PHY 207 - 1d-kinematics - J. Hedberg - 2016
Kinematics in One Dimension
1.Introduction
1.DifferentTypesofMotionWe'lllookat:
2.Dimensionalityinphysics
3.Onedimensionalkinematics
4.Particlemodel
2.DisplacementVector
1.Displacementin1-D
3.SpeedandVelocity
1....withadirection
4.Changeinvelocity.
1.Acceleration
2.Acceleration,themath.
3.Slowingdown
4.Accelerationinthenegative
5.Summaryofaccelerationsignage.
5.Kinematicequations
1.EquationsofMotion(1-D)
6.SolvingProblems
7.Plotting
1.Twoplots
2.Themotionofwhat?
8.FreeFall
1.Dropawrench
2.Howhighwasthis?
Introduction
Motion:changeinpositionororientationwithrespectto
time.
Vectorshavegivenussomebasicideasabouthowtodescribethepositionofobjectsintheuniverse/Now,we'll
continuebyextendingthoseideastoaccountforchangesinthatposition.Ofcoursetheworldwouldbeawfully
boringifthepositionofeverythingwasconstant.
Different Types of Motion We'll look at:
Linear
Circular
Projectile
Rotational
Linearmotioninvolvesthechangeinpositionofanobjectinonedirectiononly.Anexamplewouldbeatrainona
straightsectionofthetrack.Thechangeinpositionisonlyinthehorizontaldirection.
Projectilemotionoccurswhenobjectsarelaunchedinthegravitationalfieldneartheearthssurface.They
experiencemotioninboththehorizontalandtheverticaldirections.
Circularmotionoccursinafewspecificcaseswhenanobjecttravelsinaperfectcircle.Somespecialmathcanbe
usedinthesecases.
Rotationalmotionimpliesthatthebodyinquestionisrotatingaroundanaxis.Theaxisdoesn'tnecessaryneedto
passthroughtheobject.
...oracombinationofthem.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Dimensionality in physics
Preludetoadvancedphysicsandengineering:Lateron,you'llhavetoexpandyournotionofdimensionsabit.It
won'tsimplymeanstraightorcurvy,butwillinsteadbeusedtodescribethedegreesoffreedominasystem.For
example,anorbitingbody,thoughitmovesinacirclewhichrequiresxandyvaluestodescribe,canalsobe
describedbyconsideringtheradiusandtheangleofrotationinstead.Thisisjustanothercoordinatesystem:polar
coordinates(usually:randθ ).Ifwedescribetheorbitingplanetinthissystem,andsay,it'sgoingaroundina
perfectcircle,thenthervaluedoesn'tchangeandtheθ valuebecometheonlydimensionofinterest.Let'sholdoff
onthisapproachfornow,butwhenitcomesbacklateron,welcomeitwithopenarmsbecauseitallowsformuch
morepowerfulandsimpleanalysisofsystems.
One dimensional kinematics
y
Forthecaseof1-dimensionalmotion,we'llonly
considerachangeofpositioninonedirection.
Itcouldbeanyofthethreecoordinateaxes.
x
Justadescriptionofthemotion,without
attemptingtoanalyzethecause.Todescribe
motionweneed:
z
1.CoordinateSystem(origin,
orientation,scale)
2.theobjectwhichismoving
-20
-x (m)
0
20
40
60
80
100
+x (m)
1dkinematicswillbeourstartingpoint.Itisthemoststraightforwardandeasiestmathematicallytodealwithsince
onlyonepositionvariablewillbechangingwithrespecttotime.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Particle model
We'llneedtouseanabstraction:
Allrealworldobjectstakeupspace.We'llassume
thattheydon't.Inotherwords,thingslikecars,cats,
andducksarejustpoint-likeparticles.
Thisisourfirstrealabstraction.Again,sincewearetryingtopredicteverything,wewouldliketofigureoutthe
rulesthatdescribehowanyobjectwouldmove.Takeatrainforexample.Ifweaskedaquestionlike"whendoes
theCtrainenter59thstreetstation?",anaturalfollow-upwouldbe"well,doyoumeanthefrontofthetrain,orthe
middleofthetrain,ortheendofthetrain?Eachoftheseanswersmightbedifferentbyafewseconds.
