Ping Pong Ball FlightMathematical Modeling Garren Plemmons and Kevin KoteckiEnka High School and Wakefield High School2008

OutlineIntroductionProblemGoalsBackground InformationEquationsCollected DataConclusionBibliographyAcknowledgements

Problem?

How is the flight of a ping pong ball effected through the parameters of drag, lift, initial angle of the ball and velocity of the ball?

Goals Prove the input of drag, lift, angle, and speed to create a different output when changed in different situations.Model the flight of the ball through V-Python and collect informative and testable data.Learn the effects of aerodynamics in spherical objects.

Background InformationEquipment:Ball (40 mm)Hardtop table (9x5)6 netPaddleshttp://salzburgacademyusa.files.wordpress.com/2007/06/l-ping-pong-ball-and-paddle_7506.jpghttp://www.recreationcentral.com/ping-pong/images/200-ping-pong-table.jpg

Background InformationRules:Two to four player. A players must allow a ball played towards them only one bounce on their side of the table Then must return it so that it bounces on the opposite side of the playing surface. A point is scored when the opponent fails to return the ball.

Mathematical Values and Meanings

F = lift force V = velocity of the ball A = cross-sectional area of ball C = lift coefficient a= acceleration = theta

http://farm1.static.flickr.com/189/509787527_cda2ba774c_m.jpg

EquationsNewtons Second Law of MotionF=forceM=massA=acceleration

http://www.union.edu/PUBLIC/PHYDEPT/jonesc/images/Scientific/ping%20pong%20ball%20bouncing.jpg

Mathematical Equationsg=gravity 9.8 meters per secondc= drag coefficientV=velocity of the y componentK=lift coefficient

Mathematical Equationsa = accelerationdt = change in time

Mathematical Equationsa = accelerationdt = change in time

Mathematical Equationsg=gravity 9.8 meters per secondc= drag coefficientV=velocity of the y componentK=lift coefficient

Bernoullis PrincipleDue to the conservation of energy the wing is not able to separate and settle the circulation predicament. This creates a higher pressure are beneath the wing than above it. The pressure difference causes the wing to push upwards and lift is created.

http://www.hiflykites.co.za/kite-online-shop/x-zylo-flying-principles-3.gif

Mechanical and Airfoil lift similaritiesThe overall behavior of Mechanical lift is similar to that around an aerofoil with a circulation which is generated by the mechanical rotation, rather than by aerofoil action. http://upload.wikimedia.org/wikipedia/en/thumb/1/15/Magnus_effect.svg/644px-Magnus_effect.svg.pnghttp://www.furball.warbirdsiii.com/krod/images/airfoil.jpg

Forces Affecting the Flight of a Ping Pong Ball Magnus Force/Effect:http://www.symscape.com/files/images/curveball_1.img_assist_custom.jpgPerpendicular axisHigh Pressure pushes the ballSpin causes high pressure due to the pushed air

Force Diagram

Magnus ForcesRotation axis-the axis in which the ball spinsTopspinBackspinPressureThe low and high pressure areas are on opposite sides of the ball.The Magnus force is described as the low pressure area during the flight of the spinning ball.

Magnus ForcesVelocity in DirectionThe direction in which the ball is travelingPerpendicular to the Magnus force and the rotational axis of the ballEffects how radical the ball flies according to lift, drag, and spin.

Life Like TableBouncing tableBouncing netLimited bouncing

Variable PitchChanged the initial angle onlyStarted at 0 and went to 75 by 15 incrementsAll other settings were at default

Variable Pitch

Chart1

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Sheet6

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Variable DragChanged drag coefficient onlyStarted at 0.0 drag co. and went to 2.0 drag co. Increments of 0.5All other settings at default

Variable Drag

Variable LiftChanged lift coefficient onlyStarted at 0.0 lift co. and went to 2.0 lift co.Increments of 0.5All other settings at default

