Vectors-and-Scalars矢量和标量

VectorsandScalarsAPPhysicsBScalarASCALARisANYquantityinphysicsthathasMAGNITUDE,butNOTadirectionassociatedwithit.Magnitude–Anumericalvaluewithunits.ScalarExampleMagnitudeSpeed20m/sDistance10mAge15yearsHeat1000caloriesVectorAVECTORisANYquantityinphysicsthathasBOTHMAGNITUDEandDIRECTION.VectorMagnitude&DirectionVelocity20m/s,NAcceleration10m/s/s,EForce5N,WestFaxv,,,VectorsaretypicallyillustratedbydrawinganARROWabovethesymbol.Thearrowisusedtoconveydirectionandmagnitude.ApplicationsofVectorsVECTORADDITION–If2similarvectorspointintheSAMEdirection,addthem.Example:Amanwalks54.5meterseast,thenanother30meterseast.Calculatehisdisplacementrelativetowherehestarted?54.5m,E30m,E+84.5m,ENoticethattheSIZEofthearrowconveysMAGNITUDEandthewayitwasdrawnconveysDIRECTION.ApplicationsofVectorsVECTORSUBTRACTION-If2vectorsaregoinginoppositedirections,youSUBTRACT.Example:Amanwalks54.5meterseast,then30meterswest.Calculatehisdisplacementrelativetowherehestarted?54.5m,E30m,W-24.5m,ENon-CollinearVectorsWhen2vectorsareperpendicular,youmustusethePythagoreantheorem.95km,E55km,NStartFinishAmanwalks95km,Eastthen55km,north.CalculatehisRESULTANTDISPLACEMENT.ThehypotenuseinPhysicsiscalledtheRESULTANT.TheLEGSofthetrianglearecalledtheCOMPONENTSHorizontalComponentVerticalComponentkmccbacbac8.109120505595Resultant2222222BUT……whataboutthedirection?Inthepreviousexample,DISPLACEMENTwasaskedforandsinceitisaVECTORweshouldincludeaDIRECTIONonourfinalanswer.NOTE:Whendrawingarighttrianglethatconveyssometypeofmotion,youMUSTdrawyourcomponentsHEADTOTOE.NSEWNofEEofNSofWWofSNofWWofNSofEEofSNofEBUT…..whatabouttheVALUEoftheangle???JustputtingNorthofEastontheanswerisNOTspecificenoughforthedirection.WeMUSTfindtheVALUEoftheangle.NofE55km,N95km,ETofindthevalueoftheangleweuseaTrigfunctioncalledTANGENT.30)5789.0(5789.095551TansideadjacentsideoppositeTan109.8kmSotheCOMPLETEfinalansweris:109.8km,30degreesNorthofEastWhatifyouaremissingacomponent?Supposeapersonwalked65m,25degreesEastofNorth.Whatwerehishorizontalandverticalcomponents?65m25H.C.=?V.C=?Thegoal:ALWAYSMAKEARIGHTTRIANGLE!Tosolveforcomponents,weoftenusethetrigfunctionssinceandcosine.EmCHoppNmCVadjhypopphypadjhypotenusesideoppositehypotenusesideadjacent,47.2725sin65..,91.5825cos65..sincossinecosineExampleAbear,searchingforfoodwanders35meterseastthen20metersnorth.Frustrated,hewandersanother12meterswestthen6meterssouth.Calculatethebear'sdisplacement.35m,E20m,N12m,W6m,S-=23m,E-=14m,N23m,E14m,N3.31)6087.0(6087.231493.262314122TanTanmRTheFinalAnswer:26.93m,31.3degreesNORTHorEASTRExampleAboatmoveswithavelocityof15m/s,Ninariverwhichflowswithavelocityof8.0m/s,west.Calculatetheboat'sresultantvelocitywithrespecttoduenorth.15m/s,N8.0m/s,WRv1.28)5333.0(5333.0158/17158122TanTansmRvTheFinalAnswer:17m/s,@28.1degreesWestofNorthExampleAplanemoveswithavelocityof63.5m/sat32degreesSouthofEast.Calculatetheplane'shorizontalandverticalvelocitycomponents.63.5m/s32H.C.=?V.C.=?SsmCVoppEsmCHadjhypopphypadjhypotenusesideoppositehypotenusesideadjacent,/64.3332sin5.63..,/85.5332cos5.63..sincossinecosineExampleAstormsystemmoves5000kmdueeast,thenshiftscourseat40degreesNorthofEastfor1500km.Calculatethestorm'sresultantdisplacement.NkmCVoppEkmCHadjhypopphypadjhypotenusesideoppositehypotenusesideadjacent,2.96440sin1500..,1.114940cos1500..sincossinecosine5000km,E401500kmH.C.V.C.5000km+1149.1km=6149.1km6149.1km964.2kmR91.8)364.0(157.01.61492.96414.62242.9641.6149122TanTankmRTheFinalAnswer:6224.14km@8.91degrees,NorthofEast