(1) R =nd2p=nd2f(v), where(2)n=κστ,
(1) R =nd2p=nd2f(v), where(2)n=κστ,
(1) R =nd2p=nd2f(v), where
(2)n=κστ,
andnis called thecoefficient of reduction.
By means of a well-chosen value ofn, determined by a few experiments, it is possible, pending further experiment, with the most recent design, to utilize Bashforth's experimental results carried out with old-fashioned projectiles fired from muzzle-loading guns. For instance,n= 0.8 or even less is considered a good average for the modern rifle bullet.
Starting with the experimental values ofp, for a standard projectile, fired under standard conditions in air of standard density, we proceed to the construction of the ballistic table. We first determine the timetin seconds required for the velocity of a shot,dinches in diameter and weighingwlb, to fall from any initial velocity V(f/s) to any final velocityv(f/s). The shot is supposed to move horizontally, and the curving effect of gravity is ignored.
IfΔtseconds is the time during which the resistance of the air, R lb, causes the velocity of the shot to fallΔv(f/s), so that the velocity drops fromv+½Δvtov-½Δvin passing through the mean velocityv, then
(3) RΔt= loss of momentum in second-pounds,=w(v+½Δv)/g-w(v-½Δv)/g=wΔv/g
(3) RΔt= loss of momentum in second-pounds,=w(v+½Δv)/g-w(v-½Δv)/g=wΔv/g
(3) RΔt= loss of momentum in second-pounds,
=w(v+½Δv)/g-w(v-½Δv)/g=wΔv/g
so that with the value of R in (1),
(4)Δt =wΔv/nd2pg.
(4)Δt =wΔv/nd2pg.
(4)Δt =wΔv/nd2pg.
We put
(5)w/nd2= C,
(5)w/nd2= C,
(5)w/nd2= C,
and call C the ballistic coefficient (driving power) of the shot, so that
(6)Δt= CΔT, where(7)ΔT =Δv/gp,
(6)Δt= CΔT, where(7)ΔT =Δv/gp,
(6)Δt= CΔT, where
(7)ΔT =Δv/gp,
andΔT is the time in seconds for the velocity to dropΔvof the standard shot for which C=1, and for which the ballistic table is calculated.
Sincepis determined experimentally and tabulated as a function ofv, the velocity is taken as the argument of the ballistic table; and takingΔv= 10, the average value ofpin the interval is used to determineΔT.
Denoting the value of T at any velocityvby T(v), then
(8) T(v) = sum of all the preceding values ofΔT plus an arbitrary constant, expressed by the notation(9) T(v) = ∑(Δv)/gp+ a constant, or ∫dv/gp+ a constant, in whichpis supposed known as a function ofv.
(8) T(v) = sum of all the preceding values ofΔT plus an arbitrary constant, expressed by the notation(9) T(v) = ∑(Δv)/gp+ a constant, or ∫dv/gp+ a constant, in whichpis supposed known as a function ofv.
(8) T(v) = sum of all the preceding values ofΔT plus an arbitrary constant, expressed by the notation
(9) T(v) = ∑(Δv)/gp+ a constant, or ∫dv/gp+ a constant, in whichpis supposed known as a function ofv.
The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocityvand thus
(10) T(V) - T(v) = ∑vVΔv/gpor ∫vVdv/gp;
(10) T(V) - T(v) = ∑vVΔv/gpor ∫vVdv/gp;
(10) T(V) - T(v) = ∑vVΔv/gpor ∫vVdv/gp;
and for a shot whose ballistic coefficient is C
(11)t= C[T(V) - T(v)].
(11)t= C[T(V) - T(v)].
(11)t= C[T(V) - T(v)].
To save the trouble of proportional parts the value of T(v) for unit increment ofvis interpolated in a full-length extended ballistic table for T.
