chapter 2 motion along a straight line

Chapter2 MotionAlongaStraightLine GoalsforChapter2 TostudymotionalongastraightlineTodefineanddifferentiateaverageandinstantaneouslinearvelocityTodefineanddifferentiateaverageandinstantaneouslinearaccelerationToexploreapplicationsofstraight linemotionwithconstantaccelerationToexaminefreelyfallingbodiesToconsiderstraight linemotionwithvaryingacceleration 2 1Displacement time andtheaveragevelocity Displacement changeinposition itisavectorquantity Itsdirectionisfromstarttoend x x2 x1 Averagex velocity thedisplacement x dividedbythetimeinterval t Time changeintime t t2 t1 DistanceandAverageSpeed Distance lengthofthepath itdependsonthepath Itisascalarquantity Ithasnodirection Averagex speed thedistancetraveled sdividedbythetimeinterval t 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 277m 19m 4 0s 1 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 whatwasthebird saveragevelocityinm 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 t2 t3 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 withvw vr TheysetofftogetheronajourneyofdistanceD Rickwalkshalfofthedistanceandrunsthesecondhalf Timwalkshalfofthetimeandrunstheotherhalf a DrawagraphshowingthepositionsofbothTimandRickversustime b Writetwosentencesexplainingwhowinsandwhy c HowlongdoesittakeRicktocoverthedistanceD d FindRick saveragespeedforcoveringthedistanceD e HowlongdoesittakeTimtocoverthedistance TimwinsbecausehetakesshorttimetocoverthesamedistanceasRick a solution d c e VectorsV1andV2shownabovehaveequalmagnitudes Thevectorsrepresentthevelocitiesofanobjectattimest1andt2 respectively Theaverageaccelerationoftheobjectbetweentimet1andt2was ZeroDirectednorthDirectedwestDirectednorthofeastDirectednorthofwest 2 2Instantaneousvelocity Instantaneousvelocityisdefinedasthevelocityatanyspecificinstantoftimeorspecificpointalongthepath Instantaneousvelocityisavectorquantity itsmagnitudeisthespeed itsdirectionisthesameasitsmotion sdirection Howlongisaninstant Inphysics aninstantreferstoasinglevalueoftime TofindtheinstantaneousvelocityatpointP1 wemovethesecondpointP2closerandclosertothefirstpointP1andcomputetheaveragevelocityvav x x tovertheevershorterdisplacementandtimeinterval Both xand tbecomeverysmall buttheirratiodoesnotnecessarilybecomesmall Inthelanguageofcalculus thelimitof x tas tapproacheszeroiscalledthederivativeofxwiththerespecttotandiswrittendx dt P1 Theinstantaneousvelocityisthelimitoftheaveragevelocityasthetimeintervalapproacheszero itequalstheinstantaneousrateofchangeofpositionwithtime Acheetahiscrouched20mtotheeastofanobserver svehicle Attimet 0thecheetahchargesanantelopeandbeginstorunalongastraightline Duringthefirst2 0softheattack thecheetah scoordinatexvarieswithtimeaccordingtotheequationx 20m 5 0m s2 t2 Findthedisplacementofthecheetahbetweent1 1 0sandt2 2 0sFindtheaveragevelocityduringthesametimeinterval Findtheinstantaneousvelocityattimet1 1 0sbytaking t 0 1s then t 0 01s then t 0 001s Derivedageneralexpressionfortheinstantaneousvelocityasafunctionoftime andfromitfindvxatt 1 0sandt 2 0s Example2 1 Example2 1Averageandinstantaneousvelocity Averageandinstantaneousvelocitiesinx tgraph Secantline averagevelocity tangentline instantaneousvelocity example Whichcarstartslater WhendoesA Bpasseachother Whichcarreaches200kmfirst CalculateaveragespeedofAandB Theautomobilesmakea5hourtripoveratotaldistanceof200km TheDerivative aka TheSLOPE Supposeaneccentricpetantisconstrainedtomoveinonedimension Thegraphofhisdisplacementasafunctionoftimeisshownbelow Attimet theantislocatedatPointA Whilethere itspositioncoordinateisx t Attime t Dt theantislocatedatPointB Whilethere itspositioncoordinateisx t Dt Thesecantlineandtheslope SupposeasecantlineisdrawnbetweenpointsAandB Note Theslopeofthesecantlineisequaltotheriseovertherun Theslopeofthesecantlineisaveragevelocity The Tangent line