Some months ago, while shooting a bow somewhat lighter than usual, I noticed that my point of aim at 50 yards remained precisely where it had been at 60. From measurements I had made with a chronograph, I knew the arrow velocity to be about 148 feet per second. Thinking that there might possibly be a simple relationship between arrow velocity and the two ranges at which the point of aim was the same, relative to the target. I formulated the problem and found that the relationship is not at all simple. However, hiving become interested in the more general problem, I decided to see it through and to explore its possibilities for other interesting and perhaps practical relationships. This was accordingly done. Although it has represented many hours of tedious computation, the results are so interesting and valuable that the effort has been quite worth-while.
Briefly, the problem as finally formulated, was this: Assume (See Figure 1) that we know h, the height of the archer's eye above the ground level; f, the height of his eye above the axis of the arrow when drawn to his anchor point; l, the length of the arrow; and v0 the initial velocity of the arrow. We wish to find, first, the location of the point-of-aim relative to the target at each of the standard ranges (30, 40, 50, 60, 80 and 100 yards); second, the settings of the sighting point on a bow, relative to the axis of the drawn arrow, for each of the above ranges. These points of aim and sight settings are to be computed for velocities of 130, 140, 150, 160, 170 and 180 feet per second, making proper allowances for the effect of air resistance upon the trajectory of the arrow. We desire to obtain the data for a 325 grain and a 400 grain cylindrical target arrow five-sixteenths inch in diameter.
The reader will agree that this is at least a tedious, if not a difficult problem, and a large contract; but if he will consider for a moment what information its solution may yield, he will appreciate its value. For example, with the data obtained from the solution, the writer can take any bow and, with a 325 grain or 400 grain arrow, determine the location of his point of aim for a measured distance, and from this distance, determine at once the arrow velocity. Knowing this, he can, without further trial, determine the location of his point of aim for any standard target distance. If he uses a bow sight, he can, in the same way, determine its settings and can tell whether he needs a prismatic sight at any given range, or whether he can use a direct sight to aim on the gold; and if he requires a prism, what minimum strength of prism will enable him to aim on the gold at any range.