Howdowedealwiththis?Byconsideringthetraintobea'point',wecanneglecttheactuallengthofthetrainand
focusonwhat'smoreintersting:howthetrainmoves.
Thegoalistofindtheunderlyingphysicsthatdescribesalltrains.Oncewedothat,thenwecanimproveour
modelbyincludinginformationaboutthelengthoftheindividualtrainweareinterestedin.
Displacement Vector
∆x
Toquantifythemotion,we'llstartbydefining
thedisplacementvector.
x
Δx = x − x 0
Inthecaseofourwanderingbug,thiswouldbe
thedifferencebetweenthefinalpositionandthe
initialposition.
ThisfigureshowsthedisplacementvectorΔx.Thismightbedifferentthanthedistancetraveledbythebug
(showninthedottedline).
Displacement in 1-D
-20
0
20
40
60
80
-x (m)
100
+x (m)
Here'sacarthatmovesfromx 0toxcreatingadisplacementvectorof:
Δx = x − x 0 = 60m − 0m = 60m
Thecarthenreversestox = −20.
-20
0
20
40
60
80
-x (m)
100
+x (m)
TheleadstoadisplacementvectorofΔx = −80m.
Aboutnotation.Δx("deltax")referstothechangeinx.Thatis,differencebetweenafinalandinitialvalue:
Δx = x − x0
Or,inwords,thefinalxpositionminustheoriginalxpositionisequaltothechangeinx.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Speed and Velocity
AverageSpeed ≡
Distanceinagiventime
Elapsedtime
The'elapsedtime'isdeterminedinthesamewayasthedistance:
Δt = t − t0 .
Again,t0 isthestartingtime,andtisthefinaltime.
Example Problem #1:
TakingtheAtrainbetween59thand125thtakes
about8minutes.TheC,whichisalocal,takes12
minutes(onagoodday).Findtheaveragespeed
forbothofthesetrips.
...with a direction
Calculatingtheaveragespeeddidn'ttellusanythingaboutthedirectionof
travel.Forthis,we'llneedaveragevelocity.
AverageVelocity ≡
Displacement
Elapsedtime
Inmathematicalterms:
v̄¯¯ ≡
x − x0
Δx
=
t − t0
Δt
(SIunitsofaveragevelocityarem/s)
Inone-dimension,velocitycaneitherbeinthepositiveornegativedirection.
Quick Question 1
Thisisagraphshowingthepositionofanobjectwithrespectto
time.Whichchoicebestdescribesthismotion?
x(m)
3
a)Theobjectismovingat0.5m/sinthe+xdirection.
b)Theobjectismovingat1.0m/sinthe+xdirection.
c)Theobjectismovingat2.0m/sinthe+xdirection.
d)Theobjectisnotmovingatall.
2
1
0
2
4
6
t(s)
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Quick Question 2
Thisisagraphshowingthepositionofanobjectwithrespectto
time.Whichchoicebestdescribesthismotion?
x(m)
3
a)Theobjectismovingat0.5m/sinthe+xdirection.
b)Theobjectismovingat1.0m/sinthe+xdirection.
c)Theobjectismovingat2.0m/sinthe+xdirection.
d)Theobjectisnotmovingatall.
2
1
0
2
4
6
t(s)
ThinkingabouttheAtrain,it'sclearthatitsspeedandvelocitystayed
essentiallyconstantbetween59thand125thideally).However,theCtrain
hadtostartandstopat7stations.Toquantify,thisdifferenceinmotion,
we'llneedtointroducetheconceptofinstantaneousvelocity.
Ifweimaginemakingmanymeasurementsofthevelocityoverthecourseof
thetravel,byreducingtheΔxweareconsidering,thenwecanbegintosee
howwecanmoreaccuratelyassessthemotionofthetrain.
Theconceptofinstantaneousvelocityinvolvesconsideringaninfinitesimally
smallsectionofthemotion:
v = lim
Δt→0
Δx
dx
=
Δt
dt
Thiswillenableustotalkaboutthevelocityataparticle'spositionrather
thanforanentiretrip.
