(12) R[Delta]s = loss of kinetic energy in foot-pounds =w(v+[Delta]v)^2/g - w(v-[Delta]v)^2/g = wv[Delta]v/g, so that (13) [Delta]s = wv[Delta]v/nd^2pg = C[Delta]S, where (14) [Delta]S = v[Delta]v/gp = v[Delta]T,
and [Delta]S is the advance in feet of a shot for which C=1, while the velocity falls [Delta]v in passing through the average velocity v.
Denoting by S(v) the sum of all the values of [Delta]S up to any assigned velocity v,
(15) S(v) = [Sum]([Delta]S) + a constant, by which S(v) is calculated from [Delta]S, and then between two assigned velocities V and v,
(16) S(V) - S(v) = [Sum,v:V][Delta]T = [Sum]v[Delta]v/gp or [Integral,v:V]vdv/gp,
and if s feet is the advance of a shot whose ballistic coefficient is C,
(17) s = C[S(V) - S(v)].
In an extended table of S, the value is interpolated for unit increment of velocity.
A third table, due to Sir W. D. Niven, F.R.S., called the _degree_ table, determines the change of direction of motion of the shot while the velocity changes from V to v, the shot flying nearly horizontally.
To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity is v, to make an angle i radians with the horizon.
Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,
(18) v(di/dt) = g cos i,
where di denotes the infinitesimal _decrement_ of i in the infinitesimal increment of time dt_.
[v.03 p.0272] In a problem of direct fire, where the trajectory is flat enough for cos i to be undistinguishable from unity, equation (16) becomes
(19) v(di/dt) = g, or di/dt = g/v;
so that we can put
(20) [Delta]i/[Delta]t = g/v
if v denotes the mean velocity during the small finite interval of time [Delta]t, during which the direction of motion of the shot changes through [Delta]i radians.
If the inclination or change of inclination in degrees is denoted by [delta] or [Delta][delta],
(21) [delta]/180 = i/[pi], so that
(22) [Delta][delta] = 180/[pi] [Delta]i = 180g/[pi] [Delta]t/v;
and if [delta] and i change to D and I for the standard projectile,
(23) [Delta]I = g [Delta]T/v = [Delta]v/vp, [Delta]D = 180g/[pi] [Delta]T/v, and
(24) I(V) - I(v) = [Sum,v:V][Delta]v/vp or [Integral,v:V]dv/vp, D(V) - D(v) = 180/[pi] [I(V) - I(v)].
The differences [Delta]D and [Delta]I are thus calculated, while the values of D(v) and I(v) are obtained by summation with the arithmometer, and entered in their respective columns.
For some purposes it is preferable to retain the circular measure, i radians, as being undistinguishable from sin i and tan i when i is small as in direct fire.
The last function A, called the _altitude function_, will be explained when high angle fire is considered.
These functions, T, S, D, I, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above. The initial values of T, S, D, I, A must be accepted as belonging to the anterior portion of the table.
In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v^m, the integration can be effected which replaces the summation in (10), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f(v) or v^m/k or its equivalent Cr, where r is the retardation.
ABRIDGED BALLISTIC TABLE.
-----+--------+-------+---------+-------+----------+-------+-------- v. p. [Delta]T. T. [Delta]S. S. [Delta]D. D.
-----+--------+-------+---------+-------+----------+-------+-------- f/s 1600 11.416 .0271 27.5457 43.47 18587.00 .0311 49.7729 1610 11.540 .0268 27.5728 43.27 18630.47 .0306 49.8040 1620 11.662 .0265 27.5996 43.08 18673.74 .0301 49.8346 1630 11.784 .0262 27.6261 42.90 18716.82 .0296 49.8647 1640 11.909 .0260 27.6523 42.72 18759.72 .0291 49.8943 1650 12.030 .0257 27.6783 42.55 18802.44 .0287 49.9234 1660 12.150 .0255 27.7040 42.39 18844.99 .0282 49.9521 1670 12.268 .0252 27.7295 42.18 18887.38 .0277 49.9803 1680 12.404 .0249 27.7547 41.98 18929.56 .0273 50.0080 1690 12.536 .0247 27.7796 41.78 18971.54 .0268 50.0353 1700 12.666 .0244 27.8043 41.60 19013.32 .0264 50.0621 1710 12.801 .0242 27.8287 41.41 19054.92 .0260 50.0885 1720 12.900 .0239 27.8529 41.23 19096.33 .0256 50.1145 1730 13.059 .0237 27.8768 41.06 19137.56 .0252 50.1401 1740 