I have wondered whether archers who take their sport seriously haven't, like myself, desired to know with certainty how the hit in the target is affected, first, by failure to have the arrow accurately sighted, at the instant of loose, and second, by "creeping". My guess is that most of us are much interested in these questions.
It has been a pleasant process of minor mental gymnastics to dig through this problem and the results are enlightening. I shall not undertake to bore the reader with the mathematical details, but shall show how the answers to the problems are obtained, and present the answers. Among other things it seemed desireable to know a little more about point-of-aim shooting as compared with the use of a bow-sight. This question is considered also.
Consider first the process of aiming, either with a point-of-aim, or with a sight. It consists of fixing the arrow, by sighting, in such position, at full draw, that it will hit, as closely as possible, on the center of the target. If the archer succeeds in thus fixing the location of the properly drawn arrow, with a definite anchor, and with his line of sight directly on his point of-aim, he hits the mark, provided the seventeen other things than can cause him to miss are correctly executed. For the sake of simplicity we shall consider that all other points of technique are perfect, and fix our attention only on the act of aiming. If the archer holds his anchor, but allows the bow-hand to shift, or if he shifts his anchor slightly, the angle of departure of the arrow is changed by a small amount, either laterally or vertically or both. Lateral error in this angle results in a hit that is to the right or left by an amount strictly proportional to the error in angle, and the distance from the target. For example, if the tip of a 28 inch arrow is shifted sidewise from its correct position by 1-10 inch the angle is about 12 minutes of arc, or 1-5 degree. This results in an error of 5.8 inches at 40 yards, 6.4 at 50, 7.7 at 60 and 9.1 at 80, to the right or left. This is true regardless of the method of sighting, since we are considering the effect of a small angular displacement of the drawn arrow. The result applies to an arrow of any velocity.
Suppose the displacement of the tip is vertical instead of lateral. We cannot say offhand that the up-and-down errors in hits on the target will be the same as when the shift is lateral, since we are now dealing with the actual path of flight of the arrow, and we cannot be sure, without investigating, just where this path intersects the target face in relation to the center. It seems at first glance that the errors in hits at different distances, corresponding to a certain small angular error of holding, may be quite different from the lateral errors at the same distances. We should also know whether and how these errors differ for different arrow velocities.
In physics we have a simple mathematical expression which, neglecting air resistance, describes the path of the arrow. The expression is
in which v is the horizontal distance of the arrow, at some point in its path, from its point of departure (the archer's anchor) and y its corresponding vertical distance above the horizontal line, a is the angle of departure with the horizontal, y the arrow velocity, and g the acceleration of gravity. Let us assume that the horizontal line from the archer's anchor intersects the target at the center. Then we can substitute in the formula for v, x and y corresponding to actual conditions, and find a, the initial angle for each distance from shooting line to target. In each case we set this distance equal to x and take y=0, since the hit is on the horizontal line along which x is measured. Then we assign selected values to v and solve for α.