*Comment by the Bureau of Ordnance: Hawkesworth's article has been checked over by the Bureau and is correct so far as the validity of the assumptions and evaluations are concerned. In some cases the calculations are based upon the approximations of the ballistic functions, which seems a justifiable procedure.
As a concrete case, were a 6-inch shell—weight 105 pounds, coefficient of form .61, muzzle velocity 3000 ft./sec.—fired horizontally into a water tank 6 feet above its floor, how far will this shell travel under water before it strikes the floor of the tank? What will be the elapsed time in seconds? And what will be the residual or striking velocity? And the striking angle or angle of fall?
The volume of the shell is 471.098 cubic inches, and it thus displaces 17.012 pounds of water, weighing 62.4 pounds per cubic foot, or .036111 pounds per cubic inch. So that the reduced weight of the shell under water will be 87.988 pounds.
The conditions chosen are such as to make the conclusions of interest mainly from a theoretical point of view, since the shell is assumed to be fired horizontally into the water, and the effect of entry into the water at the varying angles of impact found in practice is not considered. The latter condition complicates the problem of the trajectory considerably, especially at the usual angles of fall, that is, below the neighborhood of 22½degrees.
Next, air weighs .08071 pounds per cubic foot to water's 62.4. So that water is 773.06 times denser than air; with a correspondingly magnified resistance. Hence our modified ballistic coefficient for our under-water trajectory must be Wherein W1 is our modified under-water weight of our shell; j is our coefficient of form; and d2 is the square of the shell's diameter. While 773.06 is our enlarged resistance factor.
Thus our amended ballistic coefficient is C1=.00518295.
Thirdly, to determine the amended gravity coefficient g1for under water. Now force, in this case the shell's weight, is ever the product of mass (a fixed thing) by acceleration; which here is g1. lute weight of the shell in vacuo; that is, its weight in air, plus the weight of the volume of air displaced by it; w1, is its diminished weight under water; g is the gravity coefficient in vacuo; and g1, is the desired new gravity coefficient for under water, which plainly must vary as the specific gravity of the sinking object, in this instance being 26.949 ft./sec.2.
And similarly for any medium. Thus the gravity coefficient g1 for any given object in air will be equal to the absolute coefficient g for vacuum, multiplied by the proper fraction whose numerator w1/W, is the known weight of the given object in air, and whose denominator W is the absolute weight of said object; that is, its weight in air, plus the weight of the volume of air displaced by it.
By Lieutenant Schuyler's empirical formula, deduced from an experiment at the Indian Head Naval Proving Ground, the underwater travel should be 254.723 feet; his corrected and condensed formula, after allowing for an error in a coefficient K, and also for the greater density of sea water (1.026), for the distance travelled during the reduction of the initial velocity of V1=-3000 ft./sec to the residual velocity of V=147.7757 ft./sec.; while W=105 pounds, the weight of the shell in air, will be
The shell, then, striking the bottom of the tank this glancing blow at 3° 40' 38", will ricochet upwards, at the same or nearly the same angle; describe a low trajectory arc; again strike the bottom at a much increased angle and much diminished residual velocity; again ricochet and repeat the performance, until it comes to rest. And its residual velocity at the point of second ricochet will be given by the equation
Yet this surely overlooks two considerations, one of which is very important. First, the glancing blow struck by the shell must certainly destroy some of its energy and velocity, and probably also lowers somewhat its angle of rebound or ricochet, both factors shortening the rebound and making it less than the calculated 32 feet.
Then, secondly and chiefly, the rotation of the shell to the right (as seen from its rear), the moment it touched and was reacted upon by the bottom of the tank, would throw it violently to the right, causing it to "corkscrew" erratically in that direction and smash into the right-hand wall of the tank perhaps 6 or 8 feet beyond the point where it scraped the tank's bottom.
To sum up, then, by simply employing the familiar formulas of Mayevski, Siacci, Ingals, and Alger, but using a modified ballistic coefficient C that is 773.06 times less than the ordinary coefficient for air, we will be easily able to calculate any under-water trajectory; while a similar correction will make available, for this purpose, the exterior ballistics methods of Moulton and of Gamier. So that, were one to draw the graphs of, say, the retardation or of the resistance, then the companion graphs in air and in water would be identical; save only that those in water would be "telescoped" down, with ordinates only 1/925 times the size of those in air. Nor need we be concerned here about the breaks or "stop points" in Mayevski's functions, nor even consider the change of retardation at the critical point of the velocity of sound in the medium—this case in water—since the flattening of our graph is so excessive, being but 1/925 of its previous air value, that any and all such influences are practically negligible.
The changes in resistance and thus in retardation, by changes in the temperature, and above all in the salinity of the water, can easily be allowed for by a corresponding change in the density divisor and diminished weight.
Or were we asked to determine the projectile's path upon entering the water, at the finish of a trajectory in air, then we but need to take the calculated residual velocity and angle of fall at the said point as our initial velocity and angle of departure for the required under-water trajectory.