October 2006

we receive many questions about figuring kinetic energy and
momentum in arrows. Below is information about the formulas for calculating kinetic energy and momentum, their relationship, and the derivation of these formulas. There are only two basic
formulas: one for kinetic energy and one for momentum, although there are
probably many ways to write them. Each formula has several constants that are required to make them usable in a form where the values are expressed in terms we are familiar with. Conversion to grains for the arrows and from the British gravitational units of poundals to pounds-mass are a part of that.

Determining An Arrow’s
Kinetic Energy
The basic formula for kinetic energy is:

To use the weight of the arrow in grains, Our usual unit of measurement, it is neccessary to convert from poundaIs to grains
in the formuIa, therefore:

Note: The acceleration of gravity
(g) varies with latitude. As latitude increases, “g” also increases. 32.16 feet
per second per second corresponds to about 40 degrees latitude, which is a
reasonably good average for the United States. Gravitational pull is higher at the


Dimensionally masses are measured
in poundals and velocities in feet per second. A poundal is defined as the
force which, if applied to the standard pound body, would give that body an
acceleration of one foot per second per second. One poundal equals 1 /3 2. 1 740
pound-force (lbf). These dimensions are stated in the British “absolute system” in which the basic dimensional units are: one poundal, one foot, and one second. Therefore, the basic unit of
momentum is one poundal-second.

When momentum is expressed in the British gravitational system (the system in most common use in the United States), the basic unit is one pound-second. One pound-second is equivalent
to 32.1740 poundal-seconds. Work or energy is expressed in foot- pounds in the British “gravitational system,” or as foot-poundals in the British “absolute system.” Again, the acceleration of gravity enters the picture so that: one foot-pound = 32.1740 foot-poundals.

Unfortunately the term “pound” is used ambiguously to define both “force”
and “mass” in most instances. To distinguish between these two usages, the term “pound- force” was coined to apply to the pound when it is used to express force, and the term “pound-mass” was designated to apply to pound when it is used to indicate mass.
Simply stated:
“A load that produces a vertically downward force because of the influence of gravity acting on a mass may be expressed in ‘mass’ units. Any other load is expressed in ‘force’ units.”
The kinetic energy of an arrow in flight is a function of its mass and velocity squared, as shown in the formula outlined above. It has the dimension of foot-pounds. The momentum of the same arrow is also a function of its mass and the single power of its velocity. Momentum
has the dimensions of foot»seconds. The difference between kinetic energy and momentum is a function of the velocity divided by 2 and, of course, the change in dimensions from foot-pounds to
pound-seconds. lf kinetic energy of the arrow is divided by “v/2,” then the result
is the momentum of the arrow. For example: An arrow with a weight of 450 grains and a velocity of 230 feet per second will have a kinetic energy of 52.8718 foot-pounds.
Dividing 230 by 2 yields 115. Dividing
52.8718 by 115 gives a momentum of
0.4598 pound-seconds.

To calculate momentum directly the following formula can be used:
momentum = wav/225120 Ib.-secs. 1
wa is arrow weight in grains {
v is arrow velocity in feet per second. y
For example: An arrow with a weight Y
of 450 grains and a velocity of 230 feet per
y second will have momentum equal to:
450 x 250/225120 = 0.4598 pound-seconds.
To Calculate momentum directly the following formula can be used.

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