Variable Lift

Chart1

1.999511.999511.999511.999511.99951

1.997061.9970651.997071.9970751.99708

1.992651.9926741.9926991.9927231.992748

1.986281.9863481.9864171.9864841.986552

1.977951.9780971.9782421.9783871.97853

1.967661.9679281.9681941.9684581.968719

1.955411.9558531.9562921.9567251.957153

1.94121.9418811.9425531.9432151.943869

1.925031.9260211.9269971.9279561.9289

1.90691.9082831.9096411.9109731.91228

1.886811.8886761.8905031.8922921.894042

1.864761.867211.8696021.8719381.874219

1.840751.8438931.8469551.8499371.852842

1.814781.8187361.8225791.8263141.829943

1.786851.7917461.7964921.8010921.805551

1.756961.7629341.7687111.7742961.779697

1.725111.7323091.7392521.7459491.75241

1.69131.6998791.7081331.7160751.723719

1.655531.6656541.6753691.6846961.693651

1.61781.6296421.6409791.6518351.662234

1.578111.5918531.6049771.6175151.629495

1.536461.5522951.567381.5817561.595461

1.492851.5109771.5282041.5445821.560157

1.447281.4679091.4874651.5060121.52361

1.399751.4230981.4451781.4660681.485843

1.350261.3765541.4013591.4247711.446881

1.298811.3282851.3560221.382141.406748

1.24541.27831.3091841.3381971.365468

1.190031.2266071.260861.292961.323063

1.13271.1732151.2110631.2464481.279556

1.073411.1181321.159811.1986821.234968

1.012161.0613671.1071131.149681.189323

0.948951.0029291.0529891.099461.14264

0.883780.9428250.9974511.048041.094941

0.816650.8810640.9405130.995441.046245

0.747560.8176550.8821890.9416750.996573

0.676510.7526050.8224940.8867650.945944

0.60350.6859220.761440.8307250.894378

0.528530.6176160.6990420.7735740.841893

0.45160.5476930.6353130.7153270.788507

0.372710.4761630.5702660.6560010.734239

0.291860.4030320.5039140.5956120.679106

0.209050.328310.4362710.5341770.623126

0.124280.2520040.3673490.471710.566315

0.037550.1741220.2971610.4082280.508689

0.124280.0946710.225720.3437450.450267

0.209050.1737240.1530380.2782770.391062

0.291860.2504250.0791280.2118380.331092

0.372710.3247840.1522990.1444440.27037

0.45160.3968140.2227830.0761070.208913

0.528530.4665260.2906070.1434190.146735

0.60350.5339320.3557980.2077680.08385

0.676510.5990440.4183820.2691990.020272

0.747560.6618720.4783850.327756-0.043984

0.816650.7224290.5358330.383482-0.108905

0.883780.7807250.5907510.43642-0.174478

0.948950.8367730.6431650.486611-0.240689

1.012160.8905820.6930990.534096-0.307527

1.073410.9421650.740580.578916-0.374978

1.13270.9915320.785630.621112-0.44303

1.190031.0386960.8282750.660722-0.511672

1.24541.0836650.8685380.697785-0.58089

1.298811.1264530.9064430.732339

1.350261.1670690.9420150.764423

1.399751.2055240.9752750.794072

1.447281.2418291.0062480.821325

1.492851.2759951.0349560.846216

1.536461.3080331.0614220.86878

1.578111.3379541.0856680.889054

1.61781.3657671.1077160.907071

1.655531.3914831.127589

1.69131.4151131.145308

1.725111.4366681.160895

1.756961.4561571.174371

1.786851.4735921.185757

1.814781.4889811.195074

1.840751.5023361.202343

1.864761.5136671.207583

1.886811.5229831.210817

1.90691.5302961.212063

1.925031.535614

1.94121.538948

1.955411.540308

1.967661.539704

1.977951.537145

1.986281.532641

1.992651.526202

1.997061.517838

1.999511.507558

21.495372

1.998531.48129

1.99511.46532

1.989711.447473

1.982361.427757

1.973051.406182

1.96178

1.94855

1.93336

1.91621

1.8971

1.87603

1.853

1.82801

1.80106

1.77215

1.74128

1.70845

1.67366

1.63691

1.5982

1.55753

