(25) T(V) - T(v) =k∫vVv-mdv, S(V) - S(v) =k∫vVvm+1dv, I(V) - I(v) =gk∫vVv-m-1dv,
(25) T(V) - T(v) =k∫vVv-mdv, S(V) - S(v) =k∫vVvm+1dv, I(V) - I(v) =gk∫vVv-m-1dv,
(25) T(V) - T(v) =k∫vVv-mdv, S(V) - S(v) =k∫vVvm+1dv, I(V) - I(v) =gk∫vVv-m-1dv,
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.
Example1.—Determine the timetsec. and distancesft. in which the velocity falls from 2150 to 1600 f/s.
(a) of a 6-in. shot weighing 100lb, takingn= 0.96,(b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, takingn= 0.8.
(a) of a 6-in. shot weighing 100lb, takingn= 0.96,(b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, takingn= 0.8.
(a) of a 6-in. shot weighing 100lb, takingn= 0.96,
(b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, takingn= 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.)
Example2.—Determine the remaining velocityvand time of flighttover 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 fordirect 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.
Fig. 1.Fig.1.
The equations of motion are now, the co-ordinatesxandybeing measured in feet,
The first equation leads, as before, to
(28)t= C{T(V) - T(v)},(29)x= C{S(V) - S(v)}.
(28)t= C{T(V) - T(v)},(29)x= C{S(V) - S(v)}.
(28)t= C{T(V) - T(v)},
(29)x= C{S(V) - S(v)}.
The integration of (24) gives
if T denotes the whole time of flight from O to the point B (fig. 1), where the trajectory cuts the line of sight; so that ½T is the time to the vertex A, where the shot is flying parallel to OB.
Integrating (27) again,
(31)y=g(½Tt- ½t2) = ½gt(T -t);
(31)y=g(½Tt- ½t2) = ½gt(T -t);
(31)y=g(½Tt- ½t2) = ½gt(T -t);
and denoting T -tbyt′, and takingg= 32f/s2,
(32)y= 16tt′,
(32)y= 16tt′,
(32)y= 16tt′,
which is Colonel Sladen's formula, employed in plotting ordinates of a trajectory.
At the vertex A, wherey= H, we havet=t′ = ½T, so that
(33) H = ⅛gT2,
(33) H = ⅛gT2,
(33) H = ⅛gT2,
which for practical purposes, takingg= 32, is replaced by
(34) H = 4T2, or (2T)2.
(34) H = 4T2, or (2T)2.
(34) H = 4T2, or (2T)2.
Thus, if the time of flight of a shell is 5 sec., the height of the vertex of the trajectory is about 100 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), andyor 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 velocityvis first calculated from (29) by
(35) S(v) = S(V) - X/C,
(35) S(v) = S(V) - X/C,
(35) S(v) = S(V) - X/C,
and then the time of flight T by
(36) T = C{T(V) - T(v)}.
(36) T = C{T(V) - T(v)}.
(36) T = C{T(V) - T(v)}.
Denoting the angle of departure and descent, measured in degrees and from the line of sight OB byφandβ, the total deviation in the range OB is (fig. 1)
(37)δ=φ+β= C{D(V) - D(v)}.
(37)δ=φ+β= C{D(V) - D(v)}.
(37)δ=φ+β= C{D(V) - D(v)}.
To share theδbetweenφandβ, the vertex A is taken as the point ofhalf-time(and therefore beyondhalf-range, because of the continual diminution of the velocity), and the velocityv0at A is calculated from the formula
and now the degree table for D(v) gives
(39)φ= C{D(V) - D(v0)},(40)β= C{D(v0) - D(v)}.
(39)φ= C{D(V) - D(v0)},(40)β= C{D(v0) - D(v)}.
(39)φ= C{D(V) - D(v0)},
(40)β= C{D(v0) - D(v)}.
This value ofφis the tangent elevation (T.E.); the quadrant elevation (Q.E.) isφ- S, where S is the angular depression of the line of sight OB; and if O ishft. vertical above B, the angle S at a range of R yds. is given by
(41) sin S =h/3R,
(41) sin S =h/3R,
(41) sin S =h/3R,
or, for a small angle, expressed in minutes, taking the radian as 3438′,
(42) S = 1146h/R.
(42) S = 1146h/R.
(42) S = 1146h/R.
So also the angleβmust 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 reductionn, and the muzzle velocity V.
Example3.—Determine by calculation with the abridged ballistic table the remaining velocityv, the time of flightt, angle of elevationφ, and descentβof this 6-in. gun at ranges 500, 1000, 1500, 2000 yds., taking the muzzle velocity V = 2150 f/s, and a coefficient of reductionn= 0.96. [For Table see p. 274.]
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 850 lb with velocity 2150 f/s. The complete ballistic table, and the method of high angle fire (see below) must be employed.
Range.
s.
s/C.
S(v).
v.
T(v).
t/C.
t.
