Obliczanie poprawek

redukcyjnych do długości

Tematem zadania było obliczenie poprawek redukcyjnych K

2.

K

3,

K

E

dla zadanych odległości.

Obliczenia zostały zrobione w programie Microsoft Exel do których sporządzono wykresy

zależności poprawek od długości (od 0 do 80 km, z częstotliwością co kilometr). Dodatkowo na

wykresach zaznaczono funkcję odpowiadającą ustalonej wielkości poprawek, kolejno: -1mm, -

0,1 mm, -1mm.

We wszystkich obliczeniach wartość K wynosiła 0,13.

1. Według wzoru:

obliczono wartość D

1

=

16,268km dla K

2

= -1 mm.

Punkt przecięcia się krzywych to punkt o współrzędnych (65,614; -65,614).

2. Według wzoru:

obliczono wartość D

2

=

17,928km dla K

3

=-0,1 mm.

Punkt przecięcia się krzywych to punkt o współrzędnych (75,910; 7,591).

-140

-120

-100

-80

-60

-40

-20

0

20

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

K2[mm]

D1 [km]

k2 od d1

ppm

3. Według wzoru:

obliczono wartość D

0

=

9,912km dla K

E

= -1 mm.

Punkt przecięcia wykresów to punkt o współrzędnych (31,205; 31,206).

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

[mm]

D2 [km]

K3

ppm

0

100

200

300

400

500

600

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

[mm]

D0 [km]

Ke od D0

ppm

D0 [km]

Ke [km]

Ke [mm]

0,999999999750 0,0000000010268568241 0,001027
2,000000000888 0,0000000082148545930 0,008215
2,999999993260 0,0000000277251342513 0,027725
3,999999984024 0,0000000657188367437 0,065719
4,999999968796 0,0000001283571030151 0,128357
5,999999946080 0,0000002218010740101 0,221801
6,999999914377 0,0000003522118906734 0,352212
7,999999872190 0,0000005257506939498 0,525751
8,999999818021 0,0000007485786247841 0,748579
9,999999750371 0,0000010268568241208 1,026857

10,999999667744 0,0000013667464329048 1,366746
11,999999568641 0,0000017744085920807 1,774409
12,999999451565 0,0000022560044425934 2,256004
13,999999315018 0,0000028176951253874 2,817695
14,999999157503 0,0000034656417814077 3,465642
15,999998977520 0,0000042060055515987 4,206006
16,999998773573 0,0000050449475769054 5,044948
17,999998544164 0,0000059886289982724 5,988629
18,999998287795 0,0000070432109566445 7,043211
19,999998002969 0,0000082148545929663 8,214855
20,999997688187 0,0000095097210481826 9,509721
21,999997341952 0,0000109339714632381 10,93397
22,999996962765 0,0000124937669790776 12,49377
23,999996549130 0,0000141952687366457 14,19527
24,999996099549 0,0000160446378768873 16,04464
25,999995612523 0,0000180480355407469 18,04804
26,999995086555 0,0000202116228691694 20,21162
27,999994520147 0,0000225415610030995 22,54156
28,999993911801 0,0000250440110834818 25,04401
29,999993260020 0,0000277251342512612 27,72513
30,999992563306 0,0000305910916473823 30,59109
31,999991820161 0,0000336480444127899 33,64804
32,999991029087 0,0000369021536884286 36,90215
33,999990188587 0,0000403595806152433 40,35958
34,999989297162 0,0000440264863341786 44,02649
35,999988353315 0,0000479090319861793 47,90903
36,999987355549 0,0000520133787121901 52,01338
37,999986302364 0,0000563456876531557 56,34569
38,999985192265 0,0000609121199500208 60,91212
39,999984023752 0,0000657188367437302 65,71884
40,999982795328 0,0000707719991752286

70,772

41,999981505496 0,0000760777683854607 76,07777

42,999980152757 0,0000816423055153712 81,64231

D0 [km]

Ke [km]

Ke [mm]

