(26) d^2x/dt^2 = -r = -gp/C, (27) d^2y/dt^2 = -g.
The first equation leads, as before, to
(28) t = C{T(V) - T(v)}, (29) x = C{S(V) - S(v)}.
The integration of (24) gives
(30) dy/dt = constant - gt = g(T - t),
if T denotes the whole time of flight from O to the point B (fig. 1), where the trajectory cuts the line of sight; so that T is the time to the vertex A, where the shot is flying parallel to OB.
Integrating (27) again,
(31) y = g(Tt - t^2) = gt(T - t);
and denoting T - t by t', and taking g = 32f/s^2,
(32) y = 16tt',
which is Colonel Sladen's formula, employed in plotting ordinates of a trajectory.
At the vertex A, where y = H, we have t = t' = T, so that
(33) H = 1/8gT^2,
which for practical purposes, taking g = 32, is replaced by
(34) H = 4T^2, or (2T)^2.
Thus, if the time of flight of a shell is 5 sec., the height of the vertex of the trajectory is about 100 ft.; and if the fuse is set to burst the shell one-tenth of a second short of its impact at B, the height of the burst is 7.84, say 8 ft.
The line of sight Ox, considered horizontal in range table results, may be inclined slightly to the horizon, as in shooting up or down a moderate slope, without appreciable modification of (28) and (29), and y or PM is still drawn vertically to meet OB in M.
Given the ballistic coefficient C, the initial velocity V, and a range of R yds. or X = 3R ft., the final velocity v is first calculated from (29) by
(35) S(v) = S(V) - X/C,
and then the time of flight T by
(36) T = C{T(V) - T(v)}.
Denoting the angle of departure and descent, measured in degrees and from the line of sight OB by [phi] and [beta], the total deviation in the range OB is (fig. 1)
(37) [delta] = [phi] + [beta] = C{D(V) - D(v)}.
To share the [delta] between [phi] and [beta], the vertex A is taken as the point of _half-time_ (and therefore beyond _half-range_, because of the continual diminution of the velocity), and the velocity v_0 at A is calculated from the formula
(38) T(v_0) = T(V) - T/C = {T(V) + T(v)};
and now the degree table for D(v) gives
(39) [phi] = C{D(V) - D(v_0)}, (40) [beta] = C{D(v_0) - D(v)}.
This value of [phi] is the tangent elevation (T.E.); the quadrant elevation (Q.E.) is [phi] - S, where S is the angular depression of the line of sight OB; and if O is h ft. vertical above B, the angle S at a range of R yds. is given by
(41) sin S = h/3R,
or, for a small angle, expressed in minutes, taking the radian as 3438',
(42) S = 1146h/R.
So also the angle [beta] must be increased by S to obtain the angle at which the shot strikes a horizontal plane--the water, for instance.
A systematic exercise is given here of the compilation of a range table by calculation with the ballistic table; and it is to be compared with the published official range table which follows.
A discrepancy between a calculated and tabulated result will serve to show the influence of a slight change in the coefficient of reduction n, and the muzzle velocity V.
_Example_ 3.--Determine by calculation with the abridged ballistic table the remaining velocity v, the time of flight t, angle of elevation [phi], and descent [beta] of this 6-in. gun at ranges 500, 1000, 1500, 2000 yds., taking the muzzle velocity V = 2150 f/s, and a coefficient of reduction n = 0.96. [For Table see p. 274.]
An important problem is to determine the alteration of elevation for firing up and down a slope. It is found that the alteration of the tangent elevation is almost insensible, but the quadrant elevation requires the addition or subtraction of the angle of sight.
Example.--Find the alteration of elevation required at a range of 3000 yds.
in the exchange of fire between a ship and a fort 1200 ft. high, a 12-in.
gun being employed on each side, firing a shot weighing 850 lb with velocity 2150 f/s. The complete ballistic table, and the method of high angle fire (see below) must be employed.
