Biomechanical Basis of Human Movement, 3rd Edition

Appendices

Appendix A: The Metric System and SI Units

All measurements in the biomechanics literature are expressed in terms of the metric system. This system of measurement uses units that are related to one another by some power of 10. Table A-1 presents the prefixes associated with these powers.

The standard length in the metric system is the meter. This standard measure was originally indicated by two scratches on a platinum-iridium alloy bar kept at the International Bureau of Weights and Measures in Sèvres, France. Table A-2 illustrates the use of these prefixes in common units of length.

The uniform system for the reporting of numerical values is known as the Système International d'Unites, or SI. This system was developed through international cooperation to standardize the report of scientific information. The base dimensions used in biomechanics are mass, length, time, temperature, electric current, amount of substance, and luminous intensity. The base units in SI corresponding to the base dimensions are the kilogram, the meter, the second, and the degree kelvin. Table A-3 presents the base units of SI.

Other units of measurement that are used in biomechanics are derived. These units are presented in Table A-4. Common conversions are shown in Table A-5.

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TABLE A-1 Prefixes for the Powers of 10

Prefix

Multiplier

Symbol

Example

Giga

109

G

Gigabyte (Gb)

Mega

106

M

Megawatt (MW)

Kilo

103

k

Kilogram (kg)

Centi

10–2

c

Centimeter (cm)

Milli

10–3

m

Milligram (mg)

Micro

10–6

µ

Microsecond (µs)

Nano

10–9

v

Nanosecond (ns)

TABLE A-2 Units of Length

Metric Unit

Power

Symbol

Kilometer

10–3

km

Meter

m

Decimeter

10–1

dm

Centimeter

10–2

cm

Millimeter

10–3

mm

Micrometer

10–6

µm

TABLE A-3 Base Units of Measurement

Dimension

Unit

Symbol

Mass

Kilogram

kg

Length

Meter

m

Time

Second

s

Temperature

Degree kelvin

K

Electrical current

Ampere

A

Amount of substance

Mole

Mol

Luminous intensity

Candela

cd

TABLE A-4 Derived units of measurement

Dimension

Unit

Symbol

Acceleration

Meters per second squared

m/s2

Angle

Radian

rad

Area

Meter squared

m2

Capacitance

Farad

F

Concentration

Moles per meter cubed

mol/m3

Density

Mass per unit volume

kg/m3

Energy

Joule

J

Impulse

Force × time

N5

Luminous flux

Lumen

lm

Moment of inertia

Kilogram-meters squared

kgm2

Momentum

Kilogram-meters per second

kgm/s

Power

Watts

W

Pressure

Pascal

Pa

Resistance

Ohm

Ω

Speed

Meters per second

m/s

Torque

Newton meter

Nm

Voltage

Volt

V

Volume

Meter cubed

m3

Work

Joule

J

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TABLE A-5 Common Conversions

Length

1 inch = 25.40 mm

1 inch = 2.54 cm

1 foot = 0.3048 m

1 foot= 30.48 cm

1 yard = 0.9144 m

1 mile = 1609.34 m

1 cm = 0.3937 inches

1 cm = 0.0328 feet

1 m = 39.37 inches

1 m = 3.2808 feet

1 m = 1.0936 yard

1 m = .0936 yard

1 km =0.6214 miles

Area

1 in =645.16 mm2

1 ft2 =0.0929 m2

1 yd2 =0.8361 m2

1 cm2 0.155 in2

1 m2 =10.76 ft2

Volume

1 in3 =16.387 cm3

1 ft3 =.0283 m3

1 yd3 =0.7645 m3

1 yd3 0.7645 m3

1 cm3 =0.061 in3

1 m3 =35.32 ft3

1 m3 =1.308 yd3

 

Mass

1 oz = 28.349 g

1 lb = 0.4536 kg

1 slug = 14.5939 kg

1 slug = 32.2 lb

1 g = 0.0353 oz.

1 kg = 2.2046 lb

1 kg = 0.0685 slug

 

Density

1 lb-ft3 = 16.02 kg/m3

1 slug-ft3 = 515.38 kg/m3

 

Moment of Inertia

1 slug-ft2 = 1.36 kgm2

1 lb-ft2 = 0.042 kgm2

 

Velocity

1 in/s = 24.4 mm/s

1 ft/s = 0.305 m/s

1 mph = 0.447 m/s

1 mph= 1.467 ft/s

1 m/s = 3.60 km/hr

1 m/s = 3.28 ft/s

1 m/s = 2.237 m/hr

1 cm/s = 0.0328 ft/s

Force

1 poundal = 0.1383 N

1 lb-force = 4.448 N

1 kg-force = 9.81 N

1 N = 0.102 kg
1 N = 0.2248 lb

Pressure

1 poundal/ft2 = 1.4881 Pa
1 mm mercury = 133.322 Pa

1 pound-force /ft2 = 47.889 Pa

pound-force/in2 = 6.8947 Pa

 

Work and Energy

1 foot poundal = 0.0421 J

1 foot pound-force = 1.3558 J

1 foot pound-force = 0.1383 kgm

1 kgm = 7.2307 ft-lb

1 BTU = 1.0551 kJ

1 kilocalorie = 4.1868 kJ

1 J = 0.7376 ft-lb

1 J = 0.1020 kgm

Power

1 hp (British) = 745.700 W

1 hp (metric) = 735.499 W

1 hp (metric) = 735.499 W

1 ft-lb force/s = 1.3558 W

Torque

1 Nm = 0.74 lb-ft

1 lb-ft = 1.36 Nm

1 kgm = 7.23 lb-ft

1 kg-cm = 0.0723 lb-ft

BTU, British thermal unit.

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Appendix B: Trigonometric Functions

Trigonometry is a branch of mathematics concerned with the measurements of the sides and angles of triangles and their relationships with each other. Many concepts in biomechanics require knowledge of trigonometry A triangle is composed of three sides and three angles. The sum of the three angles of a triangle equals 180°. A right triangle is a triangle in which one of the angles is a right angle, that is, one of the angles equals 90°. The sum of the remaining angles, therefore, also equals 90°. Consider the triangle with vertices A, B, and C and sides of length AB, AC, and BC in Figure B-1.

The side opposite the right angle, AB, is always the longest side in the right triangle and is referred to as the hypotenuse. The other two sides are named according to which of the other angles is under consideration. If angle A is considered, the side AC is the adjacent side, and the side BC is the opposite side. If angle B is considered (as in the triangle on the right below), BC is the adjacent side and AC is the opposite side.

Based on the right triangle, trigonometric functions may be defined. A mathematical function is a quantity whose value varies and depends on some other quantity or quantities. Trigonometric functions vary with and depend on the values of the two acute (i.e., <90°) angles in a right triangle and the lengths of the sides of the triangle. The trigonometric functions are the ratios of the lengths of the sides of the triangle based on one of the two acute angles in the triangle. There are six such functions: sine (abbreviation is sin), cosine (abbreviation is cos), tangent (abbreviation is tan), cosecant, secant, and cotangent.

In biomechanics, only the first three of these functions are important. The trigonometric functions are thus defined for angle A (Fig. B-2A) as:

  1. The sine of an angle is the ratio of the side opposite the angle to the hypotenuse.
  2. The cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse.
  3. The tangent of an angle is the ratio of the side opposite to the angle to the side adjacent to the angle.

Similarly, the ratios for angle B in Figure B-2B may be defined as:

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FIGURE B-1 A right triangle with a right angle C (90°) and two acute angles, A and B. A and B sum to 90°.

