TABLE I.
 | A table should appear at this position in the text. See Help:Table for formatting instructions. |
f 2
if
v*
p 2
1)2
v>
%
v*
IOO
log. G.
IOO
log. G.
IOO
log. G.
foo ,
log. G.
IOO
log. G.
IOO
log.G.
IOO
log. G.
IOO
log. G.
8000
9-5043
9000
9-5225
o
400
8-4354
800
8-7151
1 200
9-0661
1600
9-2282
2OOO
9-2974
6000
9-4655
IOOOO
9-5399
10
7.7244
410
8-4415
810
8-7238
1210
9-0727
1610
9-2306
2100
9-3093
6100
9-4676
IIOOO
9-5568
20
7-8655
420
8-4474
820
8-7328
I22O
9-0791
1620
9-2329
22OO
9-3.199
6200
9-4696
I2OOO
9'573i
30
7-9462
430
8-4534
830
8-7416
1230
9-0852
1630
9-2351
2300
9-3295
6300
9-4716
13000
9-5888
40
8-0025
440
8-4594
840
8-7506
1240
9-0912
1640
9-2373
2400
9-3381
6400
9-4736
14000
9-6034
50
8-0453
450
8-4654
850
8-7597
1250
9-0972
1650
9-2395
2500
9-3459
6500
9-4756
15000
9-6172
60
8-0800
460
8-4716
860
8-7688
I26O
9-1028
1660
9-2417
26OO
9-3531
6600
9-4776
16000
9-6304
70
8-1089
470
8-4776
870
8-7781
I27O
9-1083
1670
9-2438
2700
9-3598
6700
9-4796
17000
9-6429
80
8-1336
480
8-4836
880
8-7873
1280
9-II37
1680
9-2459
2800
6800
9-4815
18000
9-6549
90
8-I552
490
8-4899
890
8-7967
I29O
9-1189
1690
9-2479
29OO
9-37I5
6900
9-4835
19000
9-6662
IOO
8-1745
500
8-4959
900
8-8061
1300
9-1240
1700
9-2499
3000
9-3769
7000
9-4854
2OOOO
9-6769
no
8-1917
510
8-5021
910
8-8155
1310
9-1289
1710
9-2519
3100
9-3819
7100
9-4874
2IOOO
9-6873
I2O
8-2074
520
8-5084
920
8-8251
1320
9-1337
1720
9-2539
3200
9-3865
7200
9.4893
22OOO
9-6973
130
8-2217
530
8-5I47
930
8-8346
1330
9-1384
1730
9-2558
3300
9-3910
7300
9-4912
23OOO
9-7068
140
8-2349
540
8-5211
940
8-8442
1340
9-1430
1740
9-2576
3400
9-3951
7400
9-4931
24000
9-7159
150
8-2471
550
8-5275
950
8-8538
1350
9-1474
1750
9-2595
3500
9-3991
7500
9-4950
25OOO
9-7246
1 60
8-2586
560
8-5340
960
8-8633
1360
9-I5I7
1760
9-2613
3000
9-4029
7600
9-4969
26OOO
9-7331
170
8-2693
570
8-5405
970
8-8728
1370
9-1559
1770
9-1631
3700
9-4065
7700
9-4988
2-OOO
9-7412
1 80
8-2794
580
8-5472
980
8-8823
1380
9-1599
1780
9-2648
3800
9-4100
7800
9-5006
28OOO
9-7490
190
8-2891
590
8-5539
990
8-8919
1390
9-1639
1790
9-2665
3900
9-4I33
7900
9-5025
29OOO
9-7566
200
8-2982
600
8-5607
IOOO
8-9014
I4OO
9-1678
1800
9-2682
4OOO
9-4165
8000
9-5043
3OOOO
9-7639
2IO
8-3070
610
8-5676
IOIO
8-9107
1410
9-I7I5
1810
9-2699
4IOO
9-4196
31000
9-7710
22O
8-3I54
620
8-5745
IO2O
8-9200
I42O
9-I752
1820
9-2715
42OO
9-4226
32OOO
9-7779
230
8-3234
630
8-5816
1030
8-9293
H30
9-1788
1830
9-2731
4300
9-4254
240
8-3312
640
8-5887
1040
8-9385
1440
9-1822
1840
9-2747
44OO
9-4282
250
8-3388
650
8-5959
1050
8-9476
1450
9-I857
1850
9-2763
4500
9-4309
260
8-3461
660
8-6031
IO6O
8-9566
1460
9-1890
1860
9-2779
46OO
9-4335
27O
8-3531
670
8-6105
1070
8-9654
1470
9-1922
1870
9-2794
4700
9-4360
280
8-3601
680
8-6180
IO8O
8-9741
1480
9-1953
1880
9-2809
4800
9-4385
20X)
8-3668
690
8-6255
IOOX)
8-9826
1490
9-1984
1890
9-2824
4900
9-4410
300
8-3735
700
8-6332
IIOO
8-9910
1500
9-2014
1900
9-2838
5000
9-4434
310
8-3800
710
