`
Exterior Ballistics
of Bows and Arrows deals with the flight
of an arrow after it leaves the bow.
This article is concerned primarily with
the maximum range that can be obtained
with arrows of various designs, and with
various initial velocities. When an
arrow is shot from a bow, the distance
that it will fly depends upon the
following: `

`
1 – The velocity with
which it leaves the bow, called the
"Initial Velocity";
`

`
2 – The angle of
departure; `

`
3 – The weight of the
arrow; `

`
4 – The resistance or
friction or drag of the arrow in air;
`

`
5 – Velocity and
direction of wind. `

`
The velocity with
which an arrow leaves the bow depends
upon the force of the bow, the
efficiency of the bow, and the mass of
the arrow. The article will not deal
with the relationship between these
factors, which is called "Interior
Ballistics", but will simply assume a
given initial velocity when calculating
the effect of arrow design on maximum
range. `

`
The angle of
departure for maximum range is
approximately 45 degrees with the
horizontal. The actual angle of
departure for maximum range will be
calculated below. `

`
Since the effect of
wind on the maximum range is variable
and hard to determine, no attempt will
be made to evaluate its effects.
`

`
The resistance to
motion of an arrow during its flight
thru the air considerably reduces its
range. Since this resistance depends
upon the design of an arrow, the first
step in determining the effect of arrow
design on maximum range will be to
determine its resistance. `

`
Resistance varies
with velocity. The higher the velocity
the greater the resistance. It has been
found in wind tunnel tests, and by the
firing of a large number of projectiles,
that the resistance varies as the square
of the velocity for velocities below 800
feet per second . `

`
In archery, we deal
with velocities well below the value,
and a simple equation for resistance can
be written as follows: `

`
Rf = KV^2
`

Where Rf = the resistance in pounds

K = a coefficient depending upon arrow
design.

V = velocity in feet per second.

`
The coefficient K
applies to only one design of arrow. If
the value of K is measured for a given
arrow, any change in design, such as a
change in length, diameter, or size of
feather, would require a new wind tunnel
test to determine the new value of "K".
To obviate an infinite number of wind
tunnel tests, the separate factors in
arrow design affecting the resistance
coefficient can be evaluated, making it
possible to calculate "K" for any design
of arrow.`

`
The resistance of
arrow is made up of the following:
`

`
1 – The head-on
resistance, which includes the rear end
drag. This resistance depends upon the
head-on area of the arrow, and will
therefore vary as the square of the
diameter. The head-on resistance will
also vary in accordance with the shape
of the head of the arrow, an ogival
shaped head giving considerably less
resistance than a blunt head.
`

`
2 – Skin friction of
the arrow shaft. This varies with the
surface area of the shaft and therefore
varies with the length and diameter of
the shaft. `

`
3 – Skin friction of
the feathers. This will vary with the
total area of both sides of the
feathers. The above can be expressed in
terms of an equation as follows:
`

`
K= K'BD^2 + K" LD +
K'''F
(2)`

Where K, K', K'' and K''' are
coefficients.

B is a coefficient of form for the head
of the arrow

D is the diameter of the arrow in
inches.

L is the length of the arrow in inches.

F is the area of both sides of the
feathers in square inches.

`
By determining the
coefficients K, K', K'' and K''' in
equation (2), the value of K for any
design of arrow can be calculated. One
way to determine these values accurately
is by wind tunnel tests. `

`
The author knows of
only one wind tunnel experiment on
arrows. The May-June, 1935, issue of
Army Ordinance contained in an article
entitled "the Bow as a Missile Weapon"
by Vice Admiral W.L. Rodgers. He gives
the results of the test made by Rear
Admiral Moffet, chief of the Bureau of
Aviation, on an arrow with ogival head,
26 inches long, 5/16 inch diameter, with
3 feathers 2 1/2 inches long, having a
total feather area of 7.5 square inches.
`

`
At 200 feet per
second(FPS), the resistance of this
arrow was 0.039 pounds. Removing the
feathers, the resistance was 0.016
pounds. These two tests make it possible
to determine the value of K''' in
formula (2) as follows; `

`
0.039 - 0.016 - 0.023
pounds = the resistance of the 7 1/2
square inches of feathers at 200 FPS
velocity. `

`
Therefore K''' =
0.023/(7.5 X 2002) = 0.000000077
`

`
Since this arrow
without feathers gave a resistance of
0.016 pounds, another equation can be
written as follows; `

