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ABRIDGED BALLISTIC TABLE

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Originally appearing in Volume V03, Page 274 of the 1911 Encyclopedia Britannica.
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ABRIDGED BALLISTIC TABLE. v. p. i T. T. S. S. D. D. I. I. DA. A. f/s - 1600 11.416 •0271 27'5457 43'47 18587.00 •0311 49'7729 •000543 •868675 37'77 8470.36 1610 11.540 •0268 27.5728 43.27 18630'47 •0306 49.8040 .000534 .869218 37.63 8508.13 1620 11.662 .0265 27.5996 43.08 18673.74 '0301 49.8346 '000525 '869752 37.48 8545.76 1630 11.784 •0262 27.6261 42.90 18716.82 .0296 49.8647 .000517 .870277 37.35 8583.24 1640 11.909 •0260 27.6523 42 72 18759.72 '0291 49.8943 .000508 .870794 37.21 8620.59 1650 12.030 .0257 27.6783 42.55 1880244 .0287 49'9234 .000500 .871302 37.09 8657•8o 1660 12.150 •0255 27.7040 42.39 18844'99 '0282 49'9521 .000492 •871802 36.96 8694'89 1670 12.268 .0252 27.7295 42.18 18887.38 .0277 49'9803 •000484 .872294 36.80 8731.85 168o 12.404 .0249 27'7547 41.98 18929.56 .0273 50.0080 •000476 .872778 36'65 8768.65 1690 12.536 .0247 27.7796 41.78 18971.54 •0268 50'0353 .000468 .873254 36.50 8805.30 1700 12.666 .0244 27.8043 41.60 19013.32 .0264 50.0621 .000461 .873722 36.35 8841.80 1710 12.801 .0242 27.8287 / 41.41 19054.92 •0260 50.0885 •000453 •874183 36.21 8878.15 1720 12.900 .0239 27.8529 41.23 19096.33 .0256 50.1145 •000446 .874636 36.07 8914.36 1730 13.059 .0237 27.8768 41.06 19137.56 .0252 50.1401 •000439 .875082 35'94 8950'43 1740 13.191 •0234 27.9005 40.90 19178.62 .0248 50.1653 .000432 .875521 35.81 8986.37 1750 13.318 .0232 27'9239 40.69 19219.52 .0244 50.1901 •000425 .875953 35.65 9022.18 1760 13.466 .0230 27.9471 40'53 19260.21 .0240 50.2145 •000419 '876378 35'53 9057.83 1770 13'591 .0227 27'9701 40.33 19300'74 .0236 50.2385 •000412 •876797 35'37 9093'36 1780 13'733 '0225 27.9928 40.19 19341'07 .0233 50.2621 .000406 •877209 35'26 9128.73 1790 13.862 .0223 28.0153 40.00 19381.26 .0229 50.2854 .000400 .877615 35'11 9163'99 1800 14.002 •0221 28.0376 39.81 19421.26 .0225 50.3083 .000393 .878015 34.96 9199.10 1810 14.149 .0219 28.0597 39'68 19461.07 •0222 50.3308 .000388 .878408 34.86 9234.06 1820 14.269 .0217 28.0816 39.51 19500.75 .0219 50.3530 .000382 •878796 34'73 9268.92 1830 14.414 •0214 28.1033 39'34 .19540.26 •0216 50.3749 •000376 '879178 34.59 9303'65 1840 14.552 •0212 28.1247 139.17 19579.60' 0212 50.3965 .000370 .879554 34.46 9338'24 1850 14.696 •0210 28.1459 39.01 19618.77 .0209 50.4177 .000365 .879924 34'33 9372.70 1860 14.832 •0209 28.1669 38.90 19657.78 •0206 50.4386 .000360 •880289 34'25 9407.03 1870 14'949 .0207 28.1878 38.75 19696.68 .0203 50'4592 •000355 •880649 34'14 9441'28 1880 15.090 •0205 28.2085 38.61 19735'43 •0200 50.4795 •000350 •881004 34'02 9475.42 1890 15.224 .0203 28.2290 38.46 19774'04 •0198 50'4995 •000345 '881354 33'91 9509.44 1900 15.364 •0201 28.2493 38.32 19812.50 •0195 50.5193 •000340 •881699 33.80 9543.35 1910 15.496 .0199 28.2694 38.19 19850.82 .0192 50.5388 •000335 •882039 33.69 9577.15 1920 15.656 •0197 28.2893 38.01 19889.01 •0189 50.5580 •000330 •882374 33'55 9610.84 1930 15.