Kinematics in One Dimension - James Hedberghedberg.ccnysites.cuny.edu/content/introphysics/1d... ·...

LinearCircular

ProjectileRotational

KinematicsinOneDimension

1.Introduction1.DifferentTypesofMotionWe'lllookat:2.Dimensionalityinphysics3.Onedimensionalkinematics4.Particlemodel

2.DisplacementVector1.Displacementin1-D2.DistanceTraveled

3.SpeedandVelocity1....withadirection

4.Changeinvelocity.1.Acceleration2.Acceleration,themath.3.Slowingdown4.Accelerationinthenegative5.Summaryofaccelerationsignage.

5.Kinematicequations1.EquationsofMotion(1-D)

6.SolvingProblems7.Plotting8.FreeFall

1.Dropawrench2.Howhighwasthis?3.Everypointonalinehasatangent

Introduction

Motion:changeinpositionororientationwithrespecttotime.

DifferentTypesofMotionWe'lllookat:

...oracombinationofthem.

Vectorshavegivenussomebasicideasabouthowtodescribethepositionofobjectsintheuniverse/Now,we'llcontinuebyextendingthoseideastoaccountforchangesinthatposition.Ofcoursetheworldwouldbeawfullyboringifthepositionofeverythingwasconstant.

Linearmotioninvolvesthechangeinpositionofanobjectinonedirectiononly.Anexamplewouldbeatrainonastraightsectionofthetrack.Thechangeinpositionisonlyinthehorizontaldirection.

Projectilemotionoccurswhenobjectsarelaunchedinthegravitationalfieldneartheearthssurface.Theyexperiencemotioninboththehorizontalandtheverticaldirections.

Circularmotionoccursinafewspecificcaseswhenanobjecttravelsinaperfectcircle.Somespecialmathcanbeusedinthesecases.

Rotationalmotionimpliesthatthebodyinquestionisrotatingaroundanaxis.Theaxisdoesn'tnecessaryneedtopassthroughtheobject.

PHY 207 - 1d-kinematics - J. Hedberg - 2017

http://localhost/~james/hedberg.ccnysites/img203207/bike-flip.gif

Forthecaseof1-dimensionalmotion,we'llonlyconsiderachangeofpositioninonedirection.Itcouldbeanyofthethreecoordinateaxes.Justadescriptionofthemotion,withoutattemptingtoanalyzethecause.Todescribemotionweneed:

1.CoordinateSystem(origin,orientation,scale)2.theobjectwhichismoving

y

x

z

Dimensionalityinphysics

Onedimensionalkinematics

0 20-20 40 60 80 100

+x (m)-x (m)

Preludetoadvancedphysicsandengineering:Lateron,you'llhavetoexpandyournotionofdimensionsabit.Itwon'tsimplymeanstraightorcurvy,butwillinsteadbeusedtodescribethedegreesoffreedominasystem.Forexample,anorbitingbody,thoughitmovesinacirclewhichrequiresxandyvaluestodescribe,canalsobedescribedbyconsideringtheradiusandtheangleofrotationinstead.Thisisjustanothercoordinatesystem:polarcoordinates(usually: and ).Ifwedescribetheorbitingplanetinthissystem,andsay,it'sgoingaroundinaperfectcircle,thenthe valuedoesn'tchangeandthe valuebecometheonlydimensionofinterest.Let'sholdoffonthisapproachfornow,butwhenitcomesbacklateron,welcomeitwithopenarmsbecauseitallowsformuchmorepowerfulandsimpleanalysisofsystems.

r θr θ

1dkinematicswillbeourstartingpoint.Itisthemoststraightforwardandeasiestmathematicallytodealwithsinceonlyonepositionvariablewillbechangingwithrespecttotime.


Toquantifythemotion,we'llstartbydefiningthedisplacementvector.

Inthecaseofourwanderingbug,thiswouldbethedifferencebetweenthefinalpositionandtheinitialposition.

x∆ x

Particlemodel

We'llneedtouseanabstraction:Allrealworldobjectstakeupspace.We'llassumethattheydon't.Inotherwords,thingslikecars,cats,andducksarejustpoint-likeparticles.

