(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 pa.s.sing through the average velocity v.
Denoting by S(v) the sum of all the values of [Delta]S up to any a.s.signed velocity v,
(15) S(v) = [Sum]([Delta]S) + a constant, by which S(v) is calculated from [Delta]S, and then between two a.s.signed 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 _alt.i.tude 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 a.n.a.lysis 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 r.e.t.a.r.dation.
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.
[Ill.u.s.tration: FIG. 1.]
The equations of motion are now, the co-ordinates x and y being measured in feet,