43,999978735614 0,0000874717717059049

87,47177

44,999977252569 0,0000935723280980065

93,57233

45,999975702125 0,0000999501358326207

99,95014

46,999974082782 0,0001066113560506920

106,6114

47,999972393044 0,0001135621498931660

113,5621

48,999970631414 0,0001208086785009860

120,8087

49,999968796392 0,0001283571030150980

128,3571

50,999966886482 0,0001362135845764460

136,2136

51,999964900185 0,0001443842843259750

144,3843

52,999962836004 0,0001528753634046300

152,8754

53,999960692441 0,0001616929829533550

161,693

54,999958467999 0,0001708433041130960

170,8433

55,999956161179 0,0001803324880247960

180,3325

56,999953770484 0,0001901666958294000

190,1667

57,999951294415 0,0002003520886678550

200,3521

58,999948731476 0,0002108948276811030

210,8948

59,999946080168 0,0002218010740100890

221,8011

60,999943338994 0,0002330769887957600

233,077

61,999940506456 0,0002447287331790580

244,7287

62,999937581056 0,0002567624683009300

256,7625

63,999934561296 0,0002691843553023190

269,1844

64,999931445679 0,0002820005553241700

282,0006

65,999928232707 0,0002952172295074290

295,2172

66,999924920881 0,0003088405389930400

308,8405

67,999921508705 0,0003228766449219470

322,8766

68,999917994681 0,0003373317084350950

337,3317

69,999914377310 0,0003522118906734290

352,2119

70,999910655095 0,0003675233527778940

367,5234

71,999906826538 0,0003832722558894350

383,2723

72,999902890142 0,0003994647611489950

399,4648

73,999898844408 0,0004161070296975210

416,107

74,999894687839 0,0004332052226759560

433,2052

75,999890418937 0,0004507655012252460

450,7655

76,999886036205 0,0004687940264863340

468,794

77,999881538144 0,0004872969596001670

487,297

78,999876923257 0,0005062804617076880

506,2805

79,999872190046 0,0005257506939498420

525,7507

D1

[km]

K2 [km]

K2 [mm]

1

-0,00000000023227501

-0,000232275

2

0,00000000102685682

0,001026857

3

-0,00000000627142537

-0,006271425

4

-0,00000001486560087

-0,014865601

5

-0,00000002903437670

-0,029034377

6

-0,00000005017140294

-0,050171403

7

-0,00000007967032967

-0,07967033

8

-0,00000011892480697

-0,118924807

9

-0,00000016932848493

-0,169328485

10

-0,00000023227501362

-0,232275014

11

-0,00000030915804312

-0,309158043

12

-0,00000040137122353

-0,401371224

13

-0,00000051030820491

-0,510308205

14

-0,00000063736263736

-0,637362637

15

-0,00000078392817095

-0,783928171

16

-0,00000095139845577

-0,951398456

17

-0,00000114116714190

-1,141167142

18

-0,00000135462787941

-1,354627879

19

-0,00000159317431839

-1,593174318

20

-0,00000185820010893

-1,858200109

21

-0,00000215109890110

-2,151098901

22

-0,00000247326434498

-2,473264345

23

-0,00000282609009067

-2,826090091

24

-0,00000321096978823

-3,210969788

25

-0,00000362929708775

-3,629297088

26

-0,00000408246563932

-4,082465639

27

-0,00000457186909301

-4,571869093

28

-0,00000509890109890

-5,098901099

29

-0,00000566495530708

-5,664955307

30

-0,00000627142536764

-6,271425368

31

-0,00000691970493064

-6,919704931

32

-0,00000761118764617

-7,611187646

33

-0,00000834726716432

-8,347267164

34

-0,00000912933713517

-9,129337135

35

-0,00000995879120879

-9,958791209

36

-0,00001083702303527

-10,83702304

37

-0,00001176542626470

-11,76542626

38

-0,00001274539454714

-12,74539455

39

-0,00001377832153269

-13,77832153

40

-0,00001486560087143

-14,86560087

41

-0,00001600862621344

-16,00862621

42

-0,00001720879120879

-17,20879121

43

-0,00001846748950758

-18,46748951

D1

[km]

K2 [km]

K2 [mm]

44

-0,00001978611475988

-19,78611476

45

-0,00002116606061577

-21,16606062

46

-0,00002260872072534

-22,60872073

47

-0,00002411548873867

-24,11548874

48

-0,00002568775830583

-25,68775831

49

-0,00002732692307692

-27,32692308

50

-0,00002903437670202

-29,0343767

51

-0,00003081151283119

-30,81151283

52

-0,00003265972511454

-32,65972511

53

-0,00003458040720213

-34,5804072

54

-0,00003657495274405

-36,57495274

55

-0,00003864475539038

-38,64475539

56

-0,00004079120879121

-40,79120879

57

-0,00004301570659661

-43,0157066

58

-0,00004531964245667

-45,31964246

59

-0,00004770441002147

-47,70441002

60

-0,00005017140294108

-50,17140294

61

-0,00005272201486560

-52,72201487

62

-0,00005535763944510

-55,35763945

63

-0,00005807967032967

-58,07967033

64

-0,00006088950116938

-60,88950117

65

-0,00006378852561433

-63,78852561

66

-0,00006677813731458

-66,77813731

67

-0,00006985972992023

-69,85972992

68

-0,00007303469708134

-73,03469708

69

-0,00007630443244802

-76,30443245

70

-0,00007967032967033

-79,67032967

71

-0,00008313378239836

-83,1337824

72

-0,00008669618428219

-86,69618428

73

-0,00009035892897190

-90,35892897

74

-0,00009412341011758

-94,12341012

75

-0,00009799102136930

-97,99102137

76

-0,00010196315637715

-101,9631564

77

-0,00010604120879121

-106,0412088

78

-0,00011022657226156

-110,2265723

79

-0,00011452064043828

-114,5206404

80

-0,00011892480697145

-118,924807

D2 [km]

K3 [km]

K3 [mm]