DisplacementVector

NoteonNotation!

isthesamethingas

isthesamethingas

Thisisourfirstrealabstraction.Again,sincewearetryingtopredicteverything,wewouldliketofigureouttherulesthatdescribehowanyobjectwouldmove.Takeatrainforexample.Ifweaskedaquestionlike"whendoestheCtrainenter59thstreetstation?",anaturalfollow-upwouldbe"well,doyoumeanthefrontofthetrain,orthemiddleofthetrain,ortheendofthetrain?Eachoftheseanswersmightbedifferentbyafewseconds.

Howdowedealwiththis?Byconsideringthetraintobea'point',wecanneglecttheactuallengthofthetrainandfocusonwhat'smoreintersting:howthetrainmoves.

Thegoalistofindtheunderlyingphysicsthatdescribesalltrains.Oncewedothat,thenwecanimproveourmodelbyincludinginformationaboutthelengthoftheindividualtrainweareinterestedin.

Δx = x−x0

Thisfigureshowsthedisplacementvector .Thismightbedifferentthanthedistancetraveledbythebug(showninthedottedline).

Δx

xf x

xi x0

Whendescribingmotions,weusuallyhaveaninitialpositionandafinalposition.Wecancallthese andrespectively,whenwedoouralgebra.

Oranotherwayofwritingthesequantitiesistosayourinitialpositionis andourfinalpositionisjust .Thisisaslightlymoregeneralwayofwritingthings.

xi xf

x0 x


Displacementin1-D

0 20-20 40 60 80 100

+x (m)-x (m)

Here'sacarthatmovesfrom to creatingadisplacementvectorof:

Thecarthenreversesto .

0 20-20 40 60 80 100

+x (m)-x (m)

Theleadstoadisplacementvectorof .

DistanceTraveled

Togetthedistancetraveled,wejustneettotakethemagnitudeofthedisplacementduringacertainmotion.

SpeedandVelocity

The'elapsedtime'isdeterminedinthesamewayasthedistance:

Again, isthestartingtime,and isthefinaltime.

x0 x

Δx = x− = 60m−0m = 60mx0

x=−20

Δx = −80m

Aboutnotation. ("deltax")referstothechangein .Thatis,differencebetweenafinalandinitialvalue:

Or,inwords,thefinalxpositionminustheoriginalxpositionisequaltothechangeinx.

Δx x

Δx= x− x0

|Δx| = DistanceTraveled

Thisequationwillonlybetrueifthedisplacementisalwaysinthesamedirection.Ifhowever,thedisplacementvectorweretochangedirectionduringatrip,thethedistancetraveledmightnotbeequaltothetotaldisplacement.Forexample,ifyouwalk100feetforward,thenturnaroundandwalk50backwards.Youdisplacementfromtheinitialtofinalpositionwillonlybe50feet,butyouwillhavewalkedatotalof150feet.

AverageSpeed≡Distanceinagiventime

Elapsedtime

Δt = t− .t0

t0 t


x(m)

t(s)0

2

1

4

2

6

3

Thisisagraphshowingthepositionofanobjectwithrespecttotime.Whichchoicebestdescribesthismotion?

x(m)

t(s)0

2

1

4

2

6

3

Thisisagraphshowingthepositionofanobjectwithrespecttotime.Whichchoicebestdescribesthismotion?

TakingtheAtrainbetween59thand125thtakesabout8minutes.TheC,whichisalocal,takes12minutes(onagoodday).Findtheaveragespeedforbothofthesetrips.

...withadirection

Calculatingtheaveragespeeddidn'ttellusanythingaboutthedirectionoftravel.Forthis,we'llneedaveragevelocity.

Inmathematicalterms:

(SIunitsofaveragevelocityarem/s)

Quick Question 1

Quick Question 2

Example Problem #1:

AverageVelocity ≡DisplacementElapsedtime

≡ =vx−x0t− t0

ΔxΔt

Inone-dimension,velocitycaneitherbeinthepositiveornegativedirection.

a)Theobjectismovingat0.5m/sinthe+xdirection.b)Theobjectismovingat1.0m/sinthe+xdirection.c)Theobjectismovingat2.0m/sinthe+xdirection.d)Theobjectisnotmovingatall.

a)Theobjectismovingat0.5m/sinthe+xdirection.b)Theobjectismovingat1.0m/sinthe+xdirection.c)Theobjectismovingat2.0m/sinthe+xdirection.d)Theobjectisnotmovingatall.


