The angle of elevation at which a gun must be set to obtain a given range is ordinarily obtained by laying the gun at different elevations, observing the ranges and interpolating for intermediate ranges. Several shots must be fired at each degree of elevation in order to obtain a good mean range, slight variations in the conditions of fire being sufficient to make an appreciable difference in the results, so that a single shot is not to be depended upon.
The increasing cost of ammunition makes it an object to reduce the number of experimental firings as far as is consistent with accuracy. Several methods have been proposed and used in calculating trajectories, but the theoretical and experimental results do not agree as closely as can be desired. The principal reason for this discrepancy has been that the resistance of the air depends largely on the form of the head of the projectile, so that some correction must be applied to tables compiled from experimental firings to make them applicable in case a different projectile than that for which the tables are computed is used. In using Mr. Bashforth's tables this correction is made by multiplying dw/v2— by a factor, A, depending on the form of the head of the projectile, d being the diameter in inches and w the weight in pounds. The theoretical value of this constant factor A does not, however, give good results in all cases. In order to avoid this difficulty and at the same time to decrease the work required in computing trajectories in the ordinary way, the following method is suggested:
The connection between range and angle of elevation in a non-resisting medium is given by the expression:
sin 2a = gX/v2
in which X is the range in feet, g the force of gravity, v the initial velocity, and a the angle of departure (i. e., the angle of elevation plus the jump, if there is jump). In a resisting medium, v1, the initial velocity if substituted in this equation would give too small a value of Q, and v1, the final velocity at the range X, would give too large a value; evidently there must be some value between the limits v1 and v2 which will give the correct value of a. In a resisting medium we might expect good results from a substitution of the mean value of the square of the velocity throughout the trajectory for v in Formula 1.
An examination of Mr. Bashforth's tables of the resistance of the air shows that for high and low velocities the resistance is proportional to the square of the velocity; for velocities between 750 and 1350 f s., however, the resistance is proportional to higher powers of the velocity, the cubic law of resistance being nearer the truth between these limits than the square.
(Additional formulas are available in the PDF.)