Ingeneral,thisiswhatwe'llmeanwhenwesay'velocity'or'speed'.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Quick Question 3
x(m)
4
3
2
1
0
2
4
6
8 t(s)
Atwhichofthefollowingtimesisthespeedofthisobjectthegreatest?
a)t = 0
b)t = 2 s
c)t = 4 s
d)t = 6 s
e)t = 8 s
Change in velocity.
Naturally,inordertobeginmoving,anobjectmustchangeitsvelocity.
Here'sagraphofabicyclistridingataconstantvelocity.(Inthiscaseit's10
m/s)
0s
1s
2s
0
10
20
30
+x (m)
40
Now,here'sagraphofthesamebicyclistridingandchanginghisvelocity
duringthemotion
0s
1s
0
10
2s
20
3s
30
40
+x (m)
Intheuppermotiongraph,noticehowthelengthofthedisplacementvector d ⃗isthesameateachintervalintime.
Meaning,thatafter1secondhaspassed,thedisplacementis10m,afteranothersecondpasses,another10meters
displacementhasoccurred,makingthetotaldisplacementequalto20m.Thisismotionataconstantvelocity.This
alsoapparentinthelengthofthevelocityvectorsateachpoint.Theyarealwaysthesame.
Inthebottomgraph,thedisplacement,andvelocityvectors,changeeachtimetheyaremeasured.Thisis
representativeofmotionwithnon-constantvelocity.Thevelocityischangingastimemoveson.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Acceleration
Thischangeinvelocitywe'llcallacceleration,andwecandefineitinavery
similarwaytoourdefinitionofvelocity:
ā¯¯ =
v − v0
Δv
=
t − t0
Δt
Again,inthiscasewe'retalkingaboutaverageacceleration.
Example Problem #2:
Att = 0 ,theAtrainisatrestat59thstreet.5secondslater,it'straveling
northat19meterspersecond.Whatistheaverageaccelerationduring
thistimeinterval?
Ifweconsideredthesameverysmallchangeintime,theinfinitesimal
change,thenwecouldtalkaboutinstantaneousacceleration
Δv
dv
=
Δt→0 Δt
dt
a = lim
TheSIunitsofaccelerationaremeterspersecondpersecond,orms−2.
That'sprobablyalittlebitofaweirdunit,but,itmakessensetothinkabout
likethis:
(m
s)
s
or
vel
s
velocity (m/s)
Quick Question 4
Thisisagraphshowingthevelocityofanobjectwithrespectto
time.Whichchoicebestdescribesthismotion?
9
6
3
0
1
2
3
time (s)
a)Theobjectismovingatthesamevelocity,whichis3
m/s.
b)Theobjectstartsatrest,andincreasesitsvelocity,for
ever.
c)Theobjectstartsatrest,thenincreasesitsvelocityfora
while,thenstopsmovingafter3seconds.
d)Theobjectstartsatrest,thenincreasesitsvelocity,
thenmovesatthesamespeedaftert=3s.
e)Theobjectisnotmovingatall.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Acceleration, the math.
Toquantifytotheaccelerationofamovingbody,saythiscar,we'llneedto
knowitsinitialandfinalvelocities
Thecarhasabuildinspeedometer,sowecanlookatthatto
getthespeed,andifwedon'tchangedirection,thenthe
velocitywillbealwayspointedinthesamedirection.
Play/Pause
Forthiscaseofacarstartingfromrest,andthenincreasingvelocity,the
accelerationwillbeapositivequantity.
ā¯¯ =
v − v0
20mph − 0mph
20mph
=
=
t − t0
2s − 0s
2s
ā¯¯ =
9m/s
2s
= +4.5ms−2
Slowing down
Whatifweaskaboutacarslowingdown.Now,ourv 0 = +9m/swhile
v = 0.
Play/Pause
ā¯¯ =
Nowthemathlookslikethis:
0m/s − 9m/s
9m/s
v − v0
=
=−
= −4.5m/ s2
t − t0
2s − 0s
2s
Wenoticethattheaccelerationisnegative.