[v.03 p.0273]
Range. s. s/C. S(v). v. T(v). t/C. t. T(v_0). v_0 0 0 0 20700.53 2150 28.6891 0.0000 0.000 28.6891 215 500 1500 518 20182.53 1999 28.4399 0.2492 0.720 28.5645 207 1000 3000 1036 19664.53 1862 28.1711 0.5180 1.497 28.4301 199 1500 4500 1554 19146.53 1732 27.8815 0.8076 2.330 28.2853 191 2000 6000 2072 18628.53 1610 27.5728 1.1163 3.225 28.1310 184
Range. v_0. D(v_0). [phi]/C. [phi].[beta]/C. [beta].
0 2150 50.9219 0.0000 0.000 0.0000 0.000 500 2071 50.8132 0.1087 0.315 0.1135 0.328 1000 1994 50.6913 0.2306 0.666 0.2486 0.718 1500 1918 50.5542 0.3677 1.062 0.4085 1.181 2000 1843 50.4029 0.5190 1.500 0.5989 1.734
+---------------------+----------------------------------+ Charge weight, 13 lb 4 oz. gravimetric density, 55.01/0.504 nature, cordite, size 30 +---------------------+----------------------------------+ +---------------------+----------------------------------+ Projectile Palliser shot, Shrapnel shell. Weight, 100lb. +---------------------+----------------------------------+ +---------------------+----------------------------------+ Muzzle velocity 2154 f/s. Nature of mounting pedestal. Jump nil. +---------------------+----------------------------------+
A. Remaining Velocity.
B. To strike an object 10 ft. high range must be known to C. Slope of Descent.
D. 5' elevation or depression alters point of impact (Range ... Laterally or Vertically).
E. Elevation.
F. Range.
G. Fuse scale for T. and P. middle No. 54 Marks I., II., or III.
H. 50% of rounds should fall in: (Length Breadth Height).
I. Time of Flight.
J. Penetration into Wrought Iron.
-----+------+-----+----------+------+------+--+------------+------+----- A. B. C. D. E. F. G. H. I. J.
-----+------+-----+----------+------+------+--+------------+------+----- f/s. yds. 1 in yds. yds. ' yds. yds yds yds secs. in.
2154 .. .. .. 0.00 0 0 0 .. .. .. .. 0.00 13.6 2122 1145 687 125 0.14 0 4 100 .. 0.4 .. 0.16 13.4 2091 635 381 125 0.29 0 9 200 .. 0.4 .. 0.31 13.2 2061 408 245 125 0.43 0 13 300 1 .. 0.4 .. 0.47 13.0 2032 316 190 125 0.58 0 17 400 1 .. 0.4 .. 0.62 12.8 2003 260 156 125 0.72 0 21 500 1 .. 0.5 0.2 0.78 12.6 1974 211 127 125 0.87 0 26 600 2 .. 0.5 0.2 0.95 12.4 1946 183 110 125 1.01 0 30 700 2 .. 0.5 0.2 1.11 12.2 1909 163 98 125 1.16 0 34 800 2 .. 0.5 0.2 1.28 12.0 1883 143 85 125 1.31 0 39 900 3 .. 0.6 0.3 1.44 11.8 1857 130 78 125 1.45 0 43 1000 3 .. 0.6 0.3 1.61 11.6 1830 118 71 125 1.60 0 47 1100 3 .. 0.6 0.3 1.78 11.4 1803 110 66 125 1.74 0 51 1200 4 .. 0.6 0.3 1.95 11.2 1776 101 61 125 1.89 0 55 1300 4 .. 0.7 0.4 2.12 11.0 1749 93 56 125 2.03 0 59 1400 4 .. 0.7 0.4 2.30 10.8 1722 86 52 125 2.18 1 3 1500 5 .. 0.7 0.4 2.47 10.6 1695 80 48 125 2.32 1 7 1600 5 25 0.8 0.5 2.65 10.5 1669 71 43 125 2.47 1 11 1700 5 25 0.9 0.5 2.84 10.3 1642 67 40 100 2.61 1 16 1800 6 25 1.0 0.5 3.03 10.1 1616 61 37 100 2.76 1 22 1900 6 25 1.1 0.6 3.23 9.9 1591 57 34 100 2.91 1 27 2000 7 25 1.2 0.6 3.41 9.7 -----+------+-----+----------+------+------+--+------------+------+-----
The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article ARMOUR PLATES.