For any angle, the ratios formed by the sides of the right triangle will always be the same. For example, the sine of an angle of 32 will always equal 0.5299 regardless of the size of the sides of the triangle. The same is true for the cosine and tangent of the angle. The values for the sine, cosine, and tangent of angles can be presented in tables. Table B-1 presents these values for angles ranging from 0° to 90°.

 

FIGURE B-2 Descriptions of the sides of a right triangle based on the acute angles A (A) and B (B).

TABLE B-1 Trigonometric Functions

   

Angle

Degrees

Radian

Sine

Cosine

Tangent

         

0

0.000

0.0000

0.0000

0.0000

1

0.017

0.0175

0.9998

0.0175

2

0.035

0.0349

0.9994

0.0349

3

0.052

0.0523

0.9986

0.0524

4

0.070

0.0698

0.9976

0.0699

5

0.087

0.0872

0.9962

0.0875

6

0.105

0.1045

0.9945

0.1051

7

0.122

0.1219

0.9925

0.1228

8

0.140

0.1392

0.9903

0.1405

9

0.157

0.1564

0.9877

0.1584

10

0.175

0.1736

0.9848

0.1763

11

0.192

0.1908

0.9816

0.1944

12

0.209

0.2079

0.9781

0.2126

13

0.227

0.2250

0.9744

0.2309

14

0.244

0.2419

0.9703

0.2493

15

0.262

0.2588

0.9659

0.2679

16

0.279

0.2756

0.9613

0.2867

17

0.297

0.2924

0.9563

0.3057

18

0.314

0.3090

0.9511

0.3249

19

0.332

0.3256

0.9455

0.3443

20

0.349

0.3420

0.9397

0.3640

21

0.367

0.3584

0.9336

0.3839

22

0.384

0.3746

0.9272

0.4040

23

0.401

0.3907

0.9205

0.4245

24

0.419

0.4067

0.9135

0.4452

25

0.436

0.4226

0.9063

0.4663

26

0.454

0.4384

0.8988

0.4877

27

0.471

0.4540

0.8910

0.5095

28

0.489

0.4695

0.8829

0.5317

29

0.506

0.4848

0.8746

0.5543

30

0.524

0.5000

0.8660

0.5774

31

0.541

0.5150

0.8572

0.6009

32

0.559

0.5299

0.8480

0.6249

33

0.576

0.5446

0.8387

0.6494

34

0.593

0.5592

0.8290

0.6745

35

0.611

0.5736

0.8192

0.7002

36

0.628

0.5878

0.8090

0.7265

37

0.646

0.6018

0.7986

0.7536

38

0.663

0.6157

0.7880

0.7813

39

0.681

0.6293

0.7771

0.8098

40

0.698

0.6428

0.7660

0.8391

41

0.716

0.6561

0.7547

0.8693

42

0.733

0.6691

0.7431

0.9004

43

0.750

0.6820

0.7314

0.9325

44

0.768

0.6947

0.7193

0.9657

45

0.785

0.7071

0.7071

1.0000

46

0.803

0.7193

0.6947

1.0355

47

0.820

0.7314

0.6820

1.0724

48

0.838

0.7431

0.6691

1.1106

49

0.855

0.7547

0.6561

1.1504

50

0.873

0.7660

0.6428

1.1918

51

0.890

0.7771

0.6293

1.2349

52

0.908

0.7880

0.6157

1.2799

53

0.925

0.7986

0.6018

1.3270

54

0.942

0.8090

0.5878

1.3764

55

0.960

0.8192

0.5736

1.4281

56

0.977

0.8290

0.5592

1.4826

57

0.995

0.8387

0.5446

1.5399

58

1.012

0.8480

0.5299

1.6003

59

1.030

0.8572

0.5150

1.6643

60

1.047

0.8660

0.5000

1.7321

61

1.065

0.8746

0.4848

1.8040

62

1.082

0.8829

0.4695

1.8807

63

1.100

0.8910

0.4540

1.9626

64

1.117

0.8988

0.4384

2.0503

65

1.134

0.9063

0.4226

2.1445

66

1.152

0.9135

0.4067

2.2460

67

1.169

0.9205

0.3907

2.3559

68

1.187

0.9272

0.3746

2.4751

69

1.204

0.9336

0.3584

2.6051

70

1.222

0.9397

0.3420

2.7475

71

1.239

0.9455

0.3256

2.9042

72

1.257

0.9511

0.3090

3.0777

73

1.274

0.9563

0.2924

3.2709

74

1.292

0.9613

0.2756

3.4874

75

1.309

0.9659

0.2588

3.7321

76

1.326

0.9703

0.2419

4.0108

77

1.344

0.9744

0.2250

4.3315

78

1.361

0.9781

0.2079

4.7046

79

1.379

0.9816

0.1908

5.1446

80

1.396

0.9848

0.1736

5.6713

81

1.414

0.9877

0.1564

6.3138

82

1.431

0.9903

0.1392

7.1154

83

1.449

0.9925

0.1219

8.1443

84

1.466

0.9945

0.1045

9.5144

85

1.484

0.9962

0.0872

11.4300

86

1.501

0.9976

0.0698

14.3010

87

1.518

0.9986

0.0523

19.0810

88

1.536

0.9994

0.0349

28.6360

89

1.553

0.9998

0.0175

57.2900

90

1.571

1.0000

0.0000

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The values in this table may also be used to determine the angle when the sides of the triangle are known. Consider the triangle in Figure B-3. The length of the hypotenuse, AC, is 0.05 m, and the length of the side opposite angle C is 0.03 m. The ratio of AB to AC is the sine of angle C. Thus:

 

FIGURE B-3 A right triangle with two sides of known lengths and two unknown angles.

If the sine values in Table B-1 are examined, it can be determined that the angle whose sine is 0.6 is approximately 37°. This is the arcsin of the angle. Thus:

The ratio of the side AB to the hypotenuse AC is the cosine of angle A in Figure B-3. The ratio is still 0.6, and the angle whose cosine is 0.6 is approximately 53. This is referred to as the arccosine of an angle.

Two other useful trigonometric relationships are applicable to all triangles, not only right triangles. The first of these relationships is the law of sines, which states that the ratio of the length of any side to the sine of the angle opposite that side is equal to the ratio of any other side to

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the angle opposite that side. Consider the triangle in Figure B-4.

 

FIGURE B-4 A scalene triangle (no right angle; no two angles the same).

For this triangle, the law of sines can be stated as follows:

The other trigonometric relationship that is applicable to any triangle is the law of cosines. This relationship states that the square of the length of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle opposite the original side. Consider the triangle in Figure B-4. For side A of this triangle, the law of cosines can be stated as follows:

Similarly, the law of cosines can be stated for sides B and C.

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Appendix C: Sample Kinematic and Kinetic Data

Subject: 50-kg woman

Activity: Walking sampled at 120 Hz; one cycle from right foot touchdown to right foot touchdown

Markers: Head, R shoulder, R elbow, R wrist, R hand, R crest (IC), R greater trochanter (GT), R knee, R ankle, R heel, R fifth metatarsal (met), R toe

Stride events: Right foot touchdown (RTD), left foot touchdown (LTD), right foot toe-off (RTO)

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TABLE C-1 Kinematic Data in Millimeters

Frame Number

Head x

Head y

Shoulder x

Shoulder Y

Elbow x

Elbow Y

Wrist x

Wrist y

Hand x

Hand y

IC x

IC Y

1

28.000

1458.867

155.0670

1340.267

272.4000

1122.533

337.067

891.2001

383.333

821.1334

129.6670

959.4000

0

16.1104

1460.070

142.8280

1340.392

259.5290

1122.318

323.567

890.5505

369.112

820.3072

117.0390

960.8488

1

5.8344

1460.187

132.3700

1339.994

249.0580

1121.832

313.184

889.9189

358.828

819.6309

105.7240

960.9595

2

5.5109

1460.400

120.8120

1339.625

237.4180

1121.333

301.533

889.244

347.240

818.8842

93.2708

961.1825

3

17.5563

1460.741

108.5230

1339.332

224.9500

1120.857

288.895

888.5603

334.605

818.0917

80.1029

961.5585

4

30.0691

1461.238

95.7353

1339.154

211.8550

1120.43

275.419

887.8932

321.052

817.2701

66.4896

962.1211

5

42.8924

1461.910

82.6015

1339.121

198.2620

1120.073

261.185

887.2629

306.650

816.4315

52.6116

962.8947

6

55.9139

1462.766

69.2294

1339.257

184.2630

1119.802

246.242

886.6854

291.434

815.5863

38.5936

963.8934

7

69.0503

1463.812

55.6970

1339.575

169.9210

1119.628

230.618

886.1729

275.426

814.7437

24.5215

965.1221

8

82.2388

1465.042

42.0612

1340.085

155.2870

1119.557

214.338

885.7347

258.640

813.9131

10.4521

966.5772

9

95.4311

1466.451

28.3636

1340.787

140.4000

1119.594

197.423

885.3776

241.090

813.1049

3.5790

968.2484

10

108.5901

1468.027

14.6346

1341.677

125.2930

1119.737

179.898

885.1063

222.789

812.3306

17.5514

970.1204

11

121.6875

1469.759

0.8958

1342.746

109.9920

1119.984

161.786

884.9250

203.753

811.6043

31.4545

972.1739

12

134.7022

1471.634

12.8372

1343.984

94.5150

1120.33

143.116

884.8385

184.000

810.9424

45.2829

974.3861

13

147.6191

1473.641

26.5539

1345.378

78.8790

1120.769

123.916

884.8523

163.550

810.3641

59.0335

976.7316

14

160.4287

1475.767

40.2467

1346.915

63.0930

1121.295

104.216

884.9739

142.428

809.8916

72.7026

979.1822

15

173.1259

1478.001

53.9102

1348.583

47.1645

1121.902

84.044

885.2129

120.660

809.5491

86.2856

981.7078

16

185.7099

1480.331

67.5403

1350.371

31.0984

1122.585

63.431

885.5804

98.278

809.3627

99.7773

984.2767

17

198.1837

1482.743

81.1334

1352.268

14.8980

1123.338

42.404

886.0891

75.315

809.3590

113.1725

986.8570

18

210.5535

1485.224

94.6867

1354.265

1.4333

1124.158

20.994

886.7517

51.808

809.5640

126.4673

989.4174

19

222.8278

1487.757

108.1977

1356.350

17.8911

1125.040

0.769

887.5803

27.798

810.0020

139.6598

991.9286

20

235.0169

1490.325

121.6645

1358.513

34.4694

1125.979

22.855

888.5847

3.327

810.6936

152.7498

994.3636

21

247.1322

1492.907

135.0860

1360.738

51.1596

1126.972

45.232

889.7718

21.557

811.6552

165.7388

996.6990

22

259.1851

1495.481

148.4623

1363.009

67.9504

1128.010

67.868

891.1447

46.811

812.8972

178.6293

998.9143

23

271.1866

1498.022

161.7945

1365.305

84.8281

1129.083

90.730

892.7023

72.384

814.4236

191.4245

1000.9920

24

283.1469

1500.506

175.0846

1367.602

101.7767

1130.179

113.786

894.4395

98.228

816.2318

204.1278

1002.9180

25

295.0749

1502.905

188.3354

1369.872

118.7793

1131.283

137.003

896.3469

124.296

818.3120

216.7429

1004.6800

26

306.9781

1505.195

201.5501

1372.084

135.8182

1132.374

160.351

898.4120

150.543

820.6483

229.2738

1006.2670

27

318.8635

1507.350

214.7321

1374.207

152.8768

1133.433

183.801

900.6194

176.924

823.2191

241.7250

1007.6700

28

330.7369

1509.343

227.8844

1376.207

169.9397

1134.433

207.327

902.9514

203.398

825.9987

254.1017

1008.8810

29

342.6039

1511.150

241.0096

1378.052

186.9933

1135.350

230.903

905.3885

229.928

828.9584

266.4099

1009.8910

30

354.4694

1512.748

254.1096

1379.707

204.0269

1136.158

254.505

907.9105

256.478

832.0680

278.6570

1010.6930

31

366.3380

1514.117

267.1859

1381.145

221.0321

1136.831

278.110

910.4961

283.017

835.2977

290.8521

1011.2800

32

378.2137

1515.238

280.2396

1382.337

238.0038

1137.344

301.699

913.1249

309.516

838.6193

303.0058

1011.6470

33

390.1003

1516.097

293.2721

1383.259

254.9391

1137.677

325.249

915.7772

335.949

842.0071

315.1308

1011.7900

34

402.0008

1516.683

306.2855

1383.893

271.8378

1137.810

348.740

918.4351

362.290

845.4393

327.2408

1011.7050

35

413.9182

1516.986

319.2833

1384.222

288.7014

1137.729

372.151

921.0827

388.514

848.8983

339.3495

1011.3910

36

425.8549

1517.005

332.2706

1384.240

305.5328

1137.424

395.460

923.7070

414.599

852.3707

351.4696

1010.8470

37

437.8128

1516.737

345.2549

1383.941

322.3352

1136.889

418.643

926.2978

440.517

855.8474

363.6112

1010.0730

38

449.7935

1516.188

358.2454

1383.330

339.1115

1136.124

441.675

928.8475

466.242

859.3229

375.7819

1009.0720

39

461.7983

1515.364

371.2537

1382.416

355.8636

1135.134

464.529

931.3510

491.746

862.7949

387.9873

1007.8460

40

473.8288

1514.277

384.2926

1381.215

372.5915

1133.928

487.175

933.8054

516.996

866.2632

400.2315

1006.4040

41

485.8867

1512.942

397.3759

1379.749

389.2927

1132.521

509.583

936.2089

541.957

869.7286

412.5176

1004.7530

42

497.9743

1511.377

410.5173

1378.045

405.9619

1130.932

531.722

938.5610

566.592

873.1921

424.8483

1002.9080

43

510.0948

1509.607

423.7298

1376.137

422.5906

1129.184

553.559

940.8610

590.859

876.6538

437.2262

1000.8850

 

Frame Number

GT x

GT y

Knee x

Knee Y

Ankle x

Ankle y

R Heel x

R Heel Y

Met x

Met y

R Toe x

R Toe Y

Stride Events

–1

–151.9330

815.6667

60.0000

499.2000

179.6667

122.5333

137.0667

32.5333

318.8667

107.8000

389.6000

174.6000

0

–138.8750

817.3138

71.2735

500.1740

184.3008

122.2974

139.1965

33.7988

321.1163

101.7447

393.5727

166.1098

RTD

1

–127.3660

817.3990

81.5479

499.6656

188.6009

121.3638

140.9553

35.7224

324.5082

97.1523

399.1105

158.2565

2

–114.6950

817.6133

92.8275

499.2010

193.2326

120.3715

142.8490

37.7833

328.1142

92.1631

404.9800

149.7856

3

–101.2980

818.0074

104.7070

498.8362

197.9802

119.3710

144.7892

39.8826

331.7479

86.9928

410.8690

141.0844

4

–87.4525

818.6227

116.9140

498.6061

202.6871

118.3970

146.7124

41.9504

335.2787

81.8026

416.5578

132.4334

5

–73.3523

819.4881

129.2452

498.5237

207.2340

117.4741

148.5699

43.9356

338.6140

76.7251

421.8916

124.0495

6

–59.1365

820.6188

141.5343

498.5822

211.5301

116.6186

150.3248

45.8008

341.6912

71.8718

426.7681

116.1024

7

–44.9065

822.0160

153.6362

498.7572

215.5108

115.8410

151.9499

47.5205

344.4730

67.3340

431.1295

108.7213

8

–30.7351

823.6689

165.4220

499.0134

219.1348

115.1469

153.4265

49.0797

346.9431

63.1805

434.9549

101.9964

9

–16.6723

825.5570

176.7789

499.3102

222.3823

114.5388

154.7436

50.4725

349.1014

59.4569

438.2521

95.9820

10

–2.7509

827.6524

187.6139

499.6091

225.2517

114.0165

155.8960

51.7004

350.9600

56.1862

441.0497

90.6994

11

11.0093

829.9228

197.8582

499.8779

227.7565

113.5791

156.8838

52.7708

352.5393

53.3706

443.3898

86.1411

12

24.5970

832.3331

207.4701

500.0946

229.9220

113.2248

157.7112

53.6945

353.8654

50.9957

445.3221

82.2754

13

38.0060

834.8476

216.4356

500.2475

231.7811

112.9514

158.3862

54.4842

354.9672

49.0342

446.8991

79.0518

14

51.2330

837.4310

224.7652

500.3347

233.3704

112.7559

158.9206

55.1539

355.8746

47.4500

448.1728

76.4052

15

64.2766

840.0502

232.4894

500.3617

234.7272

112.6339

159.3292

55.7181

356.6170

46.2022

449.1921

74.2612

16

77.1368

842.6746

239.6519

500.3383

235.8862

112.5799

159.6301

56.1915

357.2221

45.2480

450.0017

72.5414

17

89.8161

845.2775

246.3037

500.2760

236.8787

112.5868

159.8431

56.5886

357.7151

44.5457

450.6412

71.1686

18

102.3196

847.8358

252.4979

500.1857

237.7323

112.6466

159.9886

56.9235

358.1183

44.0552

451.1447

70.0707

19

114.6559

850.3299

258.2861

500.0763

238.4710

112.7506

160.0858

57.2092

358.4507

43.7393

451.5414

69.1836

20

126.8370

852.7439

263.7166

499.9540

239.1157

112.8897

160.1514

57.4574

358.7282

43.5629

451.8558

68.4523

21

138.8776

855.0646

268.8339

499.8230

239.6849

113.0557

160.1989

57.6785

358.9634

43.4934

452.1079

67.8310

22

150.7940

857.2812

273.6783

499.6856

240.1949

113.2410

160.2386

57.8810

359.1665

43.5010

452.3141

67.2823

23

162.6026

859.3842

278.2859

499.5426

240.6593

113.4397

160.2780

58.0722

359.3451

43.5592

452.4875

66.7770

24

174.3181

861.3651

282.6890

499.3944

241.0892

113.6478

160.3224

58.2579

359.5049

43.6464

452.6383

66.2925

25

185.9522

863.2155

286.9158

499.2408

241.4929

113.8631

160.3755

58.4427

359.6502

43.7458

452.7746

65.8131

26

197.5134

864.9266

290.9908

499.0815

241.8764

114.0852

160.4402

58.6309

359.7843

43.8463

452.9024

65.3286

27

209.0071

866.4890

294.9362

498.9164

242.2435

114.3149

160.5187

58.8257

359.9092

43.9411

453.0261

64.8349

28

220.4373

867.8925

298.7722

498.7453

242.5968

114.5535

160.6126

59.0307

360.0264

44.0271

453.1489

64.3328

29

231.8087

869.1266

302.5183

498.5675

242.9386

114.8020

160.7236

59.2494

360.1368

44.1037

453.2730

63.8270

30

243.1287

870.1817

306.1950

498.3820

243.2710

115.0613

160.8529

59.4857

360.2404

44.1718

453.3998

63.3243

31

254.4073

871.0497

309.8237

498.1867

243.5966

115.3314

161.0014

59.7442

360.3372

44.2329

453.5304

62.8319

32

265.6574

871.7253

313.4281

497.9788

243.9186

115.6124

161.1695

60.0295

360.4265

44.2890

453.6651

62.3560

33

276.8928

872.2063

317.0341

497.7539

244.2404

115.9045

161.3575

60.3465

360.5074

44.3418

453.8039

61.9004

34

288.1276

872.4932

320.6698

497.5071

244.5661

116.2084

161.5659

60.7002

360.5789

44.3932

453.9465

61.4670

35

299.3745

872.5881

324.3649

497.2322

244.8999

116.5255

161.7948

61.0952

360.6402

44.4445

454.0924

61.0557

36

310.6444

872.4946

328.1502

496.9222

245.2467

116.8578

162.0449

61.5363

360.6906

44.4970

454.2406

60.6650

37

321.9457

872.2165

332.0572

496.5696

245.6118

117.2078

162.3167

62.0285

360.7300

44.5517

454.3904

60.2924

38

333.2847

871.7587

336.1172

496.1665

246.0014

117.5787

162.6114

62.5774

360.7588

44.6091

454.5409

59.9354

39

344.6653

871.1267

340.3614

495.7055

246.4221

117.9745

162.9301

63.1895

360.7782

44.6698

454.6915

59.5913

40

356.0901

870.3273

344.8206

495.1796

246.8816

118.4002

163.2748

63.8730

360.7903

44.7337

454.8418

59.2576

41

367.5610

869.3682

349.5252

494.5830

247.3890

118.8621

163.6482

64.6386

360.7982

44.8007

454.9916

58.9322

42

379.0804

868.2585

354.5053

493.9119

247.9545

119.3685

164.0544

65.4995

360.8056

44.8704

455.1414

58.6129

43

390.6524

867.0091

359.7906

493.1643

248.5902

119.9296

164.4993

66.4728

360.8171

44.9424

455.2919

58.2975

Frame Number

Head x

Head y

Shoulder x

Shoulder y

Elbow x

Elbow y

Wrist x

Wrist y

Hand x

Hand y

IC x

IC y

44

522.2518

1507.657

437.0246

1374.059

439.1667

1127.303

575.060

943.1083

614.715

880.1120

449.6549

998.7060

45

534.4501

1505.557

450.4108

1371.850

455.6750

