Page:EB1911 - Volume 03.djvu/291

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

Charge ${\Bigg \{}$	weight, 13 ℔ 4 oz.	55·01/0·504.
	gravimetric density,
	nature, cordite, size 30.

${\Bigg |}$

Projectile ${\bigg \{}$	Palliser shot, Shrapnel shell.
	Weight, 100 ℔.

${\Bigg |}$

Muzzle velocity, 2154 f/s.

Nature of mounting, pedestal.

Jump, nil.

Remain- ing 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.	Penetra- tion 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	11/4	. .	0·4	. .	0·62	12·8
2003	260	156	125	0·72	0	21	500	13/4	. .	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	21/4	. .	0·5	0·2	1·11	12·2
1909	163	98	125	1·16	0	34	800	23/4	. .	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	31/4	. .	0·6	0·3	1·61	11·6
1830	118	71	125	1·60	0	47	1100	33/4	. .	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	41/2	. .	0·7	0·4	2·12	11·0
1749	93	56	125	2·03	0	59	1400	43/4	. .	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	51/2	25	0·8	0·5	2·65	10·5
1669	71	43	125	2·47	1	11	1700	53/4	25	0·9	0·5	2·84	10·3
1642	67	40	100	2·61	1	16	1800	61/4	25	1·0	0·5	3·03	10·1
1616	61	37	100	2·76	1	22	1900	61/2	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)  ${\frac {d^{2}x}{dt^{2}}}=-r\cos i=-r{\frac {dx}{ds}},$

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

and eliminating r,

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

and this, in conjunction with

(46)  $\tan i={\frac {dy}{dx}}={\frac {dy}{dt}}{\Bigg /}{\frac {dx}{dt}},$

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

reduces to

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

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)  $v\cos i=q,\,$

equation (43) becomes

(50)  $dq/dt=-r\cos i,\,$

and therefore by (48)

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

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

(52)  $Cr=f(v)\,$

and now

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

an equation connecting q and i.

Now, since v=q sec i

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

and multiplying by dx/dt or q,

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

and multiplying by dx/dx or tan i,

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

also

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

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

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)  $u=q\sec \eta ,$

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