Volume 3, Part 1, Slice 2 Part 28 (1/2)

(12) R[Delta]s = loss of kinetic energy in foot-pounds =w(v+[Delta]v)^2/g - w(v-[Delta]v)^2/g = wv[Delta]v/g, so that (13) [Delta]s = wv[Delta]v/nd^2pg = C[Delta]S, where (14) [Delta]S = v[Delta]v/gp = v[Delta]T,

and [Delta]S is the advance in feet of a shot for which C=1, while the velocity falls [Delta]v in pa.s.sing through the average velocity v.

Denoting by S(v) the sum of all the values of [Delta]S up to any a.s.signed velocity v,

(15) S(v) = [Sum]([Delta]S) + a constant, by which S(v) is calculated from [Delta]S, and then between two a.s.signed velocities V and v,

(16) S(V) - S(v) = [Sum,v:V][Delta]T = [Sum]v[Delta]v/gp or [Integral,v:V]vdv/gp,

and if s feet is the advance of a shot whose ballistic coefficient is C,

(17) s = C[S(V) - S(v)].

In an extended table of S, the value is interpolated for unit increment of velocity.

A third table, due to Sir W. D. Niven, F.R.S., called the _degree_ table, determines the change of direction of motion of the shot while the velocity changes from V to v, the shot flying nearly horizontally.

To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity is v, to make an angle i radians with the horizon.

Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,

(18) v(di/dt) = g cos i,

where di denotes the infinitesimal _decrement_ of i in the infinitesimal increment of time dt_.

[v.03 p.0272] In a problem of direct fire, where the trajectory is flat enough for cos i to be undistinguishable from unity, equation (16) becomes

(19) v(di/dt) = g, or di/dt = g/v;

so that we can put

(20) [Delta]i/[Delta]t = g/v

if v denotes the mean velocity during the small finite interval of time [Delta]t, during which the direction of motion of the shot changes through [Delta]i radians.

If the inclination or change of inclination in degrees is denoted by [delta] or [Delta][delta],

(21) [delta]/180 = i/[pi], so that

(22) [Delta][delta] = 180/[pi] [Delta]i = 180g/[pi] [Delta]t/v;

and if [delta] and i change to D and I for the standard projectile,

(23) [Delta]I = g [Delta]T/v = [Delta]v/vp, [Delta]D = 180g/[pi] [Delta]T/v, and

(24) I(V) - I(v) = [Sum,v:V][Delta]v/vp or [Integral,v:V]dv/vp, D(V) - D(v) = 180/[pi] [I(V) - I(v)].

The differences [Delta]D and [Delta]I are thus calculated, while the values of D(v) and I(v) are obtained by summation with the arithmometer, and entered in their respective columns.

For some purposes it is preferable to retain the circular measure, i radians, as being undistinguishable from sin i and tan i when i is small as in direct fire.

The last function A, called the _alt.i.tude function_, will be explained when high angle fire is considered.

These functions, T, S, D, I, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above. The initial values of T, S, D, I, A must be accepted as belonging to the anterior portion of the table.

In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v^m, the integration can be effected which replaces the summation in (10), (16), and (24); and from an a.n.a.lysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f(v) or v^m/k or its equivalent Cr, where r is the r.e.t.a.r.dation.

ABRIDGED BALLISTIC TABLE.

-----+--------+-------+---------+-------+----------+-------+-------- v.

p. [Delta]T.

T. [Delta]S.

S. [Delta]D.

D.

