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The Flight of an Arrow
Part 1 of 3

At the clout, and in flight competition, one invariably hears questions and comments about the angle of elevation which will give either maximum range or some other distance. In target shooting also, at the various ranges, I have had a consuming curiosity about the path traversed by the arrow, how much it deviates from the parabola which it would follow were there is no air resistance, and the loss of range which is caused by the resistance of the air. It happens, in my case, that ordinary curiosity may be an ephemeral, evanescent, and withal, merely a temporary mental unrest, which is alleviated by reasonable explanations, unsupported by direct experiment. Usually the object of ordinary curiosity seems relatively unimportant, so that investigation and experiment are not considered worth the effort. But, when curiosity becomes consuming, there is just no getting away from it. The only way to get it satisfied is to satisfy it—and that requires getting down to fundamentals and digging until the answer is found. The flight of the arrow is one of the most fascinating things in archery; it stimulates consuming curiosity, and a knowledge of its whys and wherefores makes it increasingly fascinating.

About two years ago, in the "Journal of the Franklin Institute," Mr. F. L. English published an article entitled "The Exterior Ballistics of the Arrow." It represented a splendidly painstaking research. Not only did he work out a mathematical basis for the path of an arrow, but he tested his equations by experimental shootings, determined the coefficients used in them, and then made comparisons between observed and calculated results. Agreement was within a few per cent, which constitutes real testimony to the care with which the work was carried out.

In formulating his theory, Mr. English made two assumptions that can hardly be accepted as having a sound basis. Stokes' law [1], upon which he builds the expression for air resistance, or drag, applies only to small spheres, such as minute droplets, falling slowly in air. It does not apply to projectiles, not even to projectiles of low velocity, such as arrows. The second unjustified assumption is that of a negative acceleration of fall—in addition to drag—by which the gravitational acceleration is diminished, also proportional to the velocity[2]. The only forces encountered by the arrow during its flight are the force of gravity and the air drag along its path. The latter can, of course, be split into horizontal and vertical components. The vertical component of the drag acts downward while the arrow is rising, and upward while it is falling.

Dr. F. R. Moulton, whose book on ballistics has become a classic, stated to me in conversation that the air drag is proportional to the square of the velocity in the case of a slow-moving projectile like an arrow. His assertion is in agreement with that of Mr. George J. Higgins, professor of aeronautical engineering at the University of Detroit, and a good archer. Dr. Moulton and Mr. Higgins also agree that the explanation of the excellent agreement between the observed and computed results of Mr. English is that the coefficients, or constant factors, used in his formulas were determined from data obtained in shooting arrows which were similar to, or the same, as those later used in his experiments for verifying the formulas. When this is done, it makes little difference whether the law of resistance is assumed to follow a "first power" or a "square" law. The fact remains that, for the kinds of arrows and velocities Mr. English used, his formulas enable him to predict the flight curves, ranges, striking angles, velocities remaining after any distance of travel, etc., within a few per cent. In their application to target arrows, the English formulas thus constitute valuable close approximations to actual conditions, and for this purpose they are accurate within practical limits. It is the purpose of this discussion to make some of the results of English's work accessible to archers, and to present some of the interesting facts about the flight of an arrow. In the discussion which follows, reference will be made to the two plates accompanying this article. For both plates, trajectories have been computed for small initial angles, such as are used in target work, and for the angles of maximum range. In both plates, also, the lower section gives the re-sults for the small initial angles, and the upper section for maximum range.