13.191 .0234 27.9005 40.90 19178.62 .0248 50.1653 1750 13.318 .0232 27.9239 40.69 19219.52 .0244 50.1901 1760 13.466 .0230 27.9471 40.53 19260.21 .0240 50.2145 1770 13.591 .0227 27.9701 40.33 19300.74 .0236 50.2385 1780 13.733 .0225 27.9928 40.19 19341.07 .0233 50.2621 1790 13.862 .0223 28.0153 40.00 19381.26 .0229 50.2854 1800 14.002 .0221 28.0376 39.81 19421.26 .0225 50.3083 1810 14.149 .0219 28.0597 39.68 19461.07 .0222 50.3308 1820 14.269 .0217 28.0816 39.51 19500.75 .0219 50.3530 1830 14.414 .0214 28.1033 39.34 19540.26 .0216 50.3749 1840 14.552 .0212 28.1247 39.17 19579.60 .0212 50.3965 1850 14.696 .0210 28.1459 39.01 19618.77 .0209 50.4177 1860 14.832 .0209 28.1669 38.90 19657.78 .0206 50.4386 1870 14.949 .0207 28.1878 38.75 19696.68 .0203 50.4592 1880 15.090 .0205 28.2085 38.61 19735.43 .0200 50.4795 1890 15.224 .0203 28.2290 38.46 19774.04 .0198 50.4995 1900 15.364 .0201 28.2493 38.32 19812.50 .0195 50.5193 1910 15.496 .0199 28.2694 38.19 19850.82 .0192 50.5388 1920 15.656 .0197 28.2893 38.01 19889.01 .0189 50.5580 1930 15.809 .0196 28.3090 37.83 19927.02 .0186 50.5769 1940 15.968 .0194 28.3286 37.66 19964.85 .0184 50.5955 1950 16.127 .0192 28.3480 37.48 20002.51 .0181 50.6139 1960 16.302 .0190 28.3672 37.26 20039.99 .0178 50.6320 1970 16.484 .0187 28.3862 36.99 20077.25 .0175 50.6498 1980 16.689 .0185 28.4049 36.73 20114.24 .0172 50.6673 1990 16.888 .0183 28.4234 36.47 20150.97 .0169 50.6845 2000 17.096 .0181 28.4417 36.21 20187.44 .0166 50.7014 2010 17.305 .0178 28.4598 35.95 20223.65 .0163 50.7180 2020 17.515 .0176 28.4776 35.65 20259.60 .0160 50.7343 2030 17.752 .0174 28.4952 35.35 20295.25 .0158 50.7503 2040 17.990 .0171 28.5126 35.06 20330.60 .0155 50.7661 2050 18.229 .0169 28.5297 34.77 20365.66 .0152 50.7816 2060 18.463 .0167 28.5466 34.49 20400.43 .0149 50.7968 2070 18.706 .0165 28.5633 34.21 20434.92 .0147 50.8117 2080 18.978 .0163 28.5798 33.93 20469.13 .0144 50.8264 2090 19.227 .0160 28.5961 33.60 20503.06 .0141 50.8408 2100 19.504 .0158 28.6121 33.34 20536.66 .0139 50.8549 2110 19.755 .0156 28.6279 33.02 20570.00 .0136 50.8688 2120 20.010 .0154 28.6435 32.76 20603.02 .0134 50.8824 2130 20.294 .0152 28.6589 32.50 20635.78 .0132 50.8958 2140 20.551 .0150 28.6741 32.25 20688.28 .0129 50.9090 -----+--------+-------+---------+-------+----------+-------+--------
-----+--------+---------+---------+-------+--------- v. p. [Delta]I. I. [Delta]A. A.
-----+--------+---------+---------+-------+--------- f/s 1600 11.416 .000543 .868675 37.77 8470.36 1610 11.540 .000534 .869218 37.63 8508.13 1620 11.662 .000525 .869752 37.48 8545.76 1630 11.784 .000517 .870277 37.35 8583.24 1640 11.909 .000508 .870794 37.21 8620.59 1650 12.030 .000500 .871302 37.09 8657.80 1660 12.150 .000492 .871802 36.96 8694.89 1670 12.268 .000484 .872294 36.80 8731.85 1680 12.404 .000476 .872778 36.65 8768.65 1690 12.536 .000468 .873254 36.50 8805.30 1700 12.666 .000461 .873722 36.35 8841.80 1710 12.801 .000453 .874183 36.21 8878.15 1720 12.900 .000446 .874636 36.07 8914.36 1730 13.059 .000439 .875082 35.94 8950.43 1740 13.191 .000432 .875521 35.81 8986.37 1750 13.318 .000425 .875953 35.65 9022.18 1760 13.466 .000419 .876378 35.53 9057.83 1770 13.591 .000412 .876797 35.37 9093.36 1780 13.733 .000406 .877209 35.26 9128.73 1790 13.862 .000400 .877615 35.11 9163.99 1800 14.002 .000393 .878015 34.96 9199.10 1810 14.149 .000388 .878408 34.86 9234.06 1820 14.269 .000382 .878796 34.73 9268.92 1830 14.414 .000376 .879178 34.59 9303.65 1840 14.552 .000370 .879554 34.46 9338.24 1850 14.696 .000365 .879924 34.33 9372.70 1860 14.832 .000360 .880289 34.25 9407.03 1870 14.949 .000355 .880649 34.14 9441.28 1880 15.090 .000350 .881004 34.02 9475.42 1890 15.224 .000345 .881354 33.91 9509.44 1900 15.364 .000340 .881699 33.80 9543.35 1910 15.496 .000335 .882039 33.69 9577.15 1920 15.656 .000330 .882374 33.55 9610.84 1930 15.809 .000325 .882704 33.40 9644.39 1940 15.968 .000320 .883029 33.26 9677.79 1950 16.127 .000316 .883349 33.12 9711.05 1960 16.302 .000311 .883665 32.94 9744.17 1970 16.484 .000305 .883976 32.71 