`
(K'BD^2 + K''LD) =
0.016
(3)`

`
However this equation
has three unknowns, K', K'', and B. By
determining any two of these, the third
can be calculated from the equation.
`

`
Wind tunnel tests on
various shaped objects give a clue to
the head-on resistance represented by
the factor K'BD2. For example, K'B for a
cone varies from 0.0000033 for a 60
degree included angle to 0.0000015 for a
20 degree included angle. The cone on
most parallel piles used on arrows has
about a 60 degree included angle.
However, the above values are based on a
cone with a flat base, whereas arrows
are usually tapered towards the rear,
and the ends are slightly rounded. This
should reduce the resistance coefficient
somewhat, but in an arrow this is
largely offset by the actual nocks,
which create a disturbance in the
streamline flow. This value of K'B =
0.000003 for an arrow with a parallel
pile is probably a fair approximation.
`

`
For a bullet shaped
or ogival point, a still smaller value
can be expected. The best known formula
for the resistance of a bullet is that
of Mayeski, which for velocities below
800 FPS gives K'B = 0.00000135, and
where B, the coefficient of form, equals
1, for the ogival head. `

`
When used on an
arrow, this coefficient can again be
slightly reduced because of slightly
improved rear end conditions as follows;
`

`
K'B = 0.0000013
`

If B = 1 for ogivals heads and

B = 2.3 for parallel piles, the value of
K' becomes 0.0000013.

`
The value of B for a
blunt head is approximately 7.0. Having
established values for K' and B, these
values can be substituted in equation
(3) as follows:`

`
0.016 = (0.0000013 X
1 X 0.312^2 + K'' X .312 X26) X 200^2
and solving for K''`

K'' = 0.000000035

`
Having established
values for K', K'', K''', and B,
equation (2) can be written as follows:`

`
K = 0.0000013BD^2 +
0.000000035LD + 0.000000077F
(4)`

B = 1 for ogival or bullet shaped heads

B = 2.3 for parallel pile heads

B = 7.0 for blunt heads

D = Maximum diameter of arrow in inches

L = Length of shaft in inches

F = Area of both sides of feather's in
square inches and equation (1) becomes:

Rf = (0.000001.3BD^2 + 0.000000035LD +
0.000000077F)V^2 (5)

`
Inspection of the
above formula will indicate that the
feathers are responsible for the large
part of drag.`

`
Formula (5) also
indicates the advantage of celluloid
vanes as compared to feathers. Both the
resistance of the body of the arrow and
of the feathers depend upon their
respective surface areas. Therefore, ,
the coefficients for these two factors
as determined by the wind tunnel tests
show the difference in resistance
between a smooth surface, such the body
of an arrow, and a rough surface, such
as a feather. The coefficient for the
surface of the feather was found to be
more than double the coefficient for the
surface of the arrow. By using a smooth
celluloid vane with a surface similar to
the body surface of an arrow, the
feather resistance can be reduced by
more than 50%.`

`
In order to check
formula (5), we can make use of a method
commonly used in artillery for measuring
the resistance of projectiles. This
consists of measuring the velocity of
the projectile at points some distance
apart, either by chronograph or a
ballistic pendulum. The formula for
resistance then becomes:`

`
Rf = (W/2gx)(V1^2 -
V2^2)
(6)`

Where W = weight of the projectile in
pounds

g = Acceleration of gravity - 32.2
ft/s^2

x = Distance in feet between points of
measurement of V1 and V2.

V1 = Velocity in feet per second at the
initial point of measurement

V2 = Velocity in feet per second at
distance x from first point

`
English, in his
article on "Exterior Ballistics of
Arrows" in December 1930 issue of the
Journal of the Franklin Institute, gives
the results of tests he made with a
ballistic pendulum on two weights of
arrows with different initial
velocities. He determined the initial
velocity at the bow and the striking
velocity fifty yards from the bow for
the first arrow, and forty yards from
the bow for the second arrow.`

`
For the first test he
used an arrow weighing 0.0615 lbs. this
arrow had an initial velocity of 155.4
feet per second and a striking velocity
of 141.8 feet per second at a distance
of 150 feet from the bow.`

`
From formula (6)`

`
Rf = (0.0615/64.4 X
150) X (155.4^2 - 141.8^2) = 0.0257 lb`

`
In order to compare
this value with formula (5), we have the
length of the arrow given as 31 inches
and the diameter as 0.355 inches.
English also gave the areas of three
different size feathers used in his
various tests, but unfortunately failed
to designate which size was used on any
given test. Assuming that he used
feathers with an area of 1.11 inches per
feather, then from formula (4) -
`