-----+--------+-------+---------+-------+----------+-------+-------- f/s

1600

11.416

.0271

27.5457

43.47

18587.00

.0311

49.7729 1610

11.540

.0268

27.5728

43.27

18630.47

.0306

49.8040 1620

11.662

.0265

27.5996

43.08

18673.74

.0301

49.8346 1630

11.784

.0262

27.6261

42.90

18716.82

.0296

49.8647

1640

11.909

.0260

27.6523

42.72

18759.72

.0291

49.8943 1650

12.030

.0257

27.6783

42.55

18802.44

.0287

49.9234 1660

12.150

.0255

27.7040

42.39

18844.99

.0282

49.9521 1670

12.268

.0252

27.7295

42.18

18887.38

.0277

49.9803

1680

12.404

.0249

27.7547

41.98

18929.56

.0273

50.0080 1690

12.536

.0247

27.7796

41.78

18971.54

.0268

50.0353 1700

12.666

.0244

27.8043

41.60

19013.32

.0264

50.0621 1710

12.801

.0242

27.8287

41.41

19054.92

.0260

50.0885

1720

12.900

.0239

27.8529

41.23

19096.33

.0256

50.1145 1730

13.059

.0237

27.8768

41.06

19137.56

.0252

50.1401 1740

13.191

.0234

27.9005

40.90

19178.62

.0248

50.1653 1750

13.318

.0232

27.9239

40.69

19219.52

.0244

50.1901

1760

13.466

.0230

27.9471

40.53

19260.21

.0240

50.2145 1770

13.591

.0227

27.9701

40.33

19300.74

.0236

50.2385 1780

13.733

.0225

27.9928

40.19

19341.07

.0233

50.2621 1790

13.862

.0223

28.0153

40.00

19381.26

.0229

50.2854

1800

14.002

.0221

28.0376

39.81

19421.26

.0225

50.3083 1810

14.149

.0219

28.0597

39.68

19461.07

.0222

50.3308 1820

14.269

.0217

28.0816

39.51

19500.75

.0219

50.3530 1830

14.414

.0214

28.1033

39.34

19540.26

.0216

50.3749

1840

14.552

.0212

28.1247

39.17

19579.60

.0212

50.3965 1850

14.696

.0210

28.1459

39.01

19618.77

.0209

50.4177 1860

14.832

.0209

28.1669

38.90

19657.78

.0206

50.4386 1870

14.949

.0207

28.1878

38.75

19696.68

.0203

50.4592

1880

15.090

.0205

28.2085

38.61

19735.43

.0200

50.4795 1890

15.224

.0203

28.2290

38.46

19774.04

.0198

50.4995 1900

15.364

.0201

28.2493

38.32

19812.50

.0195

50.5193 1910

15.496

.0199

28.2694

38.19

19850.82

.0192

50.5388

1920

15.656

.0197

28.2893

38.01

19889.01

.0189

50.5580 1930

15.809

.0196

28.3090

37.83

19927.02

.0186

50.5769 1940

15.968

.0194

28.3286

37.66

19964.85

.0184

50.5955 1950

16.127

.0192

28.3480

37.48

20002.51

.0181

50.6139

1960

16.302

.0190

28.3672

37.26

20039.99

.0178

50.6320 1970

16.484

.0187

28.3862

36.99

20077.25

.0175

50.6498 1980

16.689

.0185

28.4049

36.73

20114.24

.0172

50.6673 1990

16.888

.0183

28.4234

36.47

20150.97

.0169

50.6845

2000

17.096

.0181

28.4417

36.21

20187.44

.0166

50.7014 2010

17.305

.0178

28.4598

35.95

20223.65

.0163

50.7180 2020

17.515

.0176

28.4776

35.65

20259.60

.0160

50.7343 2030

17.752

.0174

28.4952

35.35

20295.25

.0158

50.7503

2040

17.990

.0171

28.5126

35.06

20330.60

.0155

50.7661 2050

18.229

.0169

28.5297

34.77

20365.66

.0152

50.7816 2060

18.463

.0167

28.5466

34.49

20400.43

.0149

50.7968 2070

18.706

.0165

28.5633

34.21

20434.92

.0147

50.8117

2080

18.978

.0163

28.5798

33.93

20469.13

.0144

50.8264 2090

19.227

.0160

28.5961

33.60

20503.06

.0141

50.8408 2100

19.504

.0158

28.6121

33.34

20536.66

.0139

50.8549 2110

19.755

.0156

28.6279

33.02

20570.00

.0136

50.8688

2120

20.010

.0154

28.6435

32.76

20603.02

.0134

50.8824 2130

20.294

.0152

28.6589

32.50