chapter 2 motion along a straight lineVIP

Chapter2,MotionAlongaStraightLine,GoalsforChapter2,TostudymotionalongastraightlineTodefineanddifferentiateaverageandinstantaneouslinearvelocityTodefineanddifferentiateaverageandinstantaneouslinearaccelerationToexploreapplicationsofstraight-linemotionwithconstantaccelerationToexaminefreelyfallingbodiesToconsiderstraight-linemotionwithvaryingacceleration,2.1Displacement,time,andtheaveragevelocity,Displacement:changeinposition,itisavectorquantity.Itsdirectionisfromstarttoend.,x=x2x1,Averagex-velocity:thedisplacement,x,dividedbythetimeintervalt.,Time:changeintime,t=t2t1,DistanceandAverageSpeed,Distance:lengthofthepath,itdependsonthepath.Itisascalarquantity.Ithasnodirection.,Averagex-speed:thedistancetraveledsdividedbythetimeintervalt.Itisascalar.,Averagespeedvs.averagevelocityWhenAlexanderPopovsetaworldrecordin1994byswimming100.0min46.74sec,hisaveragespeedwas(100.0m)/(46.74s)=2.139m/s.butbecauseheswamfourlengthsina25meterpool,hestartedandendedatthesamepointandhehadzerototaldisplacementandzeroaveragevelocity!,example,Whatistheaveragevelocityofthecar?,Positionatt1=1.0s,Positionatt2=4.0s,x1=19m,x2=277m,vav-x=(277m19m)/(4.0s1.0s)=86m/sTheaveragevelocityispositivebecauseitismovinginthepositivedirection.Note:youcanchooseanywayas+.,+displacement,P-Tgraphofthecar,Checkyourunderstanding2.1,Eachofthefollowingautomobiletripstakesonehour.Thepositivex-directionistotheeast.Atravels50kmdueeast.Btravels50kmduewestCtravels60kmdueeast,thenturnsaroundandtravels10kmduewestDtravels70kmdueeast.Etravels20kmduewest,thenturnsaroundandtravels20kmdueeast.Rankthefivetripsinorderofaveragex-velocityfrommostpositivetomostnegative.Whichtrips,ifany,havethesameaveragex-velocity?Forwhichtrip,ifany,istheaveragex-velocityequaltozero?,4,1,3,5,2,1,3,5,Practice2.2,Inanexperiment,ashearwater(aseabird)wastakenfromitsnest,flown5150kmaway,andreleased.Thebirdfounditswaybacktoitsnest13.5daysafterrelease.Ifweplacetheorigininthenestandextendthe+x-axistothereleasepoint,whatwasthebirdsaveragevelocityinm/sForthereturnflight?Forthewholeepisode,fromleavingthenesttoreturning?,-4.42m/s,0m/s,Practice2.4,Startingfromapillar,yourun200meast(the+x-axis)atanaveragespeedof5.0m/s,andthenrun280mwestatanaveragespeedof4.0m/stoapost.CalculateYouraveragespeedfrompillartopost,Youaveragevelocityfrompillartopost.,4.4m/s,-0.72m/s,Practice2.6,Tworunnersstartsimultaneouslyfromthesamepointonacircular200mtrackandruninthesamedirection.Onerunsataconstantspeedof6.20m/s,andtheotherrunsataconstantspeedof5.50m/s.Whenwillthefastonefirst“lap”thesloweroneandhowfarfromthestartingpointwilleachhaverun?Whenwillthefastoneovertakethesloweroneforthesecondtime,andhowfarfromthestartingpointwilltheybeatthatinstant?,286s,1770m,1570m,572s,3540m,3140m,Practice2.8,AHondaCivictravelsinastraightlinealongaroad.Itsdistancexfromastopsignisgivenasafunctionoftimetbytheequationx(t)=t2t3,where=1.50m/s2and=0.0500m/s3.Calculatetheaveragevelocityofthecarforeachtimeinterval:t=0tot=2.00s;t=0tot=4.00st=2.00stot=4.00s.,example,Acatrunsalongastraightline(thex-axis)frompointAtopointBtopointC,asshown.ThedistancebetweenpointsAandCis5.00m,thedistancebetweenpointsBandCis10.0m,andthepositivedirectionofthex-axispointstotheright.ThetimetorunfromAtoBis20.0s,andthetimefromBtoCis8.00s.,WhatistheaveragespeedofthecatbetweenpointsAandC?WhatistheaveragevelocityofthecatbetweenpointsAandC?,Example-Walking1/2thetimevs.Walking1/2thedistance,TimandRickbothcanrunatspeedvrandwalkatspeedvw,withvwvr.TheysetofftogetheronajourneyofdistanceD.Rickwalkshalfofthedistanceandrunsthesecondhalf.Timwalkshalfofthetimeandrunstheotherhalf.a)DrawagraphshowingthepositionsofbothTimandRickversustime.b)Writetwosentencesexplainingwhowinsandwhy.c)HowlongdoesittakeRicktocoverthedistanceD?d)FindRicksaveragespeedforcoveringthedistanceD.e)HowlongdoesittakeTimtocoverthedistance?,TimwinsbecausehetakesshorttimetocoverthesamedistanceasRick.,a.,solution,d.,c.,e.,.,VectorsV1andV2shownabovehaveequalmagnitudes.Thevectorsrepresentthevelocitiesofanobjectattimest1andt2,respectively.Theaverageaccelerationoftheobjectbetweentimet1andt2was,ZeroDirectednorthDirectedwestDirectednorthofeastDirectednorthofwest,2.2Instantaneousvelocity,Instantaneousvelocityisdefinedasthevelocityatanyspecificinstantoftimeorspecificpointalongthepath.Instantaneousvelocityisavectorquantity,itsmagnitudeisthespeed,itsdirectionisthesameasitsmotionsdirection.Howlongisaninstant?Inphysics,aninstantreferstoasinglevalueoftime.,TofindtheinstantaneousvelocityatpointP1,wemovethesecondpointP2closerandclosertothefirstpointP1andcomputetheaveragevelocityvav-x=x/tovertheevershorterdisplacementandtimeinterval.Bothxandtbecomeverysmall,buttheirratiodoesnotnecessarilybecomesmall.Inthelanguageofcalculus,thelimitofx/tastapproacheszeroiscalledthederivativeofxwiththerespecttotandiswrittendx/dt.,P1,Theinstantaneousvelocityisthelimitoftheaveragevelocityasthetimeintervalapproacheszero;itequalstheinstantaneousrateofchangeofpositionwithtime.,Acheetahiscrouched20mtotheeastofanobserversvehicle.Attimet=0thecheetahchargesanantelopeandbeginstorunalongastraightline.Duringthefirst2.0softheattack,thecheetahscoordinatexvarieswithtimeaccordingtotheequationx=20m+(5.0m/s2)t2.Findthedisplacementofthecheetahbetweent1=1.0sandt2=2.0sFindtheaveragevelocityduringthesametimeinterval.Findtheinstantaneousvelocityattimet1=1.0sbytakingt=0.1s,thent=0.01s,thent=0.001s.Derivedageneralexpressionfortheinstantaneousvelocityasafunctionoftime,andfromitfindvxatt=1.0sandt=2.0s,Example2.1,Example2.1Averageandinstantaneousvelocity,Averageandinstantaneousvelocitiesinx-tgraph,Secantlineaveragevelocity,tangentlineinstantaneousvelocity,example,Whichcarstartslater?WhendoesAWhatisthetimerateofchangeofthefunction(velocity)?Thisisactuallyveryeasy!Theentireequationislinearandlookslikey=mx+b.Thusweknowfromthebeginningthattheslope(thederivative)ofthisisequalto3.,WedidntevenneedtoINVOKEthelimitbecausethetiscancelout.Regardless,weseethatwegetaconstant.,Example,Considerthefunctionx(t)=kt3,wherek=proportionalityconstant.,Whathappenedtoallthets?TheywenttoZEROwhenweinvokedthelimit!Whatdoesthisallmean?,TheMEANING?,Forexample,ift=2seconds,usingx(t)=kt3=(1)(2)3=8meters.Thederivative,however,tellushowourDISPLACEMENT(x)changesasafunctionofTIME(t).TherateatwhichDisplacementchangesisalsocalledVELOCITY.Thusifweuseourderivativewecanfindouthowfasttheobjectistravelingatt=2second.Sincedx/dt=3kt2=3(1)(2)2=12m/s,THEREISAPATTERNHERE!,Examplex=5,Derivativeofaconstant,Why?,.,PowerRule,x=t5,Example,x=t-5,x=t,.,ConstantMultiplier,Example,x=4t5,.,AdditionandSubtractionRule,Thederivativeofthesum(ordifference)oftwoormorefunctionsisthesum(ordifference)ofthederivativesofthefunctions.,x=2t5+3t-1,Example,Chainrule,Ifxisafunctionoff,andfisafunctionoft,soindirectly,xisafunctionoft:x(f(t),Example,Classwork,Findthederivatives(dx/dt)ofthefollowingfunctionx=t3x=1/t=t-1x=(6t3+2/t)-2x=16t216t+4,Averagevelocityvs.instantaneousvelocityExample,AHondaCivictravelsinastraightlinealongaroad.Itsdistancexfromastopsignisgivenasafunctionoftimetbytheequationx(t)=t2t3,where=1.50m/s2and=0.0500