1.5149

1.47031

1.42376

1.37525

1.32478

1.27235

1.21796

1.16161

1.1033

1.04303

lift=0

lift=0.5

lift=1.0

lift=1.5

lift=2.0

Distance

Height

Sheet1

4.381.99951

4.261.99706

4.141.99265

4.021.98628

3.91.97795

3.781.96766

3.661.95541

3.541.9412

3.421.92503

3.31.9069

3.181.88681

3.061.86476

2.941.84075

2.821.81478

2.71.78685

2.581.75696

2.461.72511

2.341.6913

2.221.65553

2.11.6178

1.981.57811

1.861.53646

1.741.49285

1.621.44728

1.51.39975

1.381.35026

1.261.29881

1.141.2454

1.021.19003

0.91.1327

0.781.07341

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0.540.94895

0.420.88378

0.30.81665

0.180.74756

0.060.67651

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-0.180.52853

-0.30.4516

-0.420.37271

-0.540.29186

-0.660.20905

-0.780.12428

-0.90.03755

-1.020.12428

-1.140.20905

-1.260.29186

-1.380.37271

-1.50.4516

-1.620.52853

-1.740.6035

-1.860.67651

-1.980.74756

-2.10.81665

-2.220.88378

-2.340.94895

-2.461.01216

-2.581.07341

-2.71.1327

-2.821.19003

-2.941.2454

-3.061.29881

-3.181.35026

-3.31.39975

-3.421.44728

-3.541.49285

-3.661.53646

-3.781.57811

-3.91.6178

-4.021.65553

-4.141.6913

-4.261.72511

-4.381.75696

-4.51.78685

-4.621.81478

-4.741.84075

-4.861.86476

-4.981.88681

-5.11.9069

-5.221.92503

-5.341.9412

-5.461.95541

-5.581.96766

-5.71.97795

-5.821.98628

-5.941.99265

-6.061.99706

-6.181.99951

-6.32

-6.421.99853

-6.541.9951

-6.661.98971

-6.781.98236

-6.91.97305

-7.021.96178

-7.141.94855

-7.261.93336

-7.381.91621

-7.51.8971

-7.621.87603

-7.741.853

-7.861.82801

-7.981.80106

-8.11.77215

-8.221.74128

-8.341.70845

-8.461.67366

-8.581.63691

-8.71.5982

-8.821.55753

-8.941.5149

-9.061.47031

-9.181.42376

-9.31.37525

-9.421.32478

-9.541.27235

-9.661.21796

-9.781.16161

-9.91.1033

-10.021.04303

Sheet1

1.99951

1.99706

1.99265

1.98628

1.97795

1.96766

1.95541

1.9412

1.92503

1.9069

1.88681

1.86476

1.84075

1.81478

1.78685

1.75696

1.72511

1.6913

1.65553

1.6178

1.57811

1.53646

1.49285

1.44728

1.39975

1.35026

1.29881

1.2454

1.19003

1.1327

1.07341

1.01216

0.94895

0.88378

0.81665

0.74756

0.67651

0.6035

0.52853

0.4516

0.37271

0.29186

0.20905

0.12428

0.03755

0.12428

0.20905

0.29186

0.37271

0.4516

0.52853

0.6035

0.67651

0.74756

0.81665

0.88378

0.94895

1.01216

1.07341

1.1327

1.19003

1.2454

1.29881

1.35026

1.39975

1.44728

1.49285

1.53646

1.57811

1.6178

1.65553

1.6913

1.72511

1.75696

1.78685

1.81478

1.84075

1.86476

1.88681

1.9069

1.92503

1.9412

1.95541

1.96766

1.97795

1.98628

1.99265

1.99706

1.99951

2

1.99853

1.9951

1.98971

1.98236

1.97305

1.96178

1.94855

1.93336

1.91621

1.8971

1.87603

1.853

1.82801

1.80106

1.77215

1.74128

1.70845

1.67366

1.63691

1.5982

1.55753

1.5149

1.47031

1.42376

1.37525

1.32478

1.27235

1.21796

1.16161

1.1033

1.04303

Distance

Height

Flight 1-Variable Lift

Sheet2

4.37971.99951

4.2587991.997065

4.1372921.992674

4.0151791.986348

3.8924551.978097

3.7691171.967928

3.6451631.955853

3.5205891.941881

3.3953911.926021

3.2695681.908283

3.1431161.888676

3.0160321.86721

2.8883121.843893

2.7599541.818736

2.6309531.791746

2.5013081.762934

2.3710151.732309

2.240071.699879

2.108471.665654

1.9762121.629642

1.8432931.591853

1.709711.552295

1.5754581.510977

1.4405361.467909

1.3049381.423098

