Meyer and John L Bohn JILA NIST and Department of Physics University of Colorado Boulder Colorado 80309 Received 26 November 2007 accepted 16 June 2008 We investigate whether storing baseballs in a controlled humidity environment signi64257cantly af ID: 60299
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aD as,D=CD Cs,D+A Asw Notethatif0,thedrierbaseballwillexperiencealargeraccelerationduetodrag.Ifweusethedatafromourexperimentsandassumeforthemomentthat=0,weÞndthatthevalueof1.12%forballsinitiallyleavingthebat.Therefore,basedsolelyonthechangeintheball,wewouldexpecthumidiÞedballstotravelslightlyfar-therthandryballsforthesameinitialconditions.Thisanalysisiscomplicatedbythefactthatthevelocity,andthus,bothchangeduringtheballÕsßight.Thevalue0overtheentirerangeoflaunchvelocitiesforthedragproÞlesofRefs..InRef.monotonicallydecreasesfromzerotosmallnegativevalues;forRef.weÞndthatitincreasesfromsmallnegativevaluestosmallpositivevalues.Thesechangescandrasti-callychangetheamountofaccelerationduetodragexperi-encedbytheballduringitsßight.Tounderstandthecom-pleteeffectofthevariationofthedragcoefÞcientafulltrajectorycalculationisneeded;theresultispresentedinFig..WeseethatthemonotonicallydecreasingvalueofinthedragproÞleofRef.causestheslopeintherangevariationtoincreasewithgreaterlaunchvelocity,thusdeviatingfromthevaluesofRef.andasmoothsphere.InthecurveofRef.weseetheslopeinrangevariationdiminish;thiseffectisduetothechangeinsignof,whichoccursaround40mWeknowthatrelativehumidityhasothereffectsonthebattedballbeyondaerodynamics.Forexample,Kaganfoundthata20%increaseintherelativehumidityreducestheballÕscoefÞcientofrestitution,resultinginabouta6ftreductioninthedistanceofabattedball.Thischangerepre-sentstheneteffectoftwocompetingtendencies.Thestan-dardballcomesoffthebatslowerthanthedryoneandsowouldnotgoasfar.Theslowerballalsoexperienceslessdecelerationduetodragandmightbeexpectedtogofarther.OnlythefulltrajectorycalculationcantellwhicheffectItturnsoutthatitisthevelocityoffthebatthatdominatesandthereforethestandardballshouldtravellessfar.Wecannaivelyexpectthetotalchangeinrangetobethesumofthecollisionalandaerodynamiceffects,whichyieldsabouta4ftoverallreductioninbattedrange.Wewillseethatthisestimateisreasonable.C.NeteffectoftherelativehumidityonbattedbaseballsCrossandNathanhaveworkedoutthedependenceofthebat-ballcollisionontheparameters,andcoefÞcientofrestitution,thediameteroftheball,andthemassoftheball,respectively.Byusingthemeasurementsofandourown,wecanextractthechangesinthelaunchparameterssuchasangle,speed,andspinrateoftheball.TodosowehavetoclearlydeÞnetheassumptionsmadebeforethecollision.Weassumetheballarrivesfromthepitcherintheexactsamelocationwiththesamevelocityandspinforadryandstandardballandthatthehittertakesthesameswing.Therefore,theonlyparametersthatchangeinthecollisionduetorelativehumiditychangesare,andWepresentinFig.thetotaleffectoftherelativehumid-ityontherangeofabattedballincludingtheeffectsofthecollision.Therangevariationisnotquitethe4ftexpectedfromaddingthecollisionalandaerodynamiceffectsindepen-dently,presumablyduetothenonlinearinteractionofthevariablesinvolved.TheresultsforthedragproÞlesofasmoothballandtheoneinRef.showlittlechangeinrangeduetovariationsin,althoughtheystarttodeviateslightlyasthelaunchvelocityisincreased.Thesmalldiminutionintherangevariationisduetotheofthedecelerationduetodrag.Ballsleavingwithalargervelocityoffthebatexperiencemoredeceleration.Thein-creaseinrangevariationofthedragproÞleofRef.isduetothelargechangeinduringitstrajectoryalthoughonlysmallerRevaluesaresampled.ThedragproÞleofRef.samplesboththedropandriseof,andthereisamaxi-mumintherangevariation,whichoccursnear40mAmoredetailedstudyofthedragproÞleforspinningbaseballsisneededtofullyunderstandtheeffectsofrelativehumidityvariation.AtCoorsFieldtheserelativehumidityvariationsareasmalleffectontherangeofabattedbaseball,3ft,anaverageofthevaluesforastandardballlaunchedat40ms.ThisvalueisconsistentwithwhatisobservedseeTable.Amoredetailedstudythatincludestheprevail-ingnortheastwinds,abreakdownofßyballstoleft,left-center,center,right-center,andrightÞeld,andathree-dimensionalmodelisneededtodeterminetherangevariations.Theßy-balldatalistedareaveragedoverallÞelddirectionsandballshittoleftÞeldwillbemoreaffectedduetothewindpatternsinCoorsFieldthanthosehittorightV.CONCLUSIONSWehaveshownthatstoringbaseballsinhumidity-controlledenvironmentscanslightlyincreasetheirsize,weight,anddensity.Thischangehassmallconsequencesfortheaerodynamicsofaballinßight.Basedsolelyonaerody-namicconsiderations,humidiÞedballswillbreakslightlymore,byabout0.2in.,whilebattedballsmaytravelfarther,ontheorderofacoupleoffeet.Botheffectsappearcounter-intuitivebutfollowfromtheaerodynamicsofspheresinair.Botheffectsarealsocommoninexperience;abaseballandwhifßeballarethesamesize,butthedenserbaseballcurves Fig.7.Relativevariationintherangeofawell-struckbaseballhitatCoorsFieldasafunctionofthevelocityofthedryballoffthebatincludingtheeffectsofanincreasedcoefÞcientofrestitution.Negativevaluesofthatthedrierbaseballßiesfurtherthanthestandardbaseball.Foreaseofcomparingtobaseballunits,thetopaxisislabeledinmilesperhourandtherightaxisislabeledinfeet.Am.J.Phys.,Vol.76,No.11,November2008EdmundR.MeyerandJohnL.Bohn