Page:EB1911 - Volume 03.djvu/291

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BALLISTICS

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 10628.53 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 ${\displaystyle {\Bigg \{}}$ weight, 13 lb 4 oz. ${\displaystyle {\frac {55.01}{0.504}}.}$ gravimetric density, nature, cordite, size 30.
${\displaystyle {\Bigg |}}$
 Projectile ${\displaystyle {\bigg \{}}$ Palliser shot, Shrapnel shell. Weight, 100 lb.
${\displaystyle {\Bigg |}}$
 Muzzle velocity, 2154 f/s. Nature of mounting, pedestal. Jump, nil.
Remaining Velocity To strike an object 10 ft. high range must be known to Slope of Descent. 5' elevation or depression alters point of impact. Elevation. Range. Fuse scale for T. and P. middle No. 54 Marks I., II., or III. 50% of rounds should fall in. Time of Flight. Penetration into Wrought Iron.
Range. Laterally or Vertically. 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 1/4 .. 0.4 .. 0.16 13.4
2091 635 381 125 0.29 0 9 200 3/4 .. 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 1 1/4 .. 0.4 .. 0.62 12.8
2003 260 156 125 0.72 0 21 500 1 3/4 .. 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 2 1/4 .. 0.5 0.2 1.11 12.2
1909 163 98 125 1.16 0 34 800 2 3/4 .. 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 3 1/4 .. 0.6 0.3 1.61 11.6
1830 118 71 125 1.60 0 47 1100 3 3/4 .. 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 4 1/2 .. 0.7 0.4 2.12 11.0
1749 93 56 125 2.03 0 59 1400 4 3/4 .. 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 5 1/2 25 0.8 0.5 2.65 10.5
1669 71 43 125 2.47 1 11 1700 5 3/4 25 0.9 0.5 2.84 10.3
1642 67 40 100 2.61 1 16 1800 6 1/4 25 1.0 0.5 3.03 10.1
1616 61 37 100 2.76 1 22 1900 6 1/2 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,

(43) ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=-r\cos i=-r{\frac {dx}{ds}},}$
(44) ${\displaystyle {\frac {d^{2}y}{dt^{2}}}=-r\sin i-g=-r{\frac {dy}{ds}}-g,}$

and eliminating r,

(45) ${\displaystyle {\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}=-g{\frac {dx}{dt}},}$

and this, in conjunction with

(46) ${\displaystyle \tan i={\frac {dy}{dx}}={\frac {dy}{dt}}{\Bigg /}{\frac {dx}{dt}},}$
(47) ${\displaystyle \sec ^{2}i{\frac {di}{dt}}=\left({\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}\right){\Bigg /}\left({\frac {dx}{dt}}\right)^{2},}$

reduces to

(48) ${\displaystyle {\frac {di}{dt}}=-{\frac {g}{v}}\cos i,{\mbox{ or }}{\frac {d\tan i}{dt}}=-g{\frac {g}{v\cos i}},}$

the equation obtained, as in (18), by resolving normally in the trajectory, but di now denoting the increment of i in the increment of time dt.

Denoting dx/dt, the horizontal component of the velocity, by q, so that

(49) ${\displaystyle v\cos i=q,\,}$

equation (43) becomes

(50) ${\displaystyle dq/dt=-r\cos i,\,}$

and therefore by (48)

(51) ${\displaystyle {\frac {dq}{di}}={\frac {dq}{dt}}{\frac {dt}{di}}={\frac {rv}{g}}}$

It is convenient to express r as a function of v in the previous notation

(52) ${\displaystyle Cr=f(v)\,}$

and now

(53) ${\displaystyle {\frac {dq}{di}}={\frac {vf(v)}{Cg}},}$

an equation connecting q and i.

Now, since v=q sec i

(54) ${\displaystyle {\frac {dt}{dq}}=-C{\frac {\sec i}{f(q\sec i)}},}$

and multiplying by dx/dt or q,

(55) ${\displaystyle {\frac {dx}{dq}}=-{\frac {Cq\sec i}{f(q\sec i)}},}$

and multiplying by dx/dx or tan i,

(56) ${\displaystyle {\frac {dy}{dq}}=-{\frac {Cq\sec i\tan i}{f(q\sec i)}};}$

also

(57) ${\displaystyle {\frac {di}{dq}}={\frac {Cg}{q\sec i.f(q\sec i)}},}$
(58) ${\displaystyle {\frac {d\tan i}{dq}}={\frac {Cg\sec i}{q.f(q\sec i)}},}$

from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).

Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v)=v2/k or v3/k.

But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values, η, cos η, and sec η, 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 angle i on the right-hand side of equations (54)-(56) by some mean value η we introduce Siacci's pseudo-velocity u defined by

(59) ${\displaystyle u=q\sec \eta ,\,}$

so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc.