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Exterior Ballistics of Bows and Arrows
Part 9 of 10

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

From formula (4) —
K = .000,001,3 x 1 x .2752 + .000,000,035 x 28 x .275 + .000,000,077 x 2.0 = .000,000,51
From formula (7) —
C0 = (.000,000,000,207,5 x 250) / .000,000,51 = .1016

From Chart No. 2 for an initial velocity of 180 feet per second, for C0 = .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 and 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 C0, 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 C0, the amount of change in range depends upon the value of the coefficient and the initial velocity. The effect of the diameter and length of arrow, etc., on the range will therefore 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, and 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 arrow for maximum range under this condition is about 260 grains. The initial velocity would have to be about 300 feet per second.