Assume therefore that the weight of arrow is 260 grains, the initial velocity is 300 feet per second, the shape of head is ogival, and that the spine remains constant, regardless of 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 all the other features constant, we can determine their effect on maximum range.

C_{0} 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 on 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.

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**CONCLUSIONS**

- An approximate formula (5) has been established for determining the resistance in air of any design arrow. The constants used in the formula were established by wind tunnel tests. Pendulum tests by English check the formula very closely.
- An accurate and simple method for solving arrow trajectories is available in Ingalls' Ballistic Tables. These tables are used by the Artillery organizations of most large countries, and are sufficiently accurate for all bow and arrow design studies.
- Use of an angle of departure of 42° will result in only a very small error in determining the maximum range for all bow and arrow conditions.
- Simple charts (2 and 3) have been developed from Ingalls' Ballistic Tables for quickly determining the maximum flight ranges of arrows of any design for any initial velocity between 100 and 500 feet per second.
- The design of an arrow has considerable effect on the maximum range.
- A change in length of the arrow affects the flight rang because of the change in surface area. See Chart No. 4.
- A change in arrow diameter affects the maximum flight range considerably because of the change in surface area and also the change in head-on resistance. See Chart No. 4 For example, increasing the arrow diameter from .250" to .300", leaving all other features the same, will decrease the maximum flight range 7%.
- The size of the feathers has a tremendous effect or maximum range. See Chart No. 4. Increasing the feather area from 2.0 square inches to 6.0 square inches decreases the maximum range 20% or 100 yards out of 500.
- Celluloid vanes increase the maximum range as compared to feathers. See Chart No. 4. The amount of increase depends upon the size of feather. With a total area of 2.C square inches for both sides of three feathers, the celluloid vanes increase the maximum flight range 12% or 60 yard; out of 500 yards.
- Further wind tunnel tests on arrows are necessary to establish more accurate coefficients in the formula for air resistance.

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