T(v0).
v0.
D(v0).
φ/C.
φ.
β/C.
β.
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
1.497
28.4301
1994
50.6913
0.2306
0.666
0.2486
0.718
1500
4500
1554
19146.53
1732
27.8815
0.8076
2.330
28.2853
1918
50.5542
0.3677
1.062
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
Range Table For 6-inch Gun.
Charge
{
weight, 13 lb 4 oz.
Projectile
{
Palliser shot, Shrapnel shell.
Muzzle velocity, 2154 f/s.
gravimetric density, 55.01/0.504.
Weight, 100lb.
Nature of mounting, pedestal.
nature, cordite, size 30
Jump, nil.
Remaining Velocity.
To strikean object10 ft.highrangemust beknown to
Slope of Descent.
5′ elevation ordepression alterspoint of impact.
Elevation.
Range.
Fusescale forT. and P.middleNo. 54Marks I.,II., or III.
50% of roundsshould fall in:
Time of Flight.
PenetrationintoWroughtIron.
Range.
LaterallyorVertically
Length.
Breadth.
Height.
f/s.
yds.
1 in
yds.
yds.
° ′
yds.
yds
yds
yds
secs.
in.
2154
..
..
..
0.00
0 0
0
..
..
..
..
0.00
13.6
2122
1145
687
125
0.14
0 4
100
¼
..
0.4
..
0.16
13.4
2091
635
381
125
0.29
0 9
200
¾
..
0.4
..
0.31
13.2
2061
408
245
125
0.43
0 13
300
1
..
0.4
..
0.47
13.0
2032
316
190
125
0.58
0 17
400
1¼
..
0.4
..
0.62
12.8
2003
260
156
125
0.72
0 21
500
1¾
..
0.5
0.2
0.78
12.6
1974
211
127
125
0.87
0 26
600
2
..
0.5
0.2
0.95
12.4
1946
183
110
125
1.01
0 30
700
2¼
..
0.5
0.2
1.11
12.2
1909
163
98
125
1.16
0 34
800
2¾
..
0.5
0.2
1.28
12.0
1883
143
85
125
1.31
0 39
900
3
..
0.6
0.3
1.44
11.8
1857
130
78
125
1.45
0 43
1000
3¼
..
0.6
0.3
1.61
11.6
1830
118
71
125
1.60
0 47
1100
3¾
..
0.6
0.3
1.78
11.4
1803
110
66
125
1.74
0 51
1200
4
..
0.6
0.3
1.95
11.2
1776
101
61
125
1.89
0 55
1300
4½
..
0.7
0.4
2.12
11.0
1749
93
56
125
2.03
0 59
1400
4¾
..
0.7
0.4
2.30
10.8
1722
86
52
125
2.18
1 3
1500
5
..
0.7
0.4
2.47
10.6
1695
80
48
125
2.32
1 7
1600
5½
25
0.8
0.5
2.65
10.5
1669
71
43
125
2.47
1 11
1700
5¾
25
0.9
0.5
2.84
10.3
1642
67
40
100
2.61
1 16
1800
6¼
25
1.0
0.5
3.03
10.1
1616
61
37
100
2.76
1 22
1900
6½
25
1.1
0.6
3.23
9.9
1591
57
34
100
2.91
1 27
2000
7
25
1.2
0.6
3.41
9.7
The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article ARMOUR PLATES.
High Angle and Curved Fire.—"High angle fire," as defined officially, "is fire at elevations greater than 15°," and "curved fire is fire from howitzers at all angles of elevation not exceeding 15°." In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.
Starting with the exact equations of motion in a resisting medium,
and eliminating r,
and this, in conjunction with
reduces to
the equation obtained, as in (18), by resolving normally in the trajectory, butdinow denoting theincrementofiin the increment of timedt.
Denotingdx/dt, the horizontal component of the velocity, byq, so that
(49)vcosi=q,
(49)vcosi=q,
(49)vcosi=q,
equation (43) becomes
(50)dq/dt= -rcosi,
(50)dq/dt= -rcosi,
(50)dq/dt= -rcosi,
and therefore by (48)
It is convenient to expressras a function ofvin the previous notation
(52) Cr=f(v),
(52) Cr=f(v),
(52) Cr=f(v),
and now
an equation connectingqandi.
Now, sincev=gseci
and multiplying bydx/dtorq,
and multiplying bydy/dxor tani,
also
from which the values oft,x,y,i, and taniare given by integration with respect toq, when seciis given as a function ofqby means of (51).
Now these integrations are quite intractable, even for a very simple mathematical assumption of the functionf(v), say the quadratic or cubic law,f(v) =v2/korv3/k.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantitiesi, cosi, and secivary so slowly that they may be replaced by theirmeanvalues,η, cosη, and secη, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.
Replacing then the angleion the right-hand side of equations (54) - (56) by some mean valueη, we introduce Siacci's pseudo-velocityudefined by