Post on 23-Aug-2020
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
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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.
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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.
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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+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.
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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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
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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
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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
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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+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
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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
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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
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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
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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
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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.
Example Problem #11:
Example Problem #12:
y0 v0
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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Dropawrench
Aworkerdropsawrenchdowntheelevatorshaftofatallbuilding.
a.Whereisthewrench1.5secondslater?b.Howfastisthewrenchfallingatthattime?
Arockisthrownupwardwithavelocityof49m/sfromapoint15mabovetheground.
a.Whendoestherockreachitsmaximumheight?b.Whatisthemaximumheightreached?c.Whendoestherockhittheground?
Drawposition,velocity,andaccelerationgraphsasafunctionsoftime,foranobjectthatisletgofromrestoffthesideofacliff.
Example Problem #13:
Example Problem #14:
Example Problem #15:
PHY 207 - 1d-kinematics - J. Hedberg - 2017
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