$09 .0196 z8.3090 37'83 19927'02 .0186 50.5769 •000325 •882704 33'40 9644'39 1940 15.968 •0194 28.3286 37.66 19964.85 •0184 50.5955 •000320 •883029 33'26 9677'79 1950 16.127 •0192 28.3480 37.48 20002.51 •0181 50.6139 •000316 •883349 33'12 9711.05 1960 16.302 •0190 28.3672 37.26 20039'99 .0178 50.6320 .000311 •883665 32'94 9744'17 1970 16.484 •0187 28.3862 136.99 20077.25 '0175 50'6498 •000305 .883976 32'71 9777'11 1980 16.689 •0185 28.4049 36'73 20114.24 .0172 50'6673 •000300 •884281 32.48 9809.82 1990 16.888 •0183 28.4234 36.47 20150.97 .0169 50.6845 .000295 •884581 32.26 9842.30 2000 17.096 •0181 28.4417 36.21 20187.44 .0166 50.7014 •000290 •884876 32'05 9874'56 2010 17.305 •0178 28.4598 35.95 20223.65 •o163 50.7180 •000285 .885166 31'83 9906.61 2020 17.515 •0176 28.4776 35'65 20259.60 •0160 50.7343 •000280 •885451 31'57 9938'44 2030 17.752 '0174 28.4952 35.35 20295.25 •o158 50.7503 •000275 •885731 31'32 9970.01 2040 17.990 •0171 28.5126 35.06 20330.60 .0155 50.7661 •000270 •886006 31.07 10001.33 34'77 20365.66 .0152 50.7816 •000265 •886276 30.82 10032.40 2060 18'463 .0167 5267 34.49 20400.43 '0149 50.7968 •000260 •886541 30.58 10063.33 28.54 2070 18.706 ! •0165 28.5633 34'21 20434'92 '0147 50.8117 •000256 •8868o1 30.34 10093.80 2080 18.978 •0163 28.5798 33'93 20469.13 .0144 50.8264 •000251 •887057 30.10 10124'14 2090 19.227 •0160 28.5961 33.60 20503.06 •0141 50.8408 •000247 •887308 29.82 10154.24 2100 19.504 •0158 28.6121 33.34 20536.66 .0139 50.8549 •000242 •887555 29'59 10184.06 21I0 19'755 •0156 28.6279 33.02 20570.00 •0136 50.8688 •000238 •887797 29.32 10213.65 2120 20.010 •0154 28.6435 32.76 20603.02 .0134 50.8824 .000234 •888035 29.10 10242'97 2130 20.294 .0152 28.6589 i 32.50 20635.78 •0132 50.8958 .000230 •888269 28.88 10272.07 2140 20.551 •0150 28.6741 32.25 20688.28 .0129 50.9090 •000226 •888499 28.66 10300.95 1 2150 .20.811 •0149 28.6891 32.00 20700.53 I •0127 50.9219 •000222 •888725 28.44 10329.61 v. m. log k. Cr =gp=f (v) =v5'/k. 3600 1'55 2'3909520 vl•55X1og-1 3.6090480 2600 1•7 2.9038022 v1.7 X log-' 3.0961978 1800 _ 2 3.8807404 v2 Xlog14.1192396 1370 _ 3 7.0190977 v3 Xlog 1 8.9809023 1230 _ 5 13'1981288 v5 X 1og114.8018712 970 _ 3 7.2265570 v3 Xlogl 8.7734430 790 2 4.330,086 v2 Xlog'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 f vv, mdv, S(V)-S(v) =k f vvmtldv I (V) - 1(v) = gk f vv`m-ldv, v 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 ioolb, taking n=0.96, (b) of a rifle bullet, 0.3o3-in. Calibre, weighing half an ounce, taking n = o•8. 