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 D