ThinkingabouttheAtrain,it'sclearthatitsspeedandvelocitystayedessentiallyconstantbetween59thand125thideally).However,theCtrainhadtostartandstopat7stations.Toquantify,thisdifferenceinmotion,we'llneedtointroducetheconceptofinstantaneousvelocity.

Ifweimaginemakingmanymeasurementsofthevelocityoverthecourseofthetravel,byreducingthe weareconsidering,thenwecanbegintoseehowwecanmoreaccuratelyassessthemotionofthetrain.

Theconceptofinstantaneousvelocityinvolvesconsideringaninfinitesimallysmallsectionofthemotion:

Thiswillenableustotalkaboutthevelocityataparticle'spositionratherthanforanentiretrip.Ingeneral,thisiswhatwe'llmeanwhenwesay'velocity'or'speed'.

Quick Question 3x(m)

t(s)0

2

1

4

2

6 8

3

4

Atwhichofthefollowingtimesisthespeedofthisobjectthegreatest?

Δx

v = =limΔt→0

ΔxΔt

dx

dt

a)t = 0b) st = 2c) st = 4d) st = 6e) st = 8


Changeinvelocity.

Naturally,inordertobeginmoving,anobjectmustchangeitsvelocity.Here'sagraphofabicyclistridingataconstantvelocity.(Inthiscaseit's10m/s)

+x (m)100 20 30 40

0s 1s 2s

Now,here'sagraphofthesamebicyclistridingandchanginghisvelocityduringthemotion

+x (m)100 20 30 40

0s 1s 2s 3s

Acceleration

Thischangeinvelocitywe'llcallacceleration,andwecandefineitinaverysimilarwaytoourdefinitionofvelocity:

Again,inthiscasewe'retalkingaboutaverageacceleration.

At ,theAtrainisatrestat59thstreet.5secondslater,it'stravelingnorthat19meterspersecond.Whatistheaverageaccelerationduringthistimeinterval?

Ifweconsideredthesameverysmallchangeintime,theinfinitesimalchange,thenwecouldtalkaboutinstantaneousacceleration

TheSIunitsofaccelerationaremeterspersecondpersecond,or .That'sprobablyalittlebitofaweirdunit,but,itmakessensetothinkaboutlikethis:

Intheuppermotiongraph,noticehowthelengthofthedisplacementvector isthesameateachintervalintime.Meaning,thatafter1secondhaspassed,thedisplacementis10m,afteranothersecondpasses,another10metersdisplacementhasoccurred,makingthetotaldisplacementequalto20m.Thisismotionataconstantvelocity.Thisalsoapparentinthelengthofthevelocityvectorsateachpoint.Theyarealwaysthesame.

Inthebottomgraph,thedisplacement,andvelocityvectors,changeeachtimetheyaremeasured.Thisisrepresentativeofmotionwithnon-constantvelocity.Thevelocityischangingastimemoveson.

d

= =av−v0t− t0

ΔvΔt

Example Problem #2:

t = 0

a = =limΔt→0

ΔvΔt

dv

dt

ms−2

or( )m

s

s

vel

s


01

3

2

6

3

9

velo

city

(m/s

)

time (s)

Thisisagraphshowingthevelocityofanobjectwithrespecttotime.Whichchoicebestdescribesthismotion?

Play/Pause

Play/Pause

Quick Question 4

Acceleration,themath.

Toquantifytotheaccelerationofamovingbody,saythiscar,we'llneedtoknowitsinitialandfinalvelocities

Thecarhasabuildinspeedometer,sowecanlookatthattogetthespeed,andifwedon'tchangedirection,thenthe

velocitywillbealwayspointedinthesamedirection.Forthiscaseofacarstartingfromrest,andthenincreasingvelocity,theaccelerationwillbeapositivequantity.

Slowingdown

Whatifweaskaboutacarslowingdown.Now,our while.

Nowthemathlookslikethis:

Wenoticethattheaccelerationisnegative.