Next, if the shot advances a distanceΔsft. in the timeΔt, during which the velocity falls fromv+½Δvtov-½Δv, we have
(12) RΔs= loss of kinetic energy in foot-pounds=w(v+½Δv)2/g-w(v-½Δv)2/g=wvΔv/g, so that(13)Δs=wvΔv/nd2pg= CΔS, where(14)ΔS =vΔv/gp=vΔT,
(12) RΔs= loss of kinetic energy in foot-pounds=w(v+½Δv)2/g-w(v-½Δv)2/g=wvΔv/g, so that(13)Δs=wvΔv/nd2pg= CΔS, where(14)ΔS =vΔv/gp=vΔT,
(12) RΔs= loss of kinetic energy in foot-pounds
=w(v+½Δv)2/g-w(v-½Δv)2/g=wvΔv/g, so that
(13)Δs=wvΔv/nd2pg= CΔS, where
(14)ΔS =vΔv/gp=vΔT,
andΔS is the advance in feet of a shot for which C=1, while the velocity fallsΔvin passing through the average velocityv.
Denoting by S(v) the sum of all the values ofΔS up to any assigned velocityv,
(15) S(v) = ∑(ΔS) + a constant, by which S(v) is calculated fromΔS, and then between two assigned velocities V andv,
(15) S(v) = ∑(ΔS) + a constant, by which S(v) is calculated fromΔS, and then between two assigned velocities V andv,
(15) S(v) = ∑(ΔS) + a constant, by which S(v) is calculated fromΔS, and then between two assigned velocities V andv,
and ifsfeet is the advance of a shot whose ballistic coefficient is C,
(17)s= C[S(V) - S(v)].
(17)s= C[S(V) - S(v)].
(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 thedegreetable, determines the change of direction of motion of the shot while the velocity changes from V tov, the shot flying nearly horizontally.
To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity isv, to make an angleiradians with the horizon.
Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,
(18)v(di/dt) =gcosi,
(18)v(di/dt) =gcosi,
(18)v(di/dt) =gcosi,
wheredidenotes the infinitesimaldecrementofiin the infinitesimal increment of timedt.
In a problem of direct fire, where the trajectory is flat enough for cosito be undistinguishable from unity, equation (16) becomes
(19)v(di/dt) =g, ordi/dt=g/v;
(19)v(di/dt) =g, ordi/dt=g/v;
(19)v(di/dt) =g, ordi/dt=g/v;
so that we can put
(20)Δi/Δt=g/v
(20)Δi/Δt=g/v
(20)Δi/Δt=g/v
ifvdenotes the mean velocity during the small finite interval of timeΔt, during which the direction of motion of the shot changes throughΔiradians.
If the inclination or change of inclination in degrees is denoted byδorΔδ,
(21)δ/180 =i/π, so that
(21)δ/180 =i/π, so that
(21)δ/180 =i/π, so that
and ifδandichange to D and I for the standard projectile,
The differencesΔD andΔI 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,iradians, as being undistinguishable from siniand taniwheniis small as in direct fire.
The last function A, called thealtitude 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 forpis 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 representpwith sufficient accuracy by an empirical formula composed of a single power ofv, sayvm, 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 indexmin a region of velocity as given in the following table, and the corresponding value ofgp, denoted byf(v) orvm/kor its equivalent Cr, whereris the retardation.
Abridged Ballistic Table.
v.
p.
ΔT.
T.
ΔS.
S.
ΔD.
D.
ΔI.
I.
ΔA.
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
18802.44
.0287
49.9234
.000500
.871302
37.09
8657.80
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
1680
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
39.17
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
15.809
.0196
28.3090
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
36.99
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
.0163
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
.0158
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
2050
18.229
.0169
28.5297
34.77
20365.66
.0152
50.7816
.000265
.886276
30.82
10032.40
2060
18.463
.0167
28.5466
34.49
20400.43
.0149
50.7968
.000260
.886541
30.58
10063.33
2070
18.706
.0165
28.5633
34.21
20434.92
.0147
50.8117
.000256
.886801
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
2110
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
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
2150
20.811
.0149
28.6891
32.00
20700.53
.0127
50.9219
.000222
.888725
28.44
10329.61
v.
m.
logk.
Cr=gp=f(v) =vm/k.
3600
1.55
2.3909520
v1.55
× log-13.6090480
2600
1.7
2.9038022
v1.7
× log-13.0961978
1800
2
3.8807404
v2
× log-14.1192596
1370
3
7.0190977
v3
× log-18.9809023
1230
5
13.1981288
v5
× log-114.8018712
970
3
7.2265570
v3
× log-18.7734430
790
2
4.3301086
v2
× log-15.6698914
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:—