m/s3.Calculatetheaveragevelocityofthecarforthetimeinterval:t=0tot=4.00s;Determinetheinstantaneousvelocityofthecaratt=2.00sandt=4.00s.,example,Anobjectismovinginonedimensionaccordingtotheformulax(t)=2t3t24.finditsvelocityatt=2s.,example,Thepositionofanobjectmovinginastraightlineisgivenbyx=(7+10t6t2)m,wheretisinseconds.Whatistheobjectsvelocityat4seconds?,Example,Anobjectmovesverticallyaccordingtoy(t)=124t+2t3.whatisitsvelocityatt=3s?,example,Anobjectsmotionisgivenbytheequationx(t)=2+4t3.whatistheequationfortheobjectsvelocity?,v(t)=12t2,Followthemotionofaparticle,Themotionoftheparticlemaybedescribedfromx-tgraph.,Questions,Thegraphaboveshowsvelocityvversustimetforanobjectinlinearmotion.Whichofthefollowingisapossiblegraphofpositionxversustimetforthisobject?,Testyourunderstanding2.2,AccordingtothegraphRankthevaluesoftheparticlesx-velocityvxatthepointsP,Q,R,andSfrommostpositivetomostnegative.Atwhichpointsisvxpositive?Atwhichpointsisvxnegative?Atwhichpointsisvxzero?RankthevaluesoftheparticlesspeedatthepointsP,Q,R,andSfromfastesttoslowest.,P,R,Q,S,R,P,Q=S,Example2.10,Aphysicsprofessorleavesherhouseandwalksalongthesidewalktowardcampus.After5minitstartstorainandshereturnshome.Accordingtothegraph,atwhichofthelabeledpointsishervelocityZero?Constantandpositive?Constantandnegative?Increasinginmagnitude?Decreasinginmagnitude?,IV,I,V,II,III,example,Whichpairofgraphsrepresentsthesame1-dimensionalmotion?,A.,B.,C.,D.,example,Thegraphrepresentstherelationshipbetweendistanceandtimeforanobject.Whatistheinstantaneousspeedoftheobjectatt=5.0seconds?t=2.0seconds?,0,1.5m/s,example,Accordingtothegraph,theaccelerationoftheobjectmustbeZeroConstantandpositiveConstantandnegativeIncreasingdecreasing,t,d,o,2.3averageandinstantaneousacceleration,TheaverageaccelerationoftheparticleasitmovesfromP1toP2isavectorquantity,whosemagnitudeequalstothechangeinvelocitydividedbythetimeinterval.,Velocitydescribeshowfastabodyspositionchangewithtime.Accelerationdescribeshowfastabodysvelocitychange,ittellshowspeedanddirectionofmotionarechanging.,Instantaneousacceleration,Theinstantaneousaccelerationisthelimitofaverageaccelerationasthetimeintervalapproacheszero.,AverageandinstantaneousaccelerationExample2.3,Supposethex-velocityvxofacaratanytimetisgivenbytheequation:vx=60m/s+(.50m/s2)t2Findthechangeinx-velocityofthecarinthetimeintervalbetweent1=1.0sandt2=3.0s.Findtheaveragex-accelerationbetweent1=1.0sandt2=3.0s.Deriveanexpressionfortheinstantaneousx-accelerationatanytime,anduseittofindthex-accelerationatt=1.0sandt=3.0s.,4.0m/s2.0m/s2a=(1.0m/s3)t;1.0m/s2;3.0m/s2,example,Thepositionofavehiclemovingonastraighttrackalongthex-axisisgivenbytheequationx(t)=t2+3t+5wherexisinmetersandtisinseconds.Whatisitsaccelerationattimet=5s?,(2m/s2),Findingaccelerationonavx-tgraphandax-tgraph,Averageaccelerationcanbedeterminedbyv-tgraph,Findingtheaccelerationonv-tgraph,Agraphofandtmaybeusedtofindtheacceleration.Averageacceleration:theslopeofsecantline.Instantaneousacceleration:theslopeofatangentlineatpoint.,Caution:Thesignofaccelerationandvelocity,aisinthesamedirectionasv,v:posa:pos.,v:neg.a:neg.,aisintheoppositedirectionasv,v:posa:neg.,v:neg.a:pos.,Wecanobtainanobjectsposition,velocityandaccelerationfromitv-tgraph,Findingaccelerationonax-tgraph,Onax-tgraph,theaccelerationisgivenbythecurvatureofthegraph.,Curvesupfromthepoint:accelerationispositive,straightornotcurvesupordown:accelerationiszero,Curvesdown:accelerationisnegative,Example,Thefigureisgraphofthecoordinateofaspidercrawlingalongthex-axis.Graphitsvelocityandaccelerationasfunctionoftime.,.,Checkyourunderstanding2.3,Refertothegraph,AtwhichofthepointsP,Q,R,andSisthex-accelerationaxpositive?Atwhichpointsisthex-accelerationaxnegative?Atwhichpointsdoesthex-accelerationappeartobezero?Ateachpointstatewhetherthespeedisincreasing,decreasing,ornotchanging.,P:visnotchange;Q:viszero,changingfrompos.toneg.,firstdecreaseinpos.thenincreaseinneg.,R:visneg.,constant;S:viszero,changingfromneg.topos.,firstdecreaseinneg.thenincreaseinpos.,S,Q,P,R,2.4motionwithconstantacceleration,Given:,derive:,vx=vx0+axt,(assumet0=0),x=x0+vx0+axt2,vx2vx02=2ax(xx0),Motionwithconstantaccelerationvx-tgraph,Ahorizontallineindicatetheslope=0,a=0,Sinceax=v/t;v=axtwhichisrepresentedbythearea.,a-tgraph,Theareaindicatethechangeinvelocityduringt,Kinematicsequationsforconstantacceleration,Example2.4,AmotorcyclistheadingeastthroughasmallIowacityacceleratesafterhepassesthesignpostmarkingthecitylimits.Hisaccelerationisaconstant4.0m/s2.Attimet=0heis5.0meastofthesignpost,movingeastat15m/s.Findhispositionandvelocityattimet=2.0s.Whereisthemotorcyclistwhenhisvelocityis25m/s?,Example2.5,Amotoristtravelingwithaconstantspeedof15m/spassesaschoolcrossingcorner,wherethespeedlimitis10m/s.Justatthemotoristpasses,apoliceofficeronamotorcyclestoppedatthecornerstartsoffinpursuitwithconstantaccelerationof3.0m/s2.Howmuchtimeelapsesbeforetheofficercatchesupwiththemotorist?Whatistheofficersspeedatthatpoint?Whatisthetotaldistanceeachvehiclehastraveledatthatpoint?,Testyourunderstanding2.4,Fourpossiblevx-tgraphsareshownforthetwovehiclesinexample2.5.whichgraphiscorrect?,Ifweignoreairfrictionandtheeffectsduetotheearthsrotation,allobjectsfallattheconstantacceleration.,Theconstantaccelerationofafreelyfallingbodyiscalledtheaccelerationduetogravity,andweuselettergtorepresentitsmagnitude.Neartheearthssurfaceg=9.81m/s/s=32ft/s/s,Onthesurfaceofthemoon,g=1.6m/s/sOnthesurfaceofthesun,g=270m/s/s,a=-g,2.5FreeFallingBodies,Example2.6,Aone-eurocoinisdroppedfromtheLeaningTowerofPisa.Itstartsfromrestandfallsfreely.Computeitspositionandvelocityafter.1.0s.2.0s,and3.0s.,Example2.7,Youthrowaballverticallyupwardfromtheroofofatallbuilding.Theballleavesyourhandatapointevenwiththeroofrailingwithanupwardspeedof15.0m/s;theballistheninfreefall.Onitswaybackdown,itjustmissestherailing.Atthelocationofthebuilding,g=9.80m/s2.findThepositionandvelocityoftheball1.00sand4.00safterleavingyourhandThevelocitywhentheballis5.00mabovetherailingThemaximumheightreachedandthetimeatwhichitisreachedTheaccelerationoftheballwhenitisatitsmaximumheight.,Velocityandaccelerationatthehighestpoint,Example2.8,FindthetimewhentheballinExample2.7is5.00mbelowtheroofrailing.,Checkyourunderstanding2.5,Ifyoutossaballupwardwithacertaininitialspeed,itfallsfreelyandreachesamaximumheighthattimetafteritleavesyourhand.Ifyouthrowtheballupwardwithdoubletheinitialspeedwhatnewmaximumheightdoestheballreach?Ifyouthrowtheballupwardwithdoubletheinitialspeed,howlongdoesittaketoreachitsmaximumheight?,4h,2t,Inthecaseofstraight-linemotion,ifthepositionxisaknownfunctionoftime,wecanfindvx=dx/dttofindx-velocity.Andwecanuseax=dvx/dttofindthex-accelerationasafunctionoftime,Inmanysituations,wecanalsofindthepositionandvelocityasfunctionoftimeifwearegivenfunctionax(t).,2.6velocityandpositionbyintegrationFindingv(t)andx(t)whengivena(t),The“AREA”,Inv-tgraph,theareaunderthelinerepresentdisplacement.,However,ifaccelerationisnotconstant,howcanwedeterminex(t)?,t1,t2,v(m/s),t(s),x=area,t1,t2,v(m/s),t(s),x=area,t1,v(m/s),t(s),t1,v(m/s),t1,v(m/s),v(t),t2,v(m/s),Zoomin,Wehavelearnedthatthe