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The Flight of an Arrow
Part 2 of 3

Plate I is prepared from computations based on an initial arrow velocity of 140 feet per second, a length of 28 inches, a diameter of 5/16 inch, 2½ inch balloon feathers, and a weight of 325 grains. The initial energy of the arrow is 14.2 foot-pounds. This represents a medium velocity arrow, shot with a bow rather lower than average in weight, e. g., a 36 pound bow of 45% efficiency. Plate II shows the corresponding results for an arrow of the same dimensions, but of 400 grains weight and 180 feet per second initial velocity. Its kinetic energy as it leaves the bow is 28.8 foot-pounds, or more than twice the energy of the arrow of Plate I. It represents, approximately, the use of a 50-pound bow of high (65%) efficiency.

These two cases have been chosen because most target arrows have weights and velocities between the values assumed. In both plates, the scale in the upper figure is double that used in the lower.

Plate I

Lower Figure: The curves marked A, B, C and E represent the actual paths of the arrow when the initial elevations are 6°, 8°, 10°and 16° respec-tively. The dotted curves, D and E, represent the parabolic paths which the arrow would follow in a vacuum, at initial angles of 10° and 16°, respectively. At 10° the actual range is 12 yards less than the parabolic ranges; at 16°, the difference is 17 yards. We notice that at the 40-yard range the elevation of about 6° is correct, as is the elevation of 8° at 50. The 60-yard range requires somewhat greater elevation than 10°, and the 100, somewhat greater than 16°. At the short ranges, especially at 40 yards, the difference in the actual path and the parabola is very slight. The time of flight for curve A is .84 seconds; for curve B, 1.06 seconds; for C, 1.26 seconds, and for E, 2.16 seconds.

Upper Figure: Curve L (dotted) is the 45° parabola giving maximum range without air resistance. Curve G is the actual trajectory for maximum range, with an initial elevation of 42°. This range is 137 yards, as against 206 in vacuum, a loss of 69 yards, or 33.7%, due to air resistance. Curve H is the descending portion of the trajectory at 40° initial elevation, and curve K at 44°. Note that 40° gives slightly greater range than 44°, but neither quite reaches the range at 42°. Since the differences are a matter of only a yard or two, we may say that in this case any elevation between 39° and 44° would give close to maximum range. This may explain the uncanny accuracy possible in clout shooting when the maximum range of the arrow is 180 yards. A variation of 5° in elevation would make a variation in range of only a yard or two. It is interesting to see also that at maximum range the 140 feet per second arrow rises more than 130 feet before it begins its descent. The time of flight is about 5.2 seconds. The average horizontal velocity is 79 feet per second.

Plate II

Lower Figure: The lower group of four curves, A, B, C and D represents the trajectories at initial elevations of 4°, 6°, 8° and 10°, with corre-sponding ranges of 44, 66, 85 and 102 yards, respectively. Curve E is the trajectory for 16° elevation, with a range of 153 yards, while the dotted curve F is the path in vacuum at 16° elevation. The loss of range is, in this case. 20 yards. The elevation for 100 yards is just under 10°. At the range corresponding to 10" elevation, the time of flight for the 102 yards is 1.81 seconds, The time for the shorter distances are approximately 1.55 seconds for 85 yards, 1.2 seconds for 66 yards, and .8 second for the 44 yard range.

Upper Figure: The dotted line H is the 45° parabola, showing a maxi-mum range in vacuum of 334 yards. The solid line G is the trajectory for the actual maximum range, 263 yards, at 40° initial angle. The loss of range in this instance is only 21.3%. The curves for other angles of elevation are not shown, although they have been computed. In this case, as in the other, a variation in elevation from 39° to 44° makes a variation in range of only a yard or two. The time of flight at maximum range is 8.0 seconds, and the maximum rise is 215 feet. The average horizontal velocity is 125.3 feet per second.