1.1686631.376554

1.0317061.328285

0.8940651.2783

0.7557351.226607

0.6167141.173215

0.4769981.118132

0.3365831.061367

0.1954651.002929

0.0536430.942825

-0.0888890.881064

-0.2321330.817655

-0.3760940.752605

-0.5207750.685922

-0.6661780.617616

-0.8123090.547693

-0.9591710.476163

-1.1067670.403032

-1.2551010.32831

-1.4041760.252004

-1.5539970.174122

-1.7045670.094671

-1.855890.173724

-2.0079690.250425

-2.1608090.324784

-2.3144130.396814

-2.4687850.466526

-2.6239290.533932

-2.7798490.599044

-2.9365480.661872

-3.0940310.722429

-3.2523010.780725

-3.4113620.836773

-3.5712190.890582

-3.7318750.942165

-3.8933350.991532

-4.0556011.038696

-4.2186791.083665

-4.3825731.126453

-4.5472861.167069

-4.7128221.205524

-4.8791861.241829

-5.0463821.275995

-5.2144141.308033

-5.3832861.337954

-5.5530021.365767

-5.7235681.391483

-5.8949851.415113

-6.067261.436668

-6.2403971.456157

-6.4143991.473592

-6.5892711.488981

-6.7650171.502336

-6.9416421.513667

-7.119151.522983

-7.2975461.530296

-7.4768341.535614

-7.6570181.538948

-7.8381031.540308

-8.0200931.539704

-8.2029941.537145

-8.3868091.532641

-8.5715431.526202

-8.7572011.517838

-8.9437871.507558

-9.1313061.495372

-9.3197621.48129

-9.5091611.46532

-9.6995071.447473

-9.8908041.427757

-10.0830581.406182

Sheet2

1.99951

1.997065

1.992674

1.986348

1.978097

1.967928

1.955853

1.941881

1.926021

1.908283

1.888676

1.86721

1.843893

1.818736

1.791746

1.762934

1.732309

1.699879

1.665654

1.629642

1.591853

1.552295

1.510977

1.467909

1.423098

1.376554

1.328285

1.2783

1.226607

1.173215

1.118132

1.061367

1.002929

0.942825

0.881064

0.817655

0.752605

0.685922

0.617616

0.547693

0.476163

0.403032

0.32831

0.252004

0.174122

0.094671

0.173724

0.250425

0.324784

0.396814

0.466526

0.533932

0.599044

0.661872

0.722429

0.780725

0.836773

0.890582

0.942165

0.991532

1.038696

1.083665

1.126453

1.167069

1.205524

1.241829

1.275995

1.308033

1.337954

1.365767

1.391483

1.415113

1.436668

1.456157

1.473592

1.488981

1.502336

1.513667

1.522983

1.530296

1.535614

1.538948

1.540308

1.539704

1.537145

1.532641

1.526202

1.517838

1.507558

1.495372

1.48129

1.46532

1.447473

1.427757

1.406182

Distance

Height

FLight 2-Variable

Sheet3

4.37941.99951

4.2575941.99707

4.134571.992699

4.0103161.986417

3.8848191.978242

3.7580671.968194

3.6300481.956292

3.5007481.942553

3.3701561.926997

3.2382571.909641

3.105041.890503

2.970491.869602

2.8345951.846955

2.6973411.822579

2.5587141.796492

2.4187021.768711

2.2772891.739252

2.1344611.708133

1.9902061.675369

1.8445081.640979

1.6973531.604977

1.5487271.56738

1.3986141.528204

1.2471.487465

1.093871.445178

0.9392091.401359

0.7830011.356022

0.6252311.309184

0.4658831.26086

0.3049421.211063

0.1423921.15981

-0.0217851.107113

-0.1876021.052989

-0.3550780.997451

-0.5242290.940513

-0.6950720.882189

-0.8676220.822494

-1.0418980.76144

-1.2179170.699042

-1.3956970.635313

-1.5752540.570266

-1.7566060.503914

-1.9397720.436271

-2.124770.367349

-2.3116180.297161

-2.5003340.22572

-2.6909370.153038

-2.8834460.079128

-3.0778810.152299

-3.274260.222783

-3.4726020.290607

-3.6729280.355798

-3.8752580.418382

-4.079610.478385

-4.2860060.535833

-4.4944660.590751

-4.7050110.643165

-4.9176610.693099

-5.1324380.74058

-5.3493620.78563

-5.5684560.828275

-5.789740.868538

