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Exterior Ballistics of Bows and Arrows
Part 3 of 10

The author knows of only one wind tunnel experiment on arrows. The May-June, 1935, issue of Army Ordnance contained an article entitled "the Bow as a Missile Weapon" by Vice Admiral W. L. Rodgers. He gives the results of a wind tunnel 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 2½ inches long, having a total feather area on both sides of 7.5 square inches.

At 200 feet per second, the resistance of this arrow was .039 pounds. Removing the feathers, the resistance was .016 pounds. These two tests make it possible to determine the value of K111in formula (2) as follows:

.039 — .016 = .023 lbs. = the resistance of 7½ square inches of feathers at 200 F.P.S. velocity.

Therefore      formula38 (5K)

Since this arrow without feathers gave a resistance of .016 pounds, another equation can be written as follows:

(K1 BD2 + K11 LD) V2 = .016 (3)

However, this equation has the three unknowns, K1, K11 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 K1 BD2. For example. K1 B for a cone varies from .000,003,3 for a 60° included angle to .000,001,5 for a 20° included angle. The cone on most parallel piles used on arrows has about a 60° 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. Thus a value of K1 B = .000,003 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 Mayevski, which for velocities below 800 F.P.S. gives K1 B = .000,001,35, and where B, the coefficient of form, equals 1, for the ogival head.