Let'sgraphicallysubtractthevelocityvectors:

Nowwe'llsubtractthemforthecarslowingdown.

a)Theobjectismovingatthesamevelocity,whichis3m/s.b)Theobjectstartsatrest,andincreasesitsvelocity,forever.c)Theobjectstartsatrest,thenincreasesitsvelocityforawhile,thenstopsmovingafter3seconds.d)Theobjectstartsatrest,thenincreasesitsvelocity,thenmovesatthesamespeedaftert=3s.e)Theobjectisnotmovingatall.

= = =av−v0t− t0

20mph−0mph2s−0s

20mph2s

= = +4.5ma9m/s

2ss−2

=+9m/sv0v = 0

= = = − =−4.5m/av−v0t− t0

0m/s−9m/s

2s−0s

9m/s

2ss2


+vx

t

+vx

t

+vx

t

+vx

t

Accelerationinthenegative

Whatifthecarstartsacceleratinginthenegativedirection?

Now,eventhespeedisincreasing,thevelocityisgettingmorenegative.Ifwedothemath,we'llseethattheaccelerationvectorpointsinthenegativedirection.

Summaryofaccelerationsignage.

Whenthesignsofanobject’svelocityandaccelerationarethesame(insamedirection),theobjectisspeedingupWhenthesignsofanobject’svelocityandaccelerationareopposite(inoppositedirections),theobjectisslowingdownandspeeddecreases

Quick Question 5Atoneparticularmoment,asubwaytrainismovingwithapositivevelocityandnegativeacceleration.Whichofthefollowingphrasesbestdescribesthemotionofthistrain?Assumethefrontofthetrainispointinginthepositivexdirection.

Quick Question 6Atoneparticularmoment,asubwaytrainismovingwithanegativevelocityandpositiveacceleration.Whichofthefollowingphrasesbestdescribesthemotionofthistrain?Assumethefrontofthetrainispointinginthepositivexdirection.

Quick Question 7Acarismovinginthenegativedirectionbutslowingdown.Whichwayistheaccelerationvectordirected?

a)Thetrainismovingforwardasitslowsdown.b)Thetrainismovinginreverseasitslowsdown.c)Thetrainismovingfasterasitmovesforward.d)Thetrainismovingfasterasitmovesinreverse.e)Thereisnowaytodeterminewhetherthetrainismovingforwardorinreverse.

a)Thetrainismovingforwardasitslowsdown.b)Thetrainismovinginreverseasitslowsdown.c)Thetrainismovingfasterasitmovesforward.d)Thetrainismovingfasterasitmovesinreverse.e)Thereisnowaytodeterminewhetherthetrainismovingforwardorinreverse.

a)Positiveb)Negaitvec)Accelerationisequalto0.


01

4

2

8

3

12

velo

city

(m/s

)

time (s)

Quick Question 8Whatistheaveragevelocityofthisobjectbetween0and3seconds?

Kinematicequations

Wecandoalotbyrearrangingtheseequations.Putting from(1)into(2)willgiveus:

or,solving(1)for ,theninsertingthatinto(2)willgiveus:

1.2.3.4.

Herewehaveanequationforvelocitywhichischangingduetoanacceleration, .Ittellsushowfastsomethingwillbegoing(andthedirection)ifhasbeenacceleratedforatime, .

Itcandetermineanobject’svelocityatanytimetwhenweknowitsinitialvelocityanditsaccelerationDoesnotrequireorgiveanyinformationaboutpositionEx:“Howfastwasthecargoingafter10secondswhileacceleratingfromrestat10m/s ”Ex:“Howlongdidittaketoreach20milesperhour”

a)0m/sb)3m/sc)4m/sd)6m/se)12m/s

1. = a = ⇒ v= +atav− v0

tv0

2. = ⇒ x− = t = ( + v)tvx−x0

t− t0x0 v

12

v0

v

3.x− = t+ ax0 v012

t2

t

4. = +2a(x− )v2 v20 x0

v= +atv0x= t = ( + v)tv 1

2v0

x= + t+ ax0 v012

t2

= +2axv2 v20

v= +atv0

a

t

2


Thisequationwilltellusthepositionofanobjectbasedontheinitialandfinalvelocities,andthetimeelapsed.Itdoesnotrequireknowing,norwillitgiveyou,theaccelerationoftheobject.

Ex:Howfardidtheduckwalkifittook10secondstoreach50milesperhourunderconstantacceleration.

GivespositionattimetintermsofinitialvelocityandaccelerationDoesn’trequireorgivefinalvelocity.Ex:“Howfarupdidtherocketgo?”