1125.319

596.194

945.3013

638.117

883.5624

462.1389

996.3943

46

546.6946

1503.340

463.8944

1369.549

472.0967

1123.261

616.927

947.4374

661.022

886.9977

474.6841

993.9764

47

558.9905

1501.042

477.4785

1367.196

488.4105

1121.160

637.231

949.5129

683.390

890.4071

487.2987

991.4807

48

571.3426

1498.698

491.1625

1364.831

504.5930

1119.047

657.077

951.5224

705.182

893.7759

499.993

988.9370

49

583.7549

1496.348

504.9425

1362.491

520.6193

1116.949

676.440

953.4589

726.364

897.0853

512.7801

986.3754

50

596.2307

1494.029

518.8112

1360.212

536.4642

1114.896

695.298

955.3139

746.906

900.3132

525.6751

983.8261

51

608.7714

1491.778

532.7583

1358.027

552.1032

1112.913

713.631

957.0778

766.783

903.4340

538.6957

981.3188

52

621.3773

1489.633

546.7705

1355.967

567.5132

1111.025

731.424

958.7397

785.976

906.4198

551.8615

978.8828

53

634.0468

1487.627

560.8318

1354.057

582.6737

1109.255

748.666

960.2883

804.471

909.2411

565.1934

976.5466

54

646.7767

1485.791

574.9236

1352.324

597.5668

1107.623

765.348

961.7119

822.262

911.8676

578.7122

974.3387

55

659.5620

1484.151

589.0253

1350.788

612.1782

1106.150

781.463

962.9990

839.345

914.2688

592.4363

972.2856

56

672.3964

1482.731

603.1137

1349.467

626.4967

1104.854

797.011

964.1389

855.722

916.4159

606.3795

970.4116

57

685.2723

1481.550

617.1643

1348.376

640.5142

1103.749

811.991

965.1218

871.399

918.2821

620.5474

968.7373

58

698.1809

1480.623

631.1516

1347.527

654.2255

1102.850

826.407

965.9398

886.383

919.8438

634.9364

967.2792

59

711.1130

1479.962

645.0503

1346.926

667.6275

1102.164

840.264

966.5865

900.683

921.0807

649.5323

966.0490

60

724.0594

1479.574

658.8367

1346.579

680.7194

1101.697

853.568

967.0580

914.311

921.9768

664.3104

965.0532

61

737.0114

1479.463

672.4901

1346.484

693.5017

1101.452

866.329

967.3524

927.277

922.5203

679.2368

964.2941

62

749.9620

1479.627

685.9935

1346.638

705.9764

1101.426

878.554

967.4700

939.592

922.7038

694.2704

963.7690

63

762.9059

1480.060

699.3351

1347.032

718.1463

1101.611

890.255

967.4127

951.268

922.5244

709.3649

963.4708

64

775.8406

1480.750

712.5073

1347.654

730.0155

1101.999

901.441

967.1836

962.314

921.9839

724.4703

963.3876

65

788.7655

1481.681

725.5068

1348.486

741.5892

1102.574

912.121

966.7866

972.741

921.0883

739.5354

963.5039

66

801.6824

1482.833

738.3333

1349.510

752.8741

1103.321

922.308

966.2256

982.556

919.8477

754.5085

963.8010

67

814.5944

1484.184

750.9880

1350.703

763.8783

1104.220

932.011

965.5046

991.770

918.2762

769.3395

964.2584

68

827.5048

1485.710

763.4733

1352.042

774.6117

1105.254

941.240

964.6271

1000.388

916.3914

783.9819

964.8554

69

840.4169

1487.389

775.7909

1353.502

785.0856

1106.401

950.004

963.5969

1008.417

914.2144

798.3947

965.5728

70

853.3326

1489.199

787.9426

1355.059

795.3122

1107.645

958.312

962.4177

1015.865

911.7689

812.5442

966.3945

71

866.2526

1491.119

799.9299

1356.693

805.3045

1108.967

966.174

961.0942

1022.737

909.0815

826.4058

967.3087

72

879.1757

1493.128

811.7548

1358.382

815.0755

1110.353

973.597

959.6320

1029.043

906.1808

839.9639

968.3082

73

892.0995

1495.209

823.4208

1360.107

824.6380

1111.790

980.589

958.0385

1034.791

903.0973

853.2113

969.3906

74

905.0204

1497.344

834.9334

1361.852

834.0043

1113.264

987.157

956.3222

1039.989

899.8622

866.1494

970.5576

75

917.9341

1499.516

846.3014

1363.601

843.1862

1114.767

993.311

954.4936

1044.651

896.5072

878.7880

971.8138

76

930.8362

1501.708

857.5371

1365.343

852.1953

1116.289

999.058

952.5641

1048.789

893.0635

891.1464

973.1654

77

943.7226

1503.906

868.6569

1367.064

861.0429

1117.822

1004.410

950.5459

1052.419

889.5609

903.2537

974.6181

78

956.5902

1506.093

879.6808

1368.753

869.7412

1119.360

1009.377

948.4516

1055.557

886.0268

915.1477

976.1752

79

969.4370

1508.252

890.6320

1370.400

878.3033

1120.899

1013.972

946.2936

1058.223

882.4857

926.8735

977.8356

80

982.2618

1510.366

901.5361

1371.995

886.7441

1122.431

1018.210

944.0840

1060.436

878.9586

938.4812

979.5927

81

995.0647

1512.416

912.4197

1373.526

895.0804

1123.954

1022.109

941.8336

1062.220

875.4634

950.0233

981.4336

82

1007.8470

1514.380

923.3101

1374.984

903.3311

1125.462

1025.690

939.5526

1063.596

872.0145

961.5525

983.3398

83

1020.6090

1516.238

934.2332

1376.355

911.5175

1126.948

1028.974

937.2496

1064.591

868.6240

973.1191

985.2875

84

1033.3530

1517.966

945.2139

1377.627

919.6627

1128.403

1031.987

934.9323

1065.230

865.3018

984.7677

987.2489

85

1046.0810

1519.540

956.2747

1378.785

927.7914

1129.817

1034.758

932.6072

1065.541

862.0560

996.5345

989.1932

86

1058.7940

1520.939

967.4357

1379.812

935.9296

1131.176

1037.315

930.2805

1065.556

858.8937

1008.4450

991.0879

87

1071.4940

1522.141

978.7147

1380.693

944.1037

1132.463

1039.691

927.9574

1065.307

855.8212

1020.5140

992.8999

88

1084.1810

1523.124

990.1266

1381.411

952.3407

1133.663

1041.920

925.6434

1064.830

852.8441

1032.7460

994.5969

89

1096.8570

1523.870

1001.6840

1381.949

960.6672

1134.758

1044.036

923.3441

1064.163

849.9679

1045.1340

996.1484

90

1109.5220

1524.363

1013.3960

1382.293

969.1091

1135.730

1046.076

921.0654

1063.349

847.1981

1057.6670

997.5264

 

Frame Number

GT x

GT y

Knee x

Knee y

Ankle x

Ankle y

R Heel x

R Heel y

Met x

Met y

R Toe x

R Toe y

Stride Events

44

402.2833

865.6323

365.4101

492.3406

249.3104

120.5581

164.9916

67.5794

360.8379

45.0161

455.4447

57.9836

 

45

413.9826

864.1425

371.3921

491.4435

250.1318

121.2689

165.5426

68.8445

360.8736

45.0907

455.6017

57.6685

46

425.7624

862.5563

377.7643

490.4779

251.0741

122.0795

166.1676

70.2976

360.9301

45.1652

455.7655

57.3491

47

437.6376

860.8923

384.5530

489.4505

252.1602

123.0099

166.8858

71.9725

361.0138

45.2383

455.9388

57.0217

48

449.6248

859.1707

391.7842

488.3699

253.4165

124.0820

167.7211

73.9066

361.1312

45.3084

456.1251

56.6824

49

461.7424

857.4121

399.4829

487.2455

254.8731

125.3197

168.7025

76.1405

361.2889

45.3743

456.3279

56.3267

50

474.0093

855.6371

407.6740

486.0878

256.5635

126.7479

169.8648

78.7167

361.4933

45.4355

456.5518

55.9497

51

486.4451

853.8651

416.3824

484.9072

258.5252

128.3919

171.2490

81.6790

361.7507

45.4938

456.8016

55.5460

52

499.0691

852.1139

425.6334

483.7140

260.7993

130.2768

172.9024

85.0714

362.0677

45.5543

457.0836

55.1095

53

511.9001

850.3997

435.4529

482.5174

263.4308

132.4263

174.8798

88.9365

362.4523

45.6266

457.4059

54.6334

54

524.9558

848.7370

445.8680

481.3257

266.4682

134.8621

177.2433

93.3147

362.9149

45.7260

457.7787

54.1101

55

538.2520

847.1390

456.9072

480.1456

269.9636

137.6026

180.0630

98.2425

363.4703

45.8735

458.2149

53.5311

56

551.8018

845.6168

468.6003

478.9818

273.9725

140.6631

183.4173

103.7511

364.1388

46.0956

458.7312

52.8864

57

565.6142

844.1796

480.9787

477.8369

278.5537

144.0548

187.3924

109.8648

364.9479

46.4237

459.3485

52.1647

58

579.6932

842.8347

494.0746

476.7118

283.7696

147.7851

192.0818

116.5996

365.9339

46.8940

460.0931

51.3535

59

594.0364

841.5867

507.9210

475.6049

289.6859

151.8575

197.5852

123.9614

367.1434

47.5465

460.9979

50.4391

LTD

60

608.6337

840.4378

522.5499

474.5135

296.3716

156.2711

204.0059

131.9451

368.6348

48.4256

462.1044

49.4077

61

623.4672

839.3875

537.9915

473.4333

303.8978

161.0196

211.4477

140.5324

370.4799

49.5791

463.4636

48.2472

62

638.5101

838.4326

554.2720

472.3594

312.3368

166.0903

220.0112

149.6902

372.7646

51.0562

465.1384

46.9498

63

653.7275

837.5678

571.4115

471.2875

321.7593

171.4627

229.7893

159.3682

375.5898

52.9052

467.2046

45.5158

64

669.0768

836.7864

589.4216

470.2143

332.2322

177.1082

240.8623

169.4967

379.0706

55.1688

469.7528

43.9582

65

684.5092

836.0806

608.3036

469.1397

343.8152

182.9895

253.2933

179.9845

383.3353

57.8796

472.8908

42.3077

66

699.9719

835.4432

628.0463

468.0684

356.5576

189.0610

267.1239

190.7183

388.5238

61.0557

476.7449

40.6152

67

715.4105

834.8681

648.6252

467.0117

370.4949

195.2689

282.3695

201.5629

394.7855

64.6967

481.4629

38.9541

68

730.7715

834.3520

670.0022

465.9895

385.6464

201.5527

299.0160

212.3643

402.2761

68.7818

487.2153

37.4164

69

746.0055

833.8949

692.1253

465.0319

402.0134

207.8452

317.0172

222.9540

411.1545

73.2680

494.1952

36.1063

70

761.0693

833.5012

714.9302

464.1787

419.5800

214.0742

336.2964

233.1561

421.5793

78.0903

502.6152

35.1289

71

775.9278

833.1797

738.3418

463.4794

438.3152

220.1630

356.7498

242.7956

433.7019

83.1627

512.6991

34.5766

72

790.5556

832.9440

762.2759

462.9903

458.1768

226.0319

378.2556

251.7072

447.6596

88.3795

524.6682

34.5153

73

804.9377

832.8115

786.6420

462.7713

479.1165

231.5979

400.6838

259.7432

463.5649

93.6183

538.7225

34.9722

74

819.0697

832.8028

811.3455

462.8813

501.0830

236.7762

423.9086

266.7791

481.4957

98.7442

555.0212

35.9293

75

832.9578

832.9400

836.2902

463.3738

524.0245

241.4812

447.8179

272.7165

501.4864

103.6165

573.6661

37.3241

76

846.6178

833.2444

861.3812

464.2920

547.8890

245.6286

472.3199

277.4831

523.5240

108.0956

594.6932

39.0584

RTO

77

860.0739

833.7345

886.5270

465.6647

572.6235

249.1396

497.3450

281.0300

547.5494

112.0514

618.0732

41.0124

78

873.3571

834.4230

911.6413

467.5040

598.1741

251.9441

522.8448

283.3286

573.4650

115.3715

643.7214

43.0613

79

886.5038

835.3152

936.6445

469.8040

624.4861

253.9848

548.7892

284.3665

601.1465

117.9675

671.5116

45.0897

80

899.5530

836.4076

961.4648

472.5408

651.5061

255.2193

575.1624

284.1450

630.4558

119.7792

701.2915

47.0030

81

912.5449

837.6876

986.0383

475.6746

679.1844

255.6219

601.9611

282.6769

661.2527

120.7769

732.8957

48.7337

82

925.5189

839.1341

1010.3090

479.1515

707.4776

255.1832

629.1921

279.9857

693.4044

120.9621

766.1557

50.2438

83

938.5117

840.7193

1034.2300

482.9071

736.3500

253.9097

656.8714

276.1059

726.7898

120.3650

800.9062

51.5241

84

951.5555

842.4100

1057.7600

486.8694

765.7741

251.8223

685.0226

271.0822

761.3018

119.0424

836.9882

52.5912

85

964.6761

844.1692

1080.8670

490.9622

795.7294

248.9548

713.6750

264.9703

796.8452

117.0729

874.2502

53.4839

86

977.8912

845.9573

1103.5240

495.1083

826.2008