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 dt =constant -gt=g(aT-t), if T denotes the whole time of flight from 0 to the point B (fig. I), where the trajectory cuts the line of sight; so that IT is the time to the vertex A, where the shot is flying parallel to OB. Integrating (27) again, (31) y = g (,Tt -It') = Igt (T -t) ; and denoting T-t by t', and taking g= 32f/s', (32) y =161t', which is Colonel Sladen's formula, employed in plotting ordinates of a trajectory. At the vertex A, where y =H, we have t =t' = IT, so that (33) H = egT", which for practical purposes, taking g =32, is replaced by (34) H=4T', or (2T)'. Thus, if the time of flight of a shell is 5 sec., the height of the vertex of the trajectory is about loo 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 ¢ and l4, the total deviation in the range OB is (fig. I) (37) S=4-f a=C{D(V)-D(v)}. To share the S between 4. and /3, 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, at A is calculated from the formula V. T(V). 1600 28.6891 tic. S(V). S(v). s/C. 1.1434 120700.53 18587.00 2113'53 V. 2150 T(v). 27.5457 S. 0.303 1/32 0.426 1.1434 0.486 2113'53 900 (300 yds.) (38) T(v,)=T(V)- C =1-{T V)+T(v)}; Example 2.—Determine the remaining velocity v and time of flight and now the degree table for D(v) gives t over a range of moo yds. of the same two shot, fired with the same (39) p=C {D(V)-D(v,)}, muzzle velocity V=2150f/s. (40)=C{D(v,)-D(v)}. d. w. C. t/C. t. s/C. s. 1(a) 6 100 2'894 I'I434 3.307 2113.53 6114 (2o38yds.) 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. The equations of motion are now, the co-ordinates x and y being measured in feet, (26) * These numbers are taken from a part omitted here of the abridged ballistic table. This value of 4. is the tangent elevation (T.E) ; the quadrant elevation (Q.E.) is 4,-S, where S is the angular depression of the line of sight OB; and if 0 is h ft. vertical above B, the angle S at a range of R yds. is given by 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 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 0, and descent /3 of this 6-in. gun at ranges 500, 1000, 150o, 2000 yds., taking the muzzle velocity V =2150 f/s, and a coefficient of reduction n=0.96. [For Table see p. 274.1 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 85o lb with velocity 2150 f/s. The complete ballistic table, and the method of high angle fire (see below) must be employed. (27) d'x d12 =-rr=-C' d'y - d12 - g. S. s/C. S(V). S(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 (41) Range. S. s/C. S(v). v. T(v). t/C. t. T(vo)• vo. D(vo). 4p/C. ik. O/C.. f. 0 0 0 20700.53 2150 28.6891 0.0000 0'000 28.6891 2150 50.9219 0.0000 0.000 0.0000 0.000 500 1500 518 20182.53 1999 28.4399 0.2492 0.720 28.5645 2071 50.8132 0.1087 0.315 0.1135 0.328 1000 3000 1036 19664.53 1862 28.1711 0.5180 1497 28.4301 1994 50.6913 0.2306 o•666 0.2486 0.718 1500 4500 1554 19146.53 1732 27.8815 0.8076 2.330 28.2853 1918 50.5542 0.3677 I•062 0.4085 1.181 2000 6000 2072 18628.53 1610 27.5728 1.1163 3.225 28.1310 1843 50.4029 0.5190 1.500 0.5989 1.734 1 55.01 . 0.504 weight, 13 lb 4 oz. Charge gravimetric density, nature, cordite, size 30.
End of Article: ABRIDGED BALLISTIC TABLE
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