GivesvelocityattimetintermsofaccelerationandpositionDoesnotrequireorgiveanyinformationaboutthetime.Ex:“Howfastwaspennygoingwhenitreachedthebottomofthewell?”

EquationsofMotion(1-D)

Thingstobeawareof:

1.Theyareonlyforsituationswheretheaccelerationisconstant.2.Thewaywehavewrittenthemisreallyjustfor1-Dmotion.

Equation MissingVariable

Goodforfinding

x a,t,v

a x,t,v

v x,a,t

t a,x,v

x= t =v(v+ )tv0

2

x= + t+x0 v0at2

2

= +2axv2 v20

v= +atv0

x=(v+ )tv0

2

x= + t+x0 v0at2

2

= +2axv2 v20


SolvingProblems

1.Diagram:drawapicture2.Characters:Considertheproblemastory.Whoarethecharacters?3.Find:clearlylistsymbolicallywhatwe'relookingfor.4.Solve:statethebasicideabehindsolution,inafewwords(physicalprinciplesused,etc.)5.Assess:doesanswermakesense?

Ataxiissittingataredlight.Thelightturnsgreenandthetaxiacceleratesat2.5m/s for3seconds.Howfardoesittravelduringthistime?

Aparticleisatrest.Whataccelerationvalueshouldwegiveitsothatitwillbe2metersawayfromitsstartingpositionafter0.4seconds?

Asubwaytrainacceleratesstartingatx=200muniformlyuntilitreachesx=350m,atauniformaccelerationvalueof0.5m/s .

a.Ifithadaninitialvelocityof0m/s,whatwillthedurationofthisaccelerationbe?b.Ifithadaninitialvelocityof8m/s,whatwillthedurationofthisaccelerationbe?

If ,find and .Also,findthetimewhenthevelocityiszero.

Ahummingbirdjustnoticedabrightredflower.Sheacceleratesinastraightlinetowardstheflower,from1.0m/sto8.5m/satarateof3.0

.Howfardoesshetraveltoreachthefinalvelocity?

Example Problem #3:

2

Example Problem #4:

Example Problem #5:

2

Example Problem #6:

x(t) = 4−27t+ t3 v(t) a(t)

Example Problem #7:

m/s2


+x (ft)0 10 20 30 40 50

1s0s 2s 3s 4s 5s 6s 7s 8s 9s

Let'slookatthemotionofahoneybadger.Aftereachsecond,wenotewherethehoneybadgerisalongthexaxis.

t[s] x[ft]

0 0.00

1 5.00

2 10.0

3 15.0

4 17.5

5 20.0

6 22.5

7 25.0

8 35.0

9 50.0

dist

ance

[ft]

0

10

20

30

40

50

time [s]1 2 3 4 5 6 7 8 9 10

+x

t0

Hereisthepositionplotforacarintraffic.Whichofthefollowingwouldbethecorrespondingvelocitygraph?

v

t(s)0

v

t(s)0

v

t(s)0

v

t(s)0

A B C D

TraianVuia,aRomanianInventor,wantedtoreach17m/sinordertotakeoffinhisflyingmachine.Hisplanecouldaccelerateat2m/s .Theonlyrunwayhehadaccesstowas80meterslong.Willhereachthenecessaryspeed?

Plotting

Quick Question 9

a b c d

+v

t

+v

t00

+v

t0

+v

t0

Quick Question 10Whichofthefollowingvelocityvs.timegraphsrepresentsanobjectwithanegativeconstantacceleration?

Example Problem #8:

2


Derivekinematicsusingcalculus.

Wecanderivenearlyallofkinematics(forcasseswithconstantacceleration)byconsideringtherelationshipsbetweenderivativesandintegrals.Let'sbeginwiththedefinitionofacceleration:

Ifwemakethe and infinitesimallysmall, and ,thenwecanrewritethisas:

Now,wecantaketheindefiniteintegralofbothsides:

Since isassumedtobeconstant,wecanremovefromtheintegrand.Performingtheindefiniteintegrals:

where istheconstantofintegration.Todeterminetheconstant ,considertheequationwhen .Thisisthe'initialcondition',thusthevelocityatthispointwillbetheinitialvelocity: .Wethereforeobtain:

byconsideringjustthedefinitionofaccelerationandtheconceptofintegration.