245.3527

742.8620

257.8366

833.3345

114.5524

912.5479

54.2587

87

991.2104

847.7333

1125.7100

499.2321

857.1769

241.0716

772.6188

249.7584

870.6895

111.5892

951.7430

54.9853

88

1004.6350

849.4551

1147.4090

503.2616

888.6476

236.1757

802.9806

240.8236

908.8328

108.2993

991.7024

55.7423

89

1018.1610

851.0812

1168.6090

507.1302

920.6024

230.7369

833.9800

231.1305

947.6873

104.8021

1032.2970

56.6135

90

1031.7800

852.5717

1189.3010

510.7773

953.0286

224.8342

865.6452

220.7872

987.1745

101.2173

1073.4010

57.6847

Frame Number

Head x

Head y

Shoulder x

Shoulder y

Elbow x

Elbow y

Wrist x

Wrist y

Hand x

Hand y

IC x

IC y

91

1122.1780

1524.589

1025.2720

1382.429

977.6909

1136.564

1048.081

918.8136

1062.432

844.5402

1070.3300

998.7059

92

1134.8250

1524.536

1037.3170

1382.347

986.4352

1137.245

1050.090

916.5961

1061.459

842.0001

1083.1050

999.6655

93

1147.4650

1524.197

1049.5320

1382.041

995.3621

1137.763

1052.148

914.4208

1060.479

839.5836

1095.9770

1000.3870

94

1160.1020

1523.572

1061.9210

1381.507

1004.4890

1138.108

1054.301

912.2968

1059.545

837.2960

1108.9330

1000.8560

95

1172.7400

1522.661

1074.4800

1380.745

1013.8300

1138.275

1056.598

910.2335

1058.710

835.1422

1121.9610

1001.0630

96

1185.3860

1521.474

1087.2070

1379.759

1023.3960

1138.259

1059.087

908.2402

1058.030

833.1259

1135.0540

1000.9990

97

1198.0500

1520.024

1100.0940

1378.556

1033.1920

1138.063

1061.813

906.3257

1057.559

831.2495

1148.2070

1000.6620

98

1210.7410

1518.332

1113.1330

1377.149

1043.2210

1137.688

1064.820

904.4973

1057.355

829.5134

1161.4160

1000.0520

99

1223.4720

1516.420

1126.3120

1375.550

1053.4790

1137.144

1068.144

902.7599

1057.472

827.9161

1174.6800

999.1736

100

1236.2490

1514.314

1139.6170

1373.776

1063.9570

1136.441

1071.818

901.1171

1057.963

826.4537

1187.9990

998.0358

101

1249.0780

1512.043

1153.0310

1371.846

1074.6410

1135.594

1075.871

899.5702

1058.880

825.1201

1201.3720

996.6514

102

1261.9590

1509.631

1166.5340

1369.781

1085.5120

1134.623

1080.325

898.1190

1060.268

823.9072

1214.7950

995.0374

103

1274.8870

1507.109

1180.1040

1367.603

1096.5420

1133.550

1085.200

896.7623

1062.169

822.8048

1228.2650

993.2150

104

1287.8500

1504.502

1193.7160

1365.336

1107.7010

1132.399

1090.508

895.4979

1064.614

821.8015

1241.7710

991.2089

105

1300.8320

1501.839

1207.3390

1363.007

1118.9540

1131.197

1096.251

894.3228

1067.629

820.8849

1255.2970

989.0475

106

1313.8080

1499.151

1220.9350

1360.641

1130.2590

1129.969

1102.426

893.2339

1071.225

820.0424

1268.8180

986.7625

107

1326.7450

1496.467

1234.4590

1358.266

1141.5710

1128.740

1109.012

892.2280

1075.401

819.2618

1282.3000

984.3880

108

1339.5990

1493.820

1247.8540

1355.909

1152.8390

1127.533

1115.977

891.3019

1080.135

818.5322

1295.6950

981.9608

109

1352.3100

1491.240

1261.0510

1353.599

1164.0020

1126.369

1123.268

890.4531

1085.389

817.8444

1308.9420

979.5194

110

1364.8020

1488.760

1273.9640

1351.363

1174.9900

1125.264

1130.814

889.6794

1091.097

817.1916

1321.9620

977.1038

111

1376.9790

1486.412

1286.4910

1349.228

1185.7180

1124.232

1138.519

888.9791

1097.170

816.5699

1334.6530

974.7546

112

1388.7200

1484.224

1298.5100

1347.222

1196.0800

1123.284

1146.261

888.3509

1103.487

815.9788

1346.8920

972.5130

113

1399.8780

1482.223

1309.8780

1345.372

1205.9500

1122.428

1153.894

887.7934

1109.898

815.4207

1358.5280

970.4190

114

1410.2790

1480.435

1320.4330

1343.704

1215.1790

1121.670

1161.243

887.3054

1116.221

814.9016

1369.3830

968.5109

115

1419.7300

1478.880

1329.9900

1342.240

1223.5910

1121.013

1168.110

886.8860

1122.249

814.4294

1379.2550

966.8234

116

1428.0220

1477.574

1338.3540

1341.002

1231.0000

1120.463

1174.283

886.5345

1127.759

814.0141

1387.9280

965.3857

117

1434.9580

1476.527

1345.3370

1340.002

1237.2200

1120.020

1179.554

886.2509

1132.526

813.6654

1395.1920

964.2192

118

1440.3830

1475.740

1350.7930

1339.246

1242.1030

1119.685

1183.750

886.0347

1136.362

813.3915

1400.8820

963.3334

119

1444.2530

1475.199

1354.6820

1338.724

1245.6000

1119.453

1186.789

885.8832

1139.165

813.1953

1404.9460

962.7195

120

1446.7480

1474.865

1357.1890

1338.399

1247.8630

1119.307

1188.778

885.7864

1141.016

813.0679

1407.5700

962.3362

121

1448.4950

1474.645

1358.9430

1338.183

1249.4580

1119.209

1190.200

885.7190

1142.352

812.9777

1409.4110

962.0807

122

1535.0000

1468.600

1446.2670

1331.267

1332.6670

1115.000

1271.000

881.6667

1222.533

807.7334

1503.2670

954.0667

Frame Number

GT x

GT y

Knee X

Knee y

Ankle x

Ankle Y

R Heel x

R Heel y

Met x

Met y

R Toe x

R Toe y

Stride Events

91

1045.4800

853.8901

1209.4790

514.1490

985.9099

218.5530

897.9977

209.9099

1027.2140

97.6617

1114.8910

59.0417

 

