It is evident that the area of the figure between any two ordinates is greater than if the known y' points were connected by a smooth continuous curve.
FIG. 3.
It has been shown mathematically that when the points to, ti, ti, etc.. are equally spaced the quantity in brackets in the value of Ay above should be reduced by 1/12 of the second difference, making it,
= = (269-9+28-2Xi-3-5Xi/i2)Xi :
'283-7
or more generally, since the same process is used in successively evaluating the other functions x', y', etc., we may write,
where f(t n } = z n ,/2(X.-i) = z n -i etc.
n represents the order of the interval, or of the tabulated values
of z, a and b. h, the uniform length of the interval
a, first differences of z
b, second differences of z.
The quantities may be arranged in tabulated form as follows: h zi
TABLE II. Continued
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t
x
x'
Ex'
y
y'
Ey'+g
8
2842-57
284-42
10-04
2575-01
220-36
I7-58
9
3122-32
275-42
8-04-
2786-90
203-73
15-75
10
3394-00
268-22
6-41
2983-00
188-73
14-31
ii
262-46
5-20
3I64-77
174-98
13-26
12
3919-26
257-73
4-32
3333-25
162-12
12-52
13
4I74-94
253-74
3489-19
149-88
11-98
H
4426-91
250-30
3-22
3633-16
138-11
11-58
15
4675-66
247-26
2-87
3765-54
126-69
11-27
16
4921-54
244-53
2-59
3886-64
"5-54
11-03
17
5164-84
242-05
2-38
3996-71
104-61
10-83
18
5405-70
239-75
2-21
4095-94
93-87
10-66
19
5644-37
237-62
2-06
4184-51
83-28
10-52
20
5880-96
235-59
95
4262-52
72-82
10-40
21
6II5-59
-85
4330-16
62-47
10-30
22
6348-37
23I-88
77
4387-50
52-22
10-20
23
6579-38
230-I5
70
4434-64
42-07
IO-II
24
6808-69
228-47
65
447I-67
32-00
10-03
25
7036-34
226-85
60
4498-67
22-01
9-95
26
7262-40
225-27
56
4I55-72
I2-IO
9-88
27
7486-90
223-73
-53
4522-89
2-25
9-82
28
7709-87
222-21
51
4520-24
-7-55
9-75
29
7931-33
22O-72
49
4507-83
17-26
9-68
30
8151-31
2I9-24
-48
4485-74
26-91
9-62
32
8586-89
216-33
44
4412-64
-46-03
9-49
34
9016-63
2I3-40
49
4301-69
-64-87
9-35
36
9440-42
2IO-38
52
4I53-35
-83-42
9-20
38
9858-11
207-29
57
3968-23
-101-65
9-03
40
10269-50
204-07
65
3747-00
-119-52
8-83
42
10674-27
200-66
75
3490-45
-136-96
8-61
44
IIO22-OI
197-04
87
3I99-50
-I53-92
8-34
46
II462-25
I93-I5
2-OI
2875-18
-170-30
8-03
48
11844-4!
188-95
2-19
2518-77
-185-99
50
I22I7-80
184-36
2-40
2131-80
200-83
7-19
52
I258I-57
179-34
2-62
1716-11
-214-69
6-66
54
I2934-85
173-85
2-8 7
1273-82
-227-41
6-05
56
13276-66
167-87
3-n
807-33
-238-85
5-38
58
13605-98
161-43
3'33
3I9-3I
-248-90
4-66
59
I3765-72
158-05
3-44
68-14
-253-37
4-28
59.269
I3808-IO
157-12
3-47
-254-49
4-18
The application of the formula will give the successive increments to be applied in evaluating I z dl.
The use of second differences in this manner permits the use of longer intervals except at the beginning when no second differences are available. In this case a shorter interval is used and a sufficient number of trials are made or a second difference is estimated by approximate methods.
The integral having been obtained by the methods described, up to any interval, Simpson's rule or other similar method may be used to check the values obtained.
Complete Solution of a Trajectory. The results of the complete solution of the following example are given in Table II. below:
Example II. A 75 mm. gun is fired at an angle of departure of 45, using a projectile of 15 Ib. weight with a form factor, 1=0-6. The muzzle velocity is 2,175 ft. per second. Determine the coordi- nates-of the trajectory and the horizontal and vertical components of the velocity and acceleration.
TABLE II.
It is to be noted that J-second intervals are used from o to I second, half-seconds from I to 5 seconds, full seconds from 5 to 30 seconds and two seconds from 30 to 58 seconds. As this was so nearly the end of the trajectory, judging from the value of y, a single second interval was next taken to 59 seconds. The exact values of the other element* corresponding to y = o, or the end of the range, are obtained by interpolation. For this purpose it may be desirable to work out the values for an additional short interval.
For the terminal velocity we have,
For the angle of fall,
dy dy dt V tan u = -=- -j- = .- dx dt dx x
The results for the end of the range and maximum ordinate are:
Range = l38o8-l m = i5ioo-7yd. Terminal Velocity 299-1 m/s = 327.1 yd /s = 98 1-3 f/s
u = tan" 1 1-61972=58 18' 55" Max. Ord. =4523-15 m. =4946-6 yd. Range to Max. Ord. =7538-2 m. =8243-9 yd.
Ballistic Tables. Using the method of numerical integration described, we may construct a series of trajectories with the values of the muzzle velocity, ballistic coefficient and angle of elevation so chosen and spaced as to cover the field of guns and ammunition in actual use. By proper arrangement of the principal elements of the trajectories thus determined, it is possible to form tables in con-
t
- = 45
x
C=2
x'
867 M Ex'
V.=2i 75
y
f/s =662.94
y'
m/s Ey'+g
vement form for use, from which by interpolation we may obtain the important elements of the trajectories corresponding to any given gun. Such tables have been constructed in France and America.
o
468-77
43-81
o
468-77
53-6i
The American tables, constructed under the supervision of A. A.
}
115-85
458-08
41-70
"5-54
455-66
51-28
Bennett, consist of two main tables. The first table is a direct tabu-
I
229-09
447-91
39-72
227-88
443-12
49-09
lation of the results of numerical integration of trajectories. For
f
339-85
438-21
37-88
337-15
431-10
47-06
this purpose it has been found most convenient and economical of
i
448-23
428-96
36-14
443-47
419-58
45-15
labour to assume a ballistic coefficient and velocity at the summit
and construct the trajectory forward and backward from that point.
'i
658-33
411-70
32-97
647-76
397-90
41-66
The arguments in this table are the ballistic coefficient, the velocity
2
860-18
395-94
30-13
841-63
377-87
38-55
at the summit and the ordinate from the summit. The table gives
2 1
1054-50
381-52
1025-87
359-30
35-78
the corresponding values of x, x', y' and t from summit forward and
3,
1241-91
368-32
25-27
1201-16
342-04
33-27
backward.
3*
1423-00
356-22
23-18
1368-12
31-01
The second table is arranged with C, and V as arguments and
4
1598-29
345-n
21-25
1527-32
310-99
28-95
gives X, T, Y and the velocity at the summit.
4*
1768-26
334-93
19-49
1679-28
296-99
27-08
Assumptions Made in Construction of Ballistic Tables. In com-
5
I933-36
325-60
17-86
1824-46
283-88
25-37
puting trajectories for use in the construction of these tables, the
following assumptions were made :
6
2250-54
309-23
14-94
2096-18
260-06
22-36
I. The earth is motionless.
7
2552-75
295-61
12-36
2345-5I
239-02
19-79
2. There is no wind.