8-6409
IIIO
8-9994
1510
9-2044
1910
9-2853
5100
9-4458
320
8-3864
720
8-6488
II2O
9-0075
1520
9-2072
1920
9-2867
5200
9-4481
33
8-3928
730
8-6568
1130
9-oi53
1530
9-2100
1930
9-2881
5300
9-4504
34
8-3989
74
8-6648
II4O
9-0232
1540
9-2128
1940
9-2895
5400
9-4526
350
8-4051
750
8-6729
1150
9-0308
1550
9-2155
1950
9-2909
5500
9-4548
360
8-4113
760
8-681 i
II60
9-0382
1560
9-2182
1960
9-2922
5600
9-4570
370
8-4174
770
8-6895
1170
9-0454
1570
9-2207
1970
9-2935
5700
9-4592
380
8-4234
780
8-6960
1180
9-0524
1580
9-2232
1980
9-2948
5800
9-4613
390
8-4294
790
8-7065
1190
9-0594
1590
9-2257
1990
9-2961
5900
9-4634
400
8-4354
800
8-7151
1 200
9-0661
I6OO
9-2282
2OOO
9-2974
60OO
ences in air resistance between projectiles now used and those with which the air resistance law was determined. Its value. I for the projectiles of the form used in determining the air- resistance law, is as low as 0-47 for modern sharp-pointed, boat- tailed projectiles. Its value can be accurately determined for any projectile by working backward from the results of firing. Such determinations show that the value may and usually does vary for the same projectile if fired at different ranges. The Differential Equations of Motion of the. Projectile. Neglecting the convergence of the action lines of gravity due to the spheroidal form of the earth and also the slight diminution in the intensity of the force of gravity due to the height which modern projectiles reach, we may write the differential equations of motion of the projectile considered as a material point, as follows:
(4) g'= -Rcos0 = *"
FIG. 2.
where, (see fig. 2), x is the abscissa of any point of the trajectory, positive to the right,
x', the horizontal component of the velocity at that point,
x", the horizontal component of the acceleration,
y, the ordinate corresponding to x, positive up,
y', the vertical component of the velocity at that point,
y", the vertical component of the acceleration,
6, the angle that the tangent to the trajectory makes with the
horizontal. Since v is the velocity of the projectile in the direction of its motion,
tf\ /- a x '
(6) Cos 9--.
(7) Sin 0=?' and if we assume
(8) E = as the ratio between retardation and velocity, we
may write (4) and (5) as follows:
(9) x"=-Ex' (10) y" = E y' g.
In this form the equations are used in the construction of trajec- tories by the method of numerical integration. By reference to (i) we see that,
In this equation, G is a function of the velocity alone, as given in Table I. H is a function of the altitude alone as given by equation (2). C is a function of the weight and form of the projectile as given in equation (3). As in the older ballistic methods, C implicitly includes unknown variations from standard conditions in such quantities as density of the air, moisture in the air, temperature of the air, yaw of the projectile, i.e. angle between the longer axis of the projectile and the tangent to the trajectory. ,