-6.0132380.906443

-6.238970.942015

-6.466960.975275

-6.6972291.006248

-6.9298021.034956

-7.16471.061422

-7.4019471.085668

-7.6415661.107716

-7.8835821.127589

-8.1280181.145308

-8.3748981.160895

-8.6242471.174371

-8.8760891.185757

-9.130451.195074

-9.3873551.202343

-9.6468281.207583

-9.9088971.210817

-10.1735861.212063

Sheet3

1.99951

1.99707

1.992699

1.986417

1.978242

1.968194

1.956292

1.942553

1.926997

1.909641

1.890503

1.869602

1.846955

1.822579

1.796492

1.768711

1.739252

1.708133

1.675369

1.640979

1.604977

1.56738

1.528204

1.487465

1.445178

1.401359

1.356022

1.309184

1.26086

1.211063

1.15981

1.107113

1.052989

0.997451

0.940513

0.882189

0.822494

0.76144

0.699042

0.635313

0.570266

0.503914

0.436271

0.367349

0.297161

0.22572

0.153038

0.079128

0.152299

0.222783

0.290607

0.355798

0.418382

0.478385

0.535833

0.590751

0.643165

0.693099

0.74058

0.78563

0.828275

0.868538

0.906443

0.942015

0.975275

1.006248

1.034956

1.061422

1.085668

1.107716

1.127589

1.145308

1.160895

1.174371

1.185757

1.195074

1.202343

1.207583

1.210817

1.212063

Distance

Height

Flight 3-Variable Lift

Sheet4

4.37911.99951

4.2563861.997075

4.1318321.992723

4.005411.986484

3.8770911.978387

3.7468471.968458

3.614651.956725

3.480471.943215

3.3442771.927956

3.2060411.910973

3.0657321.892292

2.9233181.871938

2.7787671.849937

2.6320491.826314

2.483131.801092

2.3319761.774296

2.1785561.745949

2.0228341.716075

1.8647771.684696

1.7043491.651835

1.5415141.617515

1.3762371.581756

1.208481.544582

1.0382071.506012

0.865381.466068

0.6899611.424771

0.5119111.38214

0.3311891.338197

0.1477571.29296

-0.0384271.246448

-0.2274031.198682

-0.4192141.14968

-0.6139021.09946

-0.8115111.04804

-1.0120830.99544

-1.2156650.941675

-1.42230.886765

-1.6320340.830725

-1.8449150.773574

-2.0609880.715327

-2.2803030.656001

-2.5029080.595612

-2.7288510.534177

-2.9581840.47171

-3.1909570.408228

-3.4272210.343745

-3.667030.278277

-3.9104350.211838

-4.1574920.144444

-4.4082540.076107

-4.6627780.143419

-4.9211190.207768

-5.1833360.269199

-5.4494860.327756

-5.7196290.383482

-5.9938230.43642

-6.272130.486611

-6.5546120.534096

-6.8413310.578916

-7.1323510.621112

-7.4277370.660722

-7.7275530.697785

-8.0318660.732339

-8.3407440.764423

-8.6542550.794072

-8.9724690.821325

-9.2954560.846216

-9.6232880.86878

-9.9560370.889054

-10.2937780.907071

Sheet4

1.99951

1.997075

1.992723

1.986484

1.978387

1.968458

1.956725

1.943215

1.927956

1.910973

1.892292

1.871938

1.849937

1.826314

1.801092

1.774296

1.745949

1.716075

1.684696

1.651835

1.617515

1.581756

1.544582

1.506012

1.466068

1.424771

1.38214

1.338197

1.29296

1.246448

1.198682

1.14968

1.09946

1.04804

0.99544

0.941675

0.886765

0.830725

0.773574

0.715327

0.656001

0.595612

0.534177

0.47171

0.408228

0.343745

0.278277

0.211838

0.144444

0.076107

0.143419

0.207768

0.269199

0.327756

0.383482

0.43642

0.486611

0.534096

0.578916

0.621112

0.660722

0.697785

0.732339

0.764423

0.794072

0.821325

0.846216

0.86878

0.889054

0.907071

Distance

Height

Flight 4-Variable Lift

Sheet5

4.37881.99951

4.2551761.99708

4.129081.992748

4.0004611.986552

3.869271.97853

3.7354561.968719

3.5989651.957153

3.4597441.943869

3.3177391.9289

3.1728941.91228

3.0251521.894042