92

1059.2530

855.0042

1229.1370

517.1985

1019.2250

211.9852

931.0504

198.6207

1067.7200

94.2472

1156.6450

60.7684

93

1073.0900

855.8873

1248.2710

519.8859

1052.9450

205.2283

964.8050

187.0457

1108.6050

91.0794

1198.5420

62.9454

94

1086.9820

856.5187

1266.8780

522.1782

1087.0360

198.3838

999.2498

175.3125

1149.7730

88.2566

1240.4600

65.6477

95

1100.9190

856.8826

1284.9580

524.0499

1121.4520

191.5556

1034.3570

163.5488

1191.1220

85.8689

1282.2760

68.9418

96

1114.8900

856.9678

1302.5100

525.4825

1156.1350

184.8474

1070.0800

151.8817

1232.5400

83.9973

1323.8630

72.8823

97

1128.8790

856.7668

1319.5380

526.4650

1191.0160

178.3600

1106.3500

140.4368

1273.9020

82.7122

1365.0880

77.5078

98

1142.8670

856.2758

1336.0510

526.9944

1226.0050

172.1887

1143.0730

129.3379

1315.0730

82.0706

1405.8150

82.8383

99

1156.8330

855.4942

1352.0600

527.0763

1260.9970

166.4202

1180.1300

118.7044

1355.9030

82.1127

1445.9060

88.8720

100

1170.7550

854.4260

1367.5830

526.7247

1295.8660

161.1292

1217.3720

108.6481

1396.2270

82.8590

1485.2190

95.5835

101

1184.6140

853.0799

1382.6470

525.9632

1330.4660

156.3753

1254.6250

99.2689

1435.8710

84.3064

1523.6130

102.9216

102

1198.3950

851.4703

1397.2810

524.8242

1364.6330

152.1993

1291.6890

90.6500

1474.6500

86.4260

1560.9450

110.8074

103

1212.0860

849.6176

1411.5220

523.3494

1398.1890

148.6200

1328.3410

82.8528

1512.3730

89.1621

1597.0790

119.1344

104

1225.6780

847.5482

1425.4110

521.5883

1430.9430

145.6326

1364.3410

75.9138

1548.8460

92.4315

1631.8780

127.7686

105

1239.1620

845.2944

1438.9880

519.5977

1462.7010

143.2082

1399.4320

69.8418

1583.8800

96.1250

1665.2140

136.5525

106

1252.5260

842.8939

1452.2900

517.4393

1493.2680

141.2944

1433.3560

64.6175

1617.2900

100.1100

1696.9680

145.3103

107

1265.7520

840.3885

1465.3500

515.1776

1522.4550

139.8191

1465.8530

60.1938

1648.9040

104.2365

1727.0330

153.8557

108

1278.8090

837.8237

1478.1860

512.8766

1550.0890

138.6949

1496.6760

56.4992

1678.5610

108.3453

1755.3130

162.0012

109

1291.6520

835.2469

1490.7990

510.5973

1576.0100

137.8264

1525.5960

53.4431

1706.1190

112.2787

1781.7270

169.5697

110

1304.2180

832.7062

1503.1670

508.3945

1600.0780

137.1185

1552.4090

50.9230

1731.4590

115.8918

1806.2040

176.4070

111

1316.4240

830.2491

1515.2380

506.3149

1622.1730

136.4850

1576.9420

48.8343

1754.4840

119.0627

1828.6820

182.3930

112

1328.1590

827.9207

1526.9250

504.3956

1642.1950

135.8571

1599.0540

47.0793

1775.1250

121.7017

1849.1040

187.4518

113

1339.2880

825.7624

1538.1000

502.6643

1660.0600

135.1893

1618.6440

45.5762

1793.3400

123.7588

1867.4180

191.5581

114

1349.6500

823.8110

1548.5950

501.1398

1675.7030

134.4637

1635.6440

44.2660

1809.1160

125.2282

1883.5680

194.7383

115

1359.0580

822.0975

1558.2070

499.8333

1689.0760

133.6909

1650.0260

43.1152

1822.4670

126.1492

1897.5000

197.0671

116

1367.3130

820.6466

1566.7050

498.7500

1700.1520

132.9067

1661.8020

42.1148

1833.4310

126.6039

1909.1600

198.6590

117

1374.2220

819.4747

1573.8630

497.8898

1708.9330

132.1643

1671.0280

41.2741

1842.0700

126.7078

1918.5100

199.6550

118

1379.6320

818.5875

1579.4960

497.2469

1715.4760

131.5218

1677.8230

40.6102

1848.4840

126.5941

1925.5590

200.2064

119

1383.4960

817.9734

1583.5340

496.8057

1719.9390

131.0243

1682.4070

40.1330

1852.8560

126.3889

1930.4270

200.4570

120

1385.9930

817.5894

1586.1500

496.5313

1722.6820

130.6790

1685.1930

39.8256

1855.5500

126.1770

1933.4640

200.5254

121

1387.7470

817.3316

1587.9920

496.3473

1724.4700

130.4152

1686.9810

39.6145

1857.3240

125.9547

1935.4950

200.4826

RTD

122

1483.7330

808.6667

1679.0000

487.2667

1780.3330

116.5333

1731.2670

38.6666

1917.8670

95.6666

1937.5250

200.4398

P.478

 

TABLE C-2 Kinetic Data in Newtons

Number

Fx

Fy

Fz

Events

1

0

0

0

 

0

1.5647

0.2007

7.04480

RTD

1

1.2618

6.1802

53.34734

2

1.7255

9.5325

111.1378

3

9.4928

4.6627

184.3624

4

21.0518

22.5572

235.4881

5

28.9799

41.3995

289.6832

6

30.0140

54.5544

335.5472

7

26.0155

66.6458

385.7960

8

18.2023

73.6048

427.9769

9

10.7337

82.5751

459.7222

10

6.8501

85.7424

485.8550

11

5.7930

89.8577

499.7984

12

1.3808

89.5803

520.2312

13

5.6510

93.1406

529.8775

14

14.5903

92.2852

545.6625

15

19.1174

89.9271

544.9609

16

25.3220

88.2393

555.5719

17

26.9306

82.4133

549.9595

18

28.0796

76.7952

553.0288

19

25.8505

68.7728

540.5762

20

24.7705

62.7387

533.2099

21

22.6563

54.2539

514.1803

22

21.1166

49.3988

496.4661

23

19.6229

42.0237

474.8933

24

17.8075

38.0703

444.9019

25

17.2790

31.8743

428.5030

26

16.0840

28.5682

397.5470

27

17.3019

23.9212

382.2882

28

15.6474

21.3319

356.4184

29

16.3138

19.1818

341.4227

30

14.5214

17.2860

316.2545

31

14.2226

16.3843

310.0282

32

11.9476

14.4423

290.2970

33

11.2352

14.4885

288.2801

34

9.9023

12.8933

275.9152

35

9.6495

13.1014

275.0382

36

9.5346

10.3964

269.6889

37

9.4657

11.1593

271.7935

38

10.2700

7.5758

273.8105

39

10.7067

7.1134

275.6521

40

12.2693

3.5299

285.8246

41

12.0855

2.9519

288.7185

42

13.0966

0.4550

303.9773

43

13.0047

1.9262

313.3606

44

14.2915

5.1167

336.9503

45

12.2693

9.0932

351.0691

46

13.0966

12.9773

379.3943

47

12.4531

18.4103

396.6701

48

14.4984

22.5949

428.9415

49

14.9120

29.7157

451.6543

50

15.9921

34.6401

482.5227

51

16.8883

43.2636

508.2171

52

18.2211

48.9741

533.8238

53

20.1745

57.4127

558.2904

54

21.4843

63.2157

574.0754

55

23.7823

71.7005

592.9296

56

24.4717

78.2664

600.9098

57

26.5399

85.8033

610.5562

58

24.8624

93.0165

604.5930

59

24.8624

98.9813

604.4176

60

22.2427

104.5069

583.8094

61

20.4043

107.3506

570.0415

62

17.0032

110.4948

535.1392

63

14.2686

110.7029

503.3062

64

9.7185

109.1077

452.5313

65

6.2944

101.1315

404.6502

66

3.8356

91.2132

345.5443

67

3.1002

77.4110

285.2985

68

3.6287

66.4524

228.9111

69

3.0542

51.3323

171.8222

70

4.2952

39.7495

128.9397

71

4.4330

26.8258

88.1619

72

4.6399

18.1560

61.6782

73

2.3648

10.2261

34.0546

74

2.0431

5.3710

25.1975

75

0.0480

2.8972

9.3248

76

1.4916

0.9089

10.99105

RTO

P.479

 

Appendix D: Numerical Example for Calculating Projectile Motion

The equations of constant acceleration all use parameters that are basic to linear kinematics. The three equations of constant acceleration thus provide a useful method of analyzing projectile motion. Suppose a shot putter releases the shot at an angle of 40° from a height of 2.2 m with a velocity of 13.3 m/s. Figure 8-41 illustrates what is known about the conditions of the projectile at the instant of projection and the shape of the trajectory based on our previous discussion.

The problem can be solved in seven steps using the equations of constant acceleration. Remember: To find the range of a projectile, the horizontal velocity and the length of time that the shot was in the air must be known.

Step 1

Calculate the initial vertical and horizontal velocities. (Use Appendix C for sine and cosine values).

Step 2

Use the first equation of constant acceleration to calculate the time required for the projectile to reach the apex of its trajectory. Because it is necessary to calculate the time to the apex, the equation using the vertical velocity must be used. Thus, the acceleration due to gravity, a, will act, bringing the vertical velocity to zero at the apex.

P.480

 

Step 3

Use the third equation of constant acceleration to calculate the height of the apex of the trajectory above the release height. Remember that the height of the trajectory depends on the vertical velocity, the acceleration due to gravity is approximately 9.81 m/s2, and the vertical velocity at the apex is zero.

Step 4

Calculate the total height to the apex of the trajectory.

Step 5

Calculate the time for the projectile to reach the ground from the apex of the trajectory. Because the projectile will land at a level lower than the release point, time for the projectile to reach the apex of the trajectory must be less than time for the projectile to reach the ground from the apex. The time down, tdown, can be calculated using the second equation of constant acceleration. In this equation, the displacement is negative because it measures the distance from the apex to the ground. In addition, the initial vertical velocity of the downward flight is zero.

Because time cannot be negative, each side of this equation must be multiplied by 1, resulting in the following:

Step 6

Calculate the total time the projectile is in the air by adding the time to reach the apex of the trajectory to the time to reach the ground from the apex.

Step 7

Calculate the range of the projectile. tdown= 1.10 s

The distance that the shot traveled under these conditions is 20.1 m. In our initial discussion of projectiles and the equations of constant acceleration, air resistance was considered to be negligible. The mathematics of air resistance are well beyond the scope of this book because they require solving differential equations. However, it may be of interest to know that if these initial conditions were used to solve this problem taking into account air resistance, a reduction of 1.46% in the range of the throw would be found. Considering air resistance, the throw would be 0.3 m less, or a total of 19.8 m.


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