Since the velocity depends upon the time it takes for the arrow to return to its original shape after being deflected, the speed with which the arrow comes to rest as far as vibrating is concerned can be expressed by:
Where T is the time is takes the arrow to return to its true shape.
K is a constant.
W is the weight of the arrow.
l is the length of the arrow.
E is the modulus of elasticity of the material.
I is the moment of inertia, depending on the shape and size of the cross-sectional area of the arrow.
If the weight, length and diameter of a set of arrows are made constant, the speed of return will then depend upon "E" the modulus of elasticity.
The stiffness of an arrow is measured by its resistance to being deflected by a given load or pressure, and can therefore be determined by placing it on two supports, suspending a given weight from the arrow half way between the supports and measuring the maximum deflection. This deflection can be expressed by the equation:
where D is the maximum deflection.
P is the load or pressure on the arrow half way between the supports.
l is the distance between the supports.
E is the modulus of elasticity.
I is the moment of inertia.
The smaller the deflection "D" the stiffer the arrow tested.
If the Pl, and I of a set of arrows is fixed, the stiffness will then depend upon the modulus of elasticity "E."