READTHISCAREFULLY IfweholdPOINTAfixedwhileallowingDttobecomeverysmall PointBapproachesPointAandthesecantapproachestheTANGENTtothecurveatPOINTA t x t t Dt x t Dt A B WearebasicallyZOOMINGinatpointAwhereuponinspectiontheline APPEARS straight ThusthesecantlinebecomesaTANGENTLINE Theslopeofthetangentlineis velocity Thederivative Mathematically wejustfoundtheslope Limstandfor anditshowsthe tapproacheszero Asthishappensthetopnumeratorapproachesafinite Thisiswhataderivativeis AderivativeyieldsaNEWfunctionthatdefinestherateofchangeoftheoriginalfunctionwithrespecttooneofitsvariables Theaboveexampleshowstherateofchangeof x withrespecttotime InmostPhysicsbooks thederivativeiswrittenlikethis MathematicianstreatasaSINGLESYMBOLwhichmeansfindthederivative Itissimplyamathematicaloperation Thederivativeistheslopeofthelinetangenttoapointonacurve example Considerthefunctionx t 3t 2 Whatisthetimerateofchangeofthefunction velocity Thisisactuallyveryeasy Theentireequationislinearandlookslikey mx b Thusweknowfromthebeginningthattheslope thederivative ofthisisequalto3 Wedidn tevenneedtoINVOKEthelimitbecausethe tiscancelout Regardless weseethatwegetaconstant Example Considerthefunctionx t kt3 wherek proportionalityconstant Whathappenedtoallthe t s 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 16t2 16t 4 Averagevelocityvs instantaneousvelocityExample AHondaCivictravelsinastraightlinealongaroad Itsdistancexfromastopsignisgivenasafunctionoftimetbytheequationx t t2 t3 where 1 50m s2and 0 0500m s3 Calculatetheaveragevelocityofthecarforthetimeinterval t 0tot 4 00s Determinetheinstantaneousvelocityofthecaratt 2 00sandt 4 00s example Anobjectismovinginonedimensionaccordingtotheformulax t 2t3 t2 4 finditsvelocityatt 2s example Thepositionofanobjectmovinginastraightlineisgivenbyx 7 10t 6t2 m wheretisinseconds Whatistheobject svelocityat4seconds Example Anobjectmovesverticallyaccordingtoy t 12 4t 2t3 whatisitsvelocityatt 3s example Anobject smotionisgivenbytheequationx t 2 4t3 whatistheequationfortheobject svelocity v t 12t2 Followthemotionofaparticle Themotionoftheparticlemaybedescribedfromx tgraph Questions Thegraphaboveshowsvelocityvversustimetforanobjectinlinearmotion Whichofthefollowingisapossiblegraphofpositionxversustimetforthisobject Testyourunderstanding2 2 AccordingtothegraphRankthevaluesoftheparticle sx velocityvxatthepointsP Q R andSfrommostpositivetomostnegative Atwhichpointsisvxpositive Atwhichpointsisvxnegative Atwhichpointsisvxzero Rankthevaluesoftheparticle sspeedatthepointsP 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 Velocitydescribeshowfastabody spositionchangewithtime Accelerationdescribeshowfastabody svelocitychange 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 Agraphof andtmaybeusedtofindtheacceleration Averageacceleration theslopeofsecantline Instantaneousacceleration theslopeofatangentlineatpoint Caution Thesignofaccelerationandvelocity aisinthesamedirectionasv v posa pos v neg a neg aisintheoppositedirectionasv v posa neg v neg a pos Wecanobtainanobject sposition 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 vx2 vx02 2ax x x0 Motionwithconstantaccelerationvx tgraph Ahorizontallineindicatetheslope 0 a 0 Sinceax v t v ax twhichisrepresentedbythearea a tgraph Theareaindicatethechangeinvelocityduring t 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 Whatistheofficer sspeedatthatpoint Whatisthetotaldistanceeachvehiclehastraveledatthatpoint Testyourunderstanding2 4 Fourpossiblevx tgraphsareshownforthetwovehiclesinexample2 5 whichgraphiscorrect Ifweignoreairfrictionandtheeffectsduetotheearth srotation allobjectsfallattheconstantacceleration Theconstantaccelerationofafreelyfallingbodyiscalledtheaccelerationduetogravity andweuselettergtorepresentitsmagnitude Neartheearth ssurfaceg 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 ifaccelerationisnotc