Wecanlikewiseconsiderthedefinitionofinstantaneousvelocity:

Asimilaroperationleadsto:

Now,wecannotremove fromthisintegrandsinceitisnotaconstantvalue.However,wejustfiguredoutarelationbetweenvelocityandtimeabove,so:

Inthiscase, and arebothconstants.Sotheindefiniteintegralcanbesolved:

Again,wehaveaconstantofintegrationtosolvefor: .Let'sagainconsider ,i.e.theinitialcondition.When,theobjectwillbelocatedattheinitial position, .Thus .Finally,wehaveanequationfor asa

functionoftimegivenalltheinitialconditionsofpositionandvelocity:

Thisisourfundamentalquadraticequationthatdescribesthemotionofaparticleundergoingtranslationwithconstantacceleration.

a=Δv

Δt

Δv Δv dv dt

a= ⇒ dv = a dtdv

dt

∫ dv = ∫ a dt

a

v = at+ C1

C1 C t= 0v0

v = + atv0

v =dx

dt

∫ dx= ∫ v dt

v

∫ dx= ∫ ( + at) dtv0

v0 a

x= t+ a +v012

t2 C2

C2 t= 0t= 0 x x0 =C2 x0 x

x= + t+ ax0 v012

t2


v(t)

t0

x(t)

t0

x(t)

t0

velocityasafunctionoftime: Accelerationisconstant

positionasafunctionoftime .(vel.constant,accel=0)

positionasafunctionoftime

Aturtleandarabbitaretohavearace.Theturtle’saveragespeedis0.9m/s.Therabbit’saveragespeedis9m/s.Thedistancefromthestartinglinetothefinishlineis1500m.Therabbitdecidestolettheturtlerunbeforehestartsrunningtogivetheturtleaheadstart.What,approximately,isthemaximumtimetherabbitcanwaitbeforestartingtorunandstillwintherace?

Acarandamotorcycleareat at .Thecarmovesataconstantvelocity .Themotorcyclestartsatrestandaccelerateswithconstantaccelerationa.

a.Findthe wheretheymeet.b.Findtheposition wheretheymeet.c.Findthevelocityofthemotorcyclewhentheymeet.

v= +atv0 x= vt x= t+v0at2

2

v(t) x(t)

Example Problem #9:

Example Problem #10:

= 0x0 t = 0v0

tx

Thisproblemisaskingustodescribethekinematicsofthesituationinthemostgeneraltermspossible.Therearenonumbersgiven,sowemustdoeverythingusingsymbolicalgebra.First,let'smakesureweunderstandthesetup.Therearetwovehicles:acarandamotorcycle.Theycanbeconsideredparticlesmeaningtheyarepointlike.Theactionstartsatt=0.Atthistime,bothvehiclesarelocatedattheorigin.Themotorcycleisstationary,butthecarhasavelocity, .(* isjustasymbolthatcouldbeanumber,like10m/sor34.3mph.Butweleaveitasasymbolsothatwecansolvethisprobleminageneralway,applicabletoanycar!)Nowthecarwillmovefartherthanthemotorcycleatfirst.However,themotorcyclewillcatchupandovertakethecarbecauseitisaccelerating.

a)Findoutwhen,i.e.atwhattime,theyareatthesameposition.So,weneedfunctionsthattelluswhereeachvehicleislocatedatagiventime.Wecanstartwiththebasickinematicequationofmotion:

Forthecar,sincethereisnoacceleration, ,and ,thisequationsimplifiesto:

Forthemotorcycle,ithasnointitialvelocity, ,butitdoeshasanacceleration .Italsostartsfromtheorigin:

v0 v0

x= + t+ ax0 v012

t2

a= 0 = 0x0

= txcar v0

= 0v0 a

= amoto1 2


x

t(s)

x

t(s)

carmeeting time

motorcycle

Thequestionaskwhentheobjectsmeet?Thatis,whenarethexvaluesthesame.So,wecanjustsetthetwoequationsequaltoeachother.

andsolvethisfor .

.Nowwehaveanequationfor thatwecanusegivenanyaccelerationandinitialvelocity.

b)Wheredoesthisoccur?Wecanusethetimeexpressioninoneofthepreviouspositionequations.

