broadhead that becomes damaged during
penetration dramatically increases resistance,
and overall penetration suffers.
Though mechanical broadheads having a cut on
impact tip permit easier penetration through the very elastic skin tissues,
thus far there has been little outcome difference,
on comparable shots, in the measured overall penetration (relative to
mechanical broadheads having other types of tips and offering a similar
mechanical advantage). This is suggestive
that energy loss at the time of blade deployment is a major factor in
the reduction in tissue penetration measurable with mechanical broadheads.
(6) Arrow Shafts. With any given shafting material and
shaft finish, the larger a shaft’s diameter the greater will be the resistance
to its penetration. It will present a
larger frontal area to the tissues, displace a greater volume of tissue as it
penetrates, and present more total surface area to the tissues (which
results in a higher drag factor).
As a general rule, the arrow’s shaft should
have a diameter that is less than the broadhead’s ferrule diameter. In testing with parallel shafts (as opposed
to tapered or barrel tapered shafts), outcome data shows that when a shaft’s
diameter is greater than the broadhead’s ferrule diameter the arrow’s
penetration is reduced by and average of 30 percent, as compared to a
situation where the shaft’s diameter equals the diameter of the broadhead’s
If the shaft’s diameter is less than that of
the broadhead’s ferrule, the penetration increases by an average of 10
percent. That can equate to as much
as a 40 percent difference in measurable penetration between two arrows which
are equal in all respects except for the diameter of the shaft. This is not theory. It is what average outcome measurements from comparable shots into real tissues
show. It is a graphic demonstration of
the importance of shaft drag as a factor in the overall resistance
force when penetrating real tissues.
It is tempting to advise that one use as
small a shaft diameter as possible, but recent testing is highly suggestive
that other factors may also be at play.
In the recent tests, shafts of identical materials and nearly equal mass,
but of various profiles, were tested.
All were tested at the same distance (20 yards), from the same bow, and
with the same broadhead.
The results were, to say the least, of
interest. Averaging the results from
all comparable shots, the frequency
of shafts with a tapered profile out-penetrating those with either parallel or
barrel tapered profile was extremely high.
A definite tendency was
Of note, the tapered shafts averaged about 50
to 70 grains less mass than either the parallel or the barrel tapered
shafts. They also had a larger diameter
at the point just back of the broadhead’s ferrule than either the parallel or
tapered shafts, though ALL the shafts still had a diameter (just back of the
broadhead) which was less than the broadhead’s ferrule diameter.
What the tapered shafts did have was a
significantly higher percentage of weight forward of center (high FOC) and a
shaft profile that became steadily smaller in diameter towards the rear of the
shaft - a ‘reverse inclined plane’ which, in theory, might result
in a lower overall shaft drag factor.
It is also a feasible hypothesis that the lower mass towards the
rear of the tapered shaft arrow may cause less shaft flexion,
A new series of study ‘focal points’,
designed to isolate only the FOC as a variable between the arrows physical
structure, are planned. How much of the
(consistently significant) difference
in outcome penetration was due to the high FOC and how much to shaft profile or
reduced flexion of the shaft? Only
time will tell.
(8) Shaft and Broadhead Finish. Test data indicates that both a
shaft’s finish and a broadhead’s ‘finish’ has a noteworthy effect on
penetration. A very 'slick' finish on a
shaft increases penetration, as it reduces the ‘coefficient of friction’
between shaft and tissues.
In soft tissues, recent test data is also highly
suggestive that such metal finishes as Teflon coating aids a broadhead’s
penetration through soft tissues, though a broadhead's finish appears to
have very little, if any, significant effect on an arrow’s (or broadhead’s)
ability to penetrate hard tissues (bone).
terminal arrow performance is tested further, new information will be
learned. As it stands now, the forgoing
is the best I can recommend, and be assured it correctly reflects the outcome
results relative to arrow penetration.
All of the above
factors are things over which the bowhunter has control. The field evidence clearly shows
that wise equipment selection does result in increased
lethality of the hunting arrow. All
that remains for the bowhunter to do is sharpen his or her shooting and hunting
I hope the forgoing provides some insight into the penetration
characteristics of arrows, and provides some practical applications for the
bowhunter. For those interested in
calculating the momentum and/or kinetic energy of their own
arrows, here are the formulas in a simple to use format:
Momentum = Mass x Velocity
In other words, momentum equals the arrow’s mass, measured in
grains, multiplied by the arrow’s velocity, expressed in feet per second, and
then divided by 225218. The resultant
answer will be expressed in slug-feet per second.
Kinetic Energy = ½ Mass x Velocity2
This says, the kinetic energy equals one-half the arrow’s mass,
expressed in grains, multiplied by the arrow’s velocity (expressed in
feet-per-second), then multiplied by the arrow’s velocity again, and all of
that is then divided by 225218. The answer
will be expressed in foot-pounds.
The denominator in the above equations, 225218, converts the arrow’s
physical weight, measured in grains, into pounds, and also factors in the
gravitational constant (gc).
There are 7000 grains per pound.
The gravitational constant is 32.174 feet per second per second. Thus, 7000 x 32.174 = 225218.
A special "Thanks" to O. L. Adcock for his review and comments
regarding bow efficiency and to Erik Beiergrohslein for his professional review
of the accuracy of all Physics formulas and calculation, as well as the
exactitude of their application(s) in the above document.