0,9999999998

-0,00000000001735

-0,00001735388

2,0000000010

-0,00000000013883

-0,00013883104

2,9999999937

-0,00000000046855

-0,00046855477

3,9999999851

-0,00000000111065

-0,00111064833

4,9999999710

-0,00000000216924

-0,00216923500

5,9999999498

-0,00000000374844

-0,00374843806

6,9999999203

-0,00000000595238

-0,00595238075

7,9999998811

-0,00000000888519

-0,00888518633

8,9999998307

-0,00000001265098

-0,01265097804

9,9999997677

-0,00000001735388

-0,01735387912

10,9999996908

-0,00000002309801

-0,02309801277

11,9999995986

-0,00000002998750

-0,02998750220

12,9999994897

-0,00000003812647

-0,03812647059

13,9999993626

-0,00000004761904

-0,04761904112

14,9999992161

-0,00000005856934

-0,05856933692

15,9999990486

-0,00000007108148

-0,07108148114

16,9999988588

-0,00000008525960

-0,08525959688

17,9999986454

-0,00000010120781

-0,10120780722

18,9999984068

-0,00000011903024

-0,11903023522

19,9999981418

-0,00000013883100

-0,13883100392

20,9999978489

-0,00000016071424

-0,16071423633

21,9999975267

-0,00000018478406

-0,18478405541

22,9999971739

-0,00000021114458

-0,21114458411

23,9999967890

-0,00000023989995

-0,23989994536

24,9999963707

-0,00000027115426

-0,27115426203

25,9999959175

-0,00000030501166

-0,30501165696

26,9999954281

-0,00000034157625

-0,34157625297

27,9999949011

-0,00000038095217

-0,38095217283

28,9999943350

-0,00000042324354

-0,42324353928

29,9999937286

-0,00000046855447

-0,46855447500

30,9999930803

-0,00000051698910

-0,51698910264

31,9999923888

-0,00000056865154

-0,56865154482

32,9999916527

-0,00000062364592

-0,62364592409

33,9999908707

-0,00000068207636

-0,68207636296

34,9999900412

-0,00000074404698

-0,74404698392

35,9999891630

-0,00000080966191

-0,80966190937

36,9999882346

-0,00000087902526

-0,87902526169

37,9999872546

-0,00000095224116

-0,95224116318

38,9999862217

-0,00000102941374

-1,02941373611

39,9999851344

-0,00000111064710

-1,11064710269

40,9999839914

-0,00000119604539

-1,19604538506

41,9999827912

-0,00000128571271

-1,28571270532

42,9999815325

-0,00000137975319

-1,37975318549

Zadanie 2. Z jaką dokładnością należy mierzyć czas, aby pomiar odcinka150 m miał błąd nie

większy niż 3 mm?

Odp. Z dokładnością

.

D2 [km]

K3 [km]

K3 [mm]

43,9999802139

-0,00000147827095

-1,47827094756

44,9999788339

-0,00000158137011

-1,58137011343

45,9999773913

-0,00000168915480

-1,68915480494

46,9999758845

-0,00000180172914

-1,80172914388

47,9999743122

-0,00000191919725

-1,91919725195

48,9999726731

-0,00000204166325

-2,04166325080

49,9999709656

-0,00000216923126

-2,16923126201

50,9999691885

-0,00000230200541

-2,30200540709

51,9999673403

-0,00000244008981

-2,44008980745

52,9999654196

-0,00000258358858

-2,58358858446

53,9999634250

-0,00000273260586

-2,73260585941

54,9999613552

-0,00000288724575

-2,88724575349

55,9999592088

-0,00000304761239

-3,04761238783

56,9999569843

-0,00000321380988

-3,21380988349

57,9999546804

-0,00000338594236

-3,38594236142

58,9999522956

-0,00000356411394

-3,56411394251

59,9999498286

-0,00000374842875

-3,74842874756

60,9999472780

-0,00000393899090

-3,93899089728

61,9999446424

-0,00000413590451

-4,13590451229

62,9999419203

-0,00000433927371

-4,33927371314

63,9999391105

-0,00000454920262

-4,54920262027

64,9999362115

-0,00000476579535

-4,76579535404

65,9999332219

-0,00000498915603

-4,98915603471

66,9999301403

-0,00000521938878

-5,21938878245

67,9999269653

-0,00000545659772

-5,45659771734

68,9999236956

-0,00000570088696

-5,70088695934

69,9999203297

-0,00000595236063

-5,95236062834

70,9999168662

-0,00000621112284

-6,21112284411

71,9999133038

-0,00000647727773

-6,47727772634

72,9999096411

-0,00000675092939

-6,75092939459

73,9999058766

-0,00000703218197

-7,03218196832

74,9999020090

-0,00000732113957

-7,32113956691

75,9998980368

-0,00000761790631

-7,61790630961

76,9998939588

-0,00000792258632

-7,92258631556

77,9998897734

-0,00000823528370

-8,23528370379

78,9998854794

-0,00000855610259

-8,55610259325

79,9998810752

-0,00000888514710

-8,88514710272