Itshouldalsobethesameifweputinthetimeinthemotorcycle'spositionequation:

c)Whatisthespeedofthemotorcycle?Wefirstneedtofindanequationforspeedofthemotorcycles.Let'stherelationshipbetweenpositionandvelocity:

So,whentimeis ,thespeedofthemotorcyclewillbe:

Noticehowtheaccelerationtermisgone.Thespeedofthemotorcyclewhenthetwoobjectmeetisindependentofitsacceleration.That'saninterestingbitofinformationthatwouldhavebeenlostifwedidthisproblemusingnumbersinsteadofletters.

Thisplotshowsgraphicallythesituation.Wecancomparetheslopesthattheintersectionandseethattheslopeofthemotorcycleisroughlytwicethatofthecar.

= axmoto12

t2

=xcar xmoto

t= av012

t2

t

t=2v0

a

t

= = 2xcar v02v0

a

v20

a

= a = a = 2xmoto12

t212

( )2v0

a

2 v20

a

v = = atdx

dt

t=2v0a

v = at= a( )= 22v0

av0


Quick Question 11BelowisthegraphofanobjectmovingalongthexaxisDuringwhichsection(s)doestheobjecthaveaconstantvelocity?

Quick Question 12Duringwhichsection(s)istheobjectspeedingup?

Quick Question 13Duringwhichsection(s)istheobjectstandingstill?

Quick Question 14Duringwhichsection(s)istheobjectmovingtotheleft?(assumeleftisnegativexdirection.)

FreeFall

Afreelyfallingobjectisanyobjectmovingfreelyundertheinfluenceofgravityalone.Objectcouldbe:

1.Dropped=releasedfromrest2.Throwndownward3.Thrownupward

Itdoesnotdependupontheinitialmotionoftheobject.

1.Theaccelerationofanobjectinfreefallisdirecteddownward(negativedirection),regardlessoftheinitialmotion.

2.Themagnitudeoffreefallaccelerationis .

3.Wecanneglectairresistance.4.We'llchooseouryaxistobepositiveupward.5.ConsidermotionnearEarth’ssurfacefornow.

9.8m/ = gs2


A

B

C

D

E

Kinematicequationinthecaseoffreefall:

1.2.3.4.

Theyarethesame.Wejustreplaced and .

Quick Question 15Anarrowislaunchedverticallyupward.Itmovesstraightuptoamaximumheight,thenfallstotheground.Thetrajectoryofthearrowisshown.Atwhichpointofthetrajectoryisthearrow’saccelerationthegreatest?Ignoreairresistance;theonlyforceactingisgravity.

Anobjectisthrownupwardat20m/s:

a.Howlongwillittaketoreachthetopb.Howhighisthetop?c.Howlongtoreachthebottom?d.Howfastwillitbegoingwhenitreachesthebottom?

Quick Question 16Anarrowislaunchedverticallyupward.Itmovesstraightuptoamaximumheight,thenfallstotheground.Whichgraphbestrepresentstheverticalvelocityofthearrowasafunctionoftime?Ignoreairresistance;thearrowisinfreefall!.

A B C D

+v

t

+v

t00

+v

t0

+v

t0

E

+v

t0

Ifanobjectisthrownupwardfromaheight withaspeed ,whenwillithittheground?

v= − gtv0y= t = ( + v)tv 1

2v0

y= + t− gy0 v012

t2

= −2gyv2 v20

x→ y a→−g

a)pointAb)pointBc)pointCd)pointDe)pointEf)Noneofthesebecauseitisthesameeverywhere.



y0 v0


Dropawrench

Aworkerdropsawrenchdowntheelevatorshaftofatallbuilding.

a.Whereisthewrench1.5secondslater?b.Howfastisthewrenchfallingatthattime?

Arockisthrownupwardwithavelocityof49m/sfromapoint15mabovetheground.

a.Whendoestherockreachitsmaximumheight?b.Whatisthemaximumheightreached?c.Whendoestherockhittheground?

Drawposition,velocity,andaccelerationgraphsasafunctionsoftime,foranobjectthatisletgofromrestoffthesideofacliff.





Kinematics in One Dimension - James Hedberghedberg.ccnysites.cuny.edu/content/introphysics/1d... ·...

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