2.8744551.874219

2.7207441.852842

2.5639591.829943

2.4040381.805551

2.2409191.779697

2.0745371.75241

1.9048281.723719

1.7317241.693651

1.5551591.662234

1.3750621.629495

1.1913631.595461

1.003991.560157

0.812871.52361

0.6179281.485843

0.4190861.446881

0.2162681.406748

0.0093931.365468

-0.2016191.323063

-0.4168511.279556

-0.6363881.234968

-0.8603161.189323

-1.0887221.14264

-1.3216971.094941

-1.5593311.046245

-1.8017170.996573

-2.0489520.945944

-2.3011310.894378

-2.5583530.841893

-2.820720.788507

-3.0883350.734239

-3.3613010.679106

-3.6397270.623126

-3.9237220.566315

-4.2133960.508689

-4.5088640.450267

-4.8102420.391062

-5.1176470.331092

-5.4311990.27037

-5.7510230.208913

-6.0772440.146735

-6.4099890.08385

-6.7493890.020272

-7.095576-0.043984

-7.448688-0.108905

-7.808862-0.174478

-8.176239-0.240689

-8.550964-0.307527

-8.933183-0.374978

-9.323047-0.44303

-9.720708-0.511672

-10.126322-0.58089

Sheet5

1.99951

1.99708

1.992748

1.986552

1.97853

1.968719

1.957153

1.943869

1.9289

1.91228

1.894042

1.874219

1.852842

1.829943

1.805551

1.779697

1.75241

1.723719

1.693651

1.662234

1.629495

1.595461

1.560157

1.52361

1.485843

1.446881

1.406748

1.365468

1.323063

1.279556

1.234968

1.189323

1.14264

1.094941

1.046245

0.996573

0.945944

0.894378

0.841893

0.788507

0.734239

0.679106

0.623126

0.566315

0.508689

0.450267

0.391062

0.331092

0.27037

0.208913

0.146735

0.08385

0.020272

-0.043984

-0.108905

-0.174478

-0.240689

-0.307527

-0.374978

-0.44303

-0.511672

-0.58089

Distance

Height

Flight 5-Variable Lift

Variable SpeedChanged the initial speedAll other settings set at default

Variable Speed

Chart2

1.999511.999511.999511.999511.999511.99951

1.997061.997061.997061.997061.997061.99706

1.992651.992651.992651.992651.992651.99265

1.986281.986281.986281.986281.986281.98628

1.977951.977951.977951.977951.977951.97795

1.967661.967661.967661.967661.967661.96766

1.955411.955411.955411.955411.955411.95541

1.94121.94121.94121.94121.94121.9412

1.925031.925031.925031.925031.925031.92503

1.90691.90691.90691.90691.90691.9069

1.886811.886811.886811.886811.886811.88681

1.864761.864761.864761.864761.864761.86476

1.840751.840751.840751.840751.840751.84075

1.814781.814781.814781.814781.814781.81478

1.786851.786851.786851.786851.786851.78685

1.756961.756961.756961.756961.756961.75696

1.725111.725111.725111.725111.725111.72511

1.69131.69131.69131.69131.69131.6913

1.655531.655531.655531.655531.655531.65553

1.61781.61781.61781.61781.61781.6178

1.578111.578111.578111.578111.578111.57811

1.536461.536461.536461.536461.536461.53646

1.492851.492851.492851.492851.492851.49285

1.447281.447281.447281.447281.447281.44728

1.399751.399751.399751.399751.399751.39975

1.350261.350261.350261.350261.350261.35026

1.298811.298811.298811.298811.298811.29881

1.24541.24541.24541.24541.24541.2454

1.190031.190031.190031.190031.190031.19003

1.13271.13271.13271.13271.13271.1327

1.073411.073411.073411.073411.073411.07341

1.012161.012161.012161.012161.012161.01216

0.948950.948950.948950.948950.948950.94895

0.883780.883780.883780.883780.883780.88378

0.816650.816650.816650.816650.816650.81665

0.747560.747560.747560.747560.747560.74756

0.676510.676510.676510.676510.676510.67651

0.60350.60350.60350.60350.60350.6035

0.528530.528530.528530.528530.528530.52853

0.45160.45160.45160.45160.45160.451