The value of the coefficient of form, i, is determined by experimental ranging at the Naval Proving Ground. A number of shots (from four to seven) are fired at each angle of elevation in a series such as the following: 5°, 15°, 20°, 30°, 45°, and again at 15 degrees. The velocity is measured at each angle. Throughout the firing, meteorological observations are made, at the surface and aloft, to determine wind, temperature, and atmospheric density. On the basis of these observations computations are made for ballistic density and ballistic wind for the predicted height of maximum ordinate.
Since the projectile travels through various altitude zones, it is clear that values of wind, temperature, and density do not remain the same throughout the trajectory. Therefore, fictitious values of wind and air density are computed which represent a constant value for each over the entire trajectory. The fictitious values are called ballistic density, ballistic wind, and ballistic temperature; they are used for correcting the trajectories derived from observed data to those which would exist in the arbitrarily defined standard atmosphere on which retardation of the standard projectile is based.
Therefore it is understood that 90 percent or more of the range table is mathematically computed for each range table.
17B1. Preparation of Range Tables
For service use, the Bureau of Ordnance publishes for each gun and for each projectile type fired by the gun a range table, which presents, in convenient form for reference, the elements of the trajectories resulting from firing that specified projectile, at a specified initial velocity at various angles of elevation under standard conditions (which will be described later). A range table tabulates, for each 100-yard increment of range, such characteristics of the trajectory as angle of elevation, time of flight, angle of fall, and striking velocity.
It would obviously be impracticable to measure directly all the elements of the various trajectories, data for which appear in a range table. Neither is there a single simple equation whose solution will give the characteristics of a trajectory at any point, principally because of complications introduced by air resistance. The actual preparation of the range table involves experimental firings of a small number of the specified projectiles to obtain data which are used in conjunction with a complex set of equations or formulas to solve the trajectory.
The equations require (in addition to values of initial velocity and angle of departure) a value known as the Ballistic Coefficient, C, for their solution. This coefficient is a measure of comparison between the retardation of the specific projectile for which the range table is being prepared and the retardation of a projectile of a specific standard form in air of an arbitrarily chosen standard density. The expression for C is:
C = w / i·d²
where w = weight of projectile in pounds, d = diameter of the projectile in inches, and i = coefficient of form (the ratio of the retardation of the given projectile to that of a projectile of standard characteristics).
The retardation of the projectile of standard characteristics which is being used as a basis for comparison is available in either the form of a drag coefficient or the form of a resistance curve. The data for such a drag coefficient or resistance curve are available from measurements obtained by actual experimental firings of the standard projectile. It should be noted that i will include not only retardation relations based on form, but any factors, other than weight and diameter, which affect retardation.
The value of the coefficient of form, i, is determined by experimental ranging at the Naval Proving Ground. A number of shots (from four to seven) are fired at each angle of elevation in a series such as the following: 5°, 15°, 20°, 30°, 45°, and again at 15 degrees. The velocity is measured at each angle. Throughout the firing, meteorological observations are made, at the surface and aloft, to determine wind, temperature, and atmospheric density. On the basis of these observations computations are made for ballistic density and ballistic wind for the predicted height of maximum ordinate. Since the projectile travels through various altitude zones, it is clear that values of wind, temperature, and density do not remain the same throughout the trajectory. Therefore, fictitious values of wind and air density are computed which represent a constant value for each over the entire trajectory. The fictitious values are called ballistic density, ballistic wind, and ballistic temperature; they are used for correcting the trajectories derived from observed data to those which would exist in the arbitrarily defined standard atmosphere on which retardation of the standard projectile is based.
Uncorrected observed ranges are recorded and averaged for each angle of elevation. These ranges are then corrected for errors due to: (1) the height of the gun above tide level, (2) the curvature of the earth, (3) jump, and (4) rotation of the earth. These corrections are in addition to the corrections for ballistic temperature, wind, and density mentioned above. The first two corrections mentioned in this paragraph permit the range table to conform to the requirement that the point of fall be in the horizontal plane tangent to the earth at the gun.
The corrected observed values are used to obtain a final coefficient of form, i, for each of the angles of elevation. From this the value of C for the designed weight and diameter of the projectile can be computed.
The observed lateral deviation is corrected for the effect of the component of wind acting at right angles to the line of fire and for rotation of the earth. The remaining deviation is the observed drift.
The formulas mentioned above can be used to solve for the elements at points on the trajectory, since the value of C is now available. Formerly, these formulas were worked out and assembled in ballistic tables for a number of combinations of the variables involved. The entering arguments of such a table are Eg, I.V. and C. From any combination of these three values it was possible to pick off the required values for the range table such as Range, Time of Flight, Angle of Fall, Maximum Ordinate, etc. The ballistic table can thus be considered as a master range table.
It should be obvious that compilation of ballistic tables and construction of range tables from them entail much complicated and tedious work.
Current practice at the Naval Proving Ground is to compute all range table entries directly from the equations by means of large mechanical and electronic computing machines. These machines are capable of handling a large number of complex solutions rapidly and accurately, thus eliminating the need for ballistic tables. Obviously, in using computing machines in the preparation of range tables instead of using ballistic tables, a different method of arriving at the desired value is used rather than a change in basic principle of solution of the trajectory.
The computation of drift for standard conditions is a separate calculation involving the use of the observed drift in a special formula.