`
K = 0.0000013 X
0.355^2 + 0.000000035 X 0.355 X 31 +
0.0000000767 X 1.11 X 6 = 0.00000106 and
Rf = 0.00000106V^2`

`
The average velocity
for the first test was 148.6 feet per
second. Therefore:`

`
Rf =
0.00000106(148.6)^2 = 0.0236 pounds`

`
The value of
resistance as determined by actual test
was accordingly 9% greater than the
resistance as computed from formula (5).`

`
Similarly, the second
test by English on an arrow weighing
0.0851 pounds with an initial velocity
of 138.5 feet per second and a striking
velocity of 131.4 feet per second, 120
feet away, gives a value of:`

`
Rf = 0.0210 pounds`

`
Formula (5) for this
arrow gives;`

`
Rf = 0.0190 pounds`

`
These two values also
check within 10%. The discrepancies are
either due to an error in the
coefficients formula (4) or an error in
the measurement of velocities by
English. Because of the absence of
sufficient wind tunnel tests on arrows,
an error in the coefficients is entirely
possible. However, since the discrepancy
would only be 4% in the maximum range of
the arrow, the check between resistance
as calculated by formula (4) and as
determined by English is remarkably
close.`

`
Therefore formula
(5), with coefficients based on wind
tunnel tests, probably the most accurate
formula for resistance of an arrow that
is available at the present time.
Additional ballistic pendulum or wind
tunnel tests will undoubtedly modify the
coefficients to some extent, but such
modification will be small and can have
very little effect on the relative
results obtained by its use in this
article.`

`
Having determined the
resistance Rf of an arrow, the next step
is to determine the ballistic
coefficient Co. This is defined as:`

`
Co = AV^2/Ra
(7)`

`
Where A = 0.00004676
for velocities less than 800 feet per
second`

V = velocity in feet per second

Ra = deceleration of the arrow due
to air resistance

`
But Ra = Rf/M = Rfg/W
(from the formula: force=mass X
acceleration`

`
Then Co = AV^2W/Rfg`

`
But Rf = KV^2 from
(1)`

`
Then Co = AW/Kg`

`
Or Co =
0.0000000002075W/K
(8)`

`
Where W = weight of
the arrow in grains and K is obtained
from formula (4)`

`
The ballistic
coefficient Co is therefore dependent
upon permanent features of a given arrow
such as size, weight, and feather area,
and can be readily computed. The
coefficient Co may be thought of as
measuring the "ranging power" of the
arrow. It varies directly with the
weight and inversely with the
coefficient of resistance K.`

`
It should be noted
here that the value of Co is based on
standard atmospheric conditions
corresponding to 59 degrees Fahrenheit
and a barometric pressure of 29.53 and
relative humidity of 78%. For any other
atmospheric conditions, a correction
factor must be applied. However, since
this series of articles deals almost
entirely with relative values, the
correction factor (which is practically
negligible in most cases) will not be
considered.`

`
Having determined the
ballistic coefficient Co, the next step
is to solve the trajectory for a given
initial velocity and given angle of
departure.`

`
The accurate solution
of an actual trajectory is complicated
and laborious because the drag is not
constant but varies as the square of
velocity. The most accurate method is to
divide the trajectory into small parts
and compute them in sequence.`

`
The next most
accurate method is known as the
Ingalls-Siacci method which, in making
certain assumptions, permits a simpler
solution of the problem. No attempt will
be made here to describe this method,
since a detailed description would fill
a book by itself. Suffice it to say that
up to very recent years the
Ingalls-Siacci method has been used
universally for the high degree of
accuracy demanded by artillery in the
solution of trajectories.`

`
In applying this
method, use is made of the Ingalls'
Ballistic Tables, by means of which the
actual work of computation is materially
reduced. These tables are so well known
that copies may be found in almost any
large public library. In recent years,
additional tables have been prepared for
velocities below 800 feet per second.,
where the resistance varies as the
square of the velocity , which simplify
the work still further. since all our
bow and arrow work deals with velocities
below 800 feet per second, these
additional table have materially reduced
the work of solving arrow trajectories.`

`
Having computed the
ballistic coefficient Co, formula (8),
for a given arrow, the complete
trajectory can then be solved for any
initial velocity and any angle of
departure, by use of these tables.
Factors such as range, maximum ordinate,
time of flight and striking velocity can
all be readily and quickly determined.
No further description of these tables
will be given here, since complete
instructions for their use accompany the
tables.`