9777.11 1980 16.689 .000300 .884281 32.48 9809.82 1990 16.888 .000295 .884581 32.26 9842.30 2000 17.096 .000290 .884876 32.05 9874.56 2010 17.305 .000285 .885166 31.83 9906.61 2020 17.515 .000280 .885451 31.57 9938.44 2030 17.752 .000275 .885731 31.32 9970.01 2040 17.990 .000270 .886006 31.07 10001.33 2050 18.229 .000265 .886276 30.82 10032.40 2060 18.463 .000260 .886541 30.58 10063.33 2070 18.706 .000256 .886801 30.34 10093.80 2080 18.978 .000251 .887057 30.10 10124.14 2090 19.227 .000247 .887308 29.82 10154.24 2100 19.504 .000242 .887555 29.59 10184.06 2110 19.755 .000238 .887797 29.32 10213.65 2120 20.010 .000234 .888035 29.10 10242.97 2130 20.294 .000230 .888269 28.88 10272.07 2140 20.551 .000226 .888499 28.66 10300.95 2150 20.811 .000222 .888725 28.44 10329.61 -----+--------+---------+---------+-------+---------
+------+---------+------------+----------------------------------+ v. m. log k. Cr = gp = f(v) = {v^m}/k. +------+---------+------------+----------------------------------+ 3600 1.55 2.3909520 v^{1.55} log^{-1} [=3].6090480 2600 1.7 2.9038022 v^{1.7} log^{-1} [=3].0961978 1800 2 3.8807404 v^2 log^{-1} [=4].1192596 1370 3 7.0190977 v^3 log^{-1} [=8].9809023 1230 5 13.1981288 v^5 log^{-1}[=14].8018712 970 3 7.2265570 v^3 log^{-1} [=8].7734430 790 2 4.3301086 v^2 log^{-1} [=5].6698914 +------+---------+------------+----------------------------------+
The numbers have been changed from kilogramme-metre to pound-foot units by Colonel Ingalls, and employed by him in the calculation of an extended ballistic table, which can be compared with the result of the abridged table. The calculation can be carried out in each region of velocity from the formulae:--
(25) T(V) - T(v) = k [Integral,v:V] v^{-m} dv, S(V) - S(v) = k [Integral,v:V] v^{m+1} dv, I(V) - I(v) = gk [Integral,v:V] v^{-m-1} dv,
and the corresponding integration.
The following exercises will show the application of the ballistic table. A slide rule should be used for the arithmetical operations, as it works to the accuracy obtainable in practice.
_Example_ 1.--Determine the time t sec. and distance s ft. in which the velocity falls from 2150 to 1600 f/s.
(a) of a 6-in. shot weighing 100lb, taking n = 0.96, (b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, taking n = 0.8.
------+------+---------+---------+--------+----------+----------+-------- V. v. T(V). T(v). t/C. S(V) S(v) s/C.
------+------+---------+---------+--------+----------+----------+-------- 2150 1600 28.6891 27.5457 1.1434 20700.53 18587.00 2113.53 ------+------+---------+---------+--------+----------+----------+--------
----+-------+------+-------+--------+-------+---------+----------------- d. w. C. t/C. t. S/C. s.
----+-------+------+-------+--------+-------+---------+----------------- (a) 6 100 2.894 1.1434 3.307 2113.53 6114 (2038 yds.) (b) 0.303 1/32 0.426 1.1434 0.486 2113.53 900 (300 yds.) ----+-------+------+-------+--------+-------+---------+-----------------
_Example_ 2.--Determine the remaining velocity v and time of flight t over a range of 1000 yds. of the same two shot, fired with the same muzzle velocity V = 2150 f/s.
---+----+-----+---------+---------+-----+--------+--------+-------+------ S. s/C. S(V). S(v). v. T(V). T(v). t/C. t.
---+----+-----+---------+---------+-----+--------+--------+-------+------ (a) 3000 1037 20700.53 19663.53 1861 28.6891 28.1690 0.5201 1.505 (b) 3000 7050 20700.53 13650.53 920* 28.6891 23.0803 5.6088 2.387 ---+----+-----+---------+---------+-----+--------+--------+-------+------
* These numbers are taken from a part omitted here of the abridged ballistic table.
In the calculation of range tables for _direct fire_, defined officially as "fire from guns with full charge at elevation not exceeding 15," the vertical component of the resistance of the air may be ignored as insensible, and the actual velocity and its horizontal component, or component parallel to the line of sight, are undistinguishable.
[Illustration: FIG. 1.]
The equations of motion are now, the co-ordinates x and y being measured in feet,