`
Since in this article
we are interested mainly in the maximum
flight ranges for a given set of
conditions, we can reduce the number of
variables and concentrate on the quick
solution of certain factors of
trajectory.`

`
The first problem is
to determine the angle of departure that
will give maximum range. By assuming a
certain ballistic coefficient Co, and
various values of initial velocity so as
to give various ranges, the maximum
ranges were computed for various angles
of departures by use of the tables. The
result are plotted in Chart No.1. This
shows that as the initial velocity
increases, the angle of departure for
maximum range decreases, but the
variation for quite a number of degrees
is so small that by using an average
angle of 42 degrees for all conditions ,
the maximum possible error is a
fractional part of 1%.`

`
If we assume standard
atmospheric conditions and an angle of
departure of 42 degrees, the ranges for
a large variation of initial velocities
and ballistic coefficients can be
compared. The results of such
computations are shown plotted in Chart
No. 2 and No. 3. Chart No. 2 shows the
maximum range plotted as a function of
the ballistic coefficient Co for every
40 feet per second velocity between 100
and 500. Chart No. 3 shows the maximum
range plotted as a function of velocity
for various values of the ballistic
coefficient Co.`

`
With these two
charts, the maximum flight range for any
size and weight arrow , and for any
initial velocity between 100 and 500
feet per second can be quickly
determined by interpolation. But it
cannot give results for the
Turkish-shaped, heavy banded arrow. For
these, special measurements in wind
tunnels seems necessary.`

`
As an example, assume
an arrow of 250 grains , .275" diameter,
28" long, and with feathers having a
total area including both sides, of 2.0
square inches.`

`
From formula (4) -
`

`
K = 0.0000013 X 1 X
.275^2 + 0.000000035 X 28 X .275 +
0.000000077 X 2.0 = 0.00000051`

`
From formula (7) -
`

`
Co = 0.0000000002075
X 250/0.00000051 = 0.1016`

`
From Chart No. 2 for
an initial velocity of 180 feet per
second, for Co = 0.1016, the maximum
range is 250 yards.`

`
These ranges
represent the maximum range
theoretically possible in absolutely
still air. Under actual conditions, the
resistance of the arrow may be increased
due to wind or due to flirting of the
arrow in flight, both of which would
tend to decrease the range. However, the
theoretical consideration is an
excellent means for determining the
relative effect of various features in
bow and arrow design on range, and also
indicates the maximum range that can be
obtained for a given set of conditions.`

`
Since equation (3)
gives the effect of various portions of
the arrow on air resistance, we can
determine from Chart No. 2 the effects
of changes in arrow design on maximum
range. The features in arrow design
which have no influence on the initial
velocity of the arrow shot from a given
bow but which affect resistance ad
therefore range, are the length, and
diameter of the arrow, the shape of the
head, and size and type of feather.
Weight affects the range by changing the
ballistic coefficient Co, but weight
also affects the velocity of the arrow
from a given bow.`

`
Referring to Chart
No. 2, it will be noted that for a given
change in the ballistic coefficient Co,
the amount of change in range depends
upon the value of the coefficient and
the initial velocity. The effect of the
diameter and the length of arrow, etc.,
on the range will depend upon the weight
and design of the arrow and on the
initial velocity. however, in order to
obtain some idea as to the effect of
these features, we can assume some
average condition.`

`
Let us consider the
effects of arrow length and diameter.
ans size and type of feathers on a
flight arrow for ranges between 450 and
500 yards, which would require a bow of
60 to 80 pounds. The weight of the arrow
for maximum range under this condition
is about 260 grains. The initial
velocity would be 300 feet per second.`

`
Assume therefore that
the weight of arrow is 260 grains, the
initial velocity is 300 feet per second,
the shape of the head is ogival, and
that the spine remains constant,
regardless of the change in length or
diameter.`

`
By varying the length
and diameter of the arrow, and the size
and type of feathers, each in turn ,
while keeping al the other features
constant, we can determine the effect on
maximum range.`

`
Co was determined for
the various conditions by use of formula
(4) and (8), and the ranges were taken
from Chart No. 2. The results of these
computations are shown in Chart No. 4.
The basic arrow was assumed to be 28"
long, .275" in diameter, with feathers
having a total area, including both
sides, of 2.0 square inches. This chart
shows the large effect feather size has
on flight range, and the increased range
to be obtained with celluloid vanes.`