Let'sgraphicallysubtractthevelocityvectors:
Nowwe'llsubtractthemforthecarslowingdown.
Acceleration in the negative
Whatifthecarstartsacceleratinginthenegativedirection?
Now,eventhespeedisincreasing,thevelocityisgettingmorenegative.
Ifwedothemath,we'llseethattheaccelerationvectorpointsinthenegative
direction.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Summary of acceleration signage.
Whenthesignsofanobject’svelocityandaccelerationarethesame(insame
direction),theobjectisspeedingup
Whenthesignsofanobject’svelocityandaccelerationareopposite(in
oppositedirections),theobjectisslowingdownandspeeddecreases
+v x
+v x
+v x
+v x
t
t
t
t
Quick Question 5
Atoneparticularmoment,asubwaytrainismovingwithapositivevelocityandnegativeacceleration.
Whichofthefollowingphrasesbestdescribesthemotionofthistrain?Assumethefrontofthetrainis
pointinginthepositivexdirection.
a)Thetrainismovingforwardasitslowsdown.
b)Thetrainismovinginreverseasitslowsdown.
c)Thetrainismovingfasterasitmovesforward.
d)Thetrainismovingfasterasitmovesinreverse.
e)Thereisnowaytodeterminewhetherthetrainismovingforwardorinreverse.
Quick Question 6
Atoneparticularmoment,asubwaytrainismovingwithanegativevelocityandpositiveacceleration.
Whichofthefollowingphrasesbestdescribesthemotionofthistrain?Assumethefrontofthetrainis
pointinginthepositivexdirection.
a)Thetrainismovingforwardasitslowsdown.
b)Thetrainismovinginreverseasitslowsdown.
c)Thetrainismovingfasterasitmovesforward.
d)Thetrainismovingfasterasitmovesinreverse.
e)Thereisnowaytodeterminewhetherthetrainismovingforwardorinreverse.
velocity (m/s)
Example Problem #3:
Whatistheaveragevelocityofthisobjectbetween
0and3seconds?
12
8
4
0
1
2
3
time (s)
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Kinematic equations
1. ā¯¯ = a =
2. ¯v̄¯ =
x − x0
t − t0
v − v0
t
⇒ v = v0 + at
1
⇒ x − x0 = ¯v̄¯t = (v0 + v)t
2
Wecandoalotbyrearrangingtheseequations.
Puttingvfrom(1)into(2)willgiveus:
1
3.x − x0 = v0 t + at2
2
or,solving(1)fort,theninsertingthatinto(2)willgiveus:
4. v2 = v20 + 2a(x − x0 )
1. v = v0 + at
2. x = ¯v̄¯t = 12 (v0 + v)t
3. x = x0 + v0 t + 12 at2
4. v2 = v20 + 2ax
v = v0 + at
Herewehaveanequationforvelocitywhichischangingduetoan
acceleration,a.
Ittellsushowfastsomethingwillbegoing(andthedirection)ifhasbeen
acceleratedforatime,t.
Itcandetermineanobject’svelocityatanytimetwhenweknowits
initialvelocityanditsacceleration
Doesnotrequireorgiveanyinformationaboutposition
Ex:“Howfastwasthecargoingafter10secondswhileaccelerating
fromrestat10m/s2”
Ex:“Howlongdidittaketoreach20milesperhour”
x=
(v + v0 )t
2
Thisequationwilltellusthepositionofanobjectbasedontheinitialand
startingvelocities,andthetimeelapsed.
Itdoesnotrequireknowing,norwillitgiveyou,theaccelerationofthe
object.
Ex:Howfardidtheduckwalkifittook10secondstoreach50miles
perhourunderconstantacceleration.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
x = x0 + v0 t +
at2
2
Givespositionattimetintermsofinitialvelocityandacceleration
Doesn’trequireorgivefinalvelocity.
Ex:“Howfarupdidtherocketgo?”
v2 = v20 + 2ax
Givesvelocityattimetintermsofaccelerationandposition
Doesnotrequireorgiveanyinformationaboutthetime.
Ex:“Howfastwaspennygoingwhenitreachedthebottomofthe
well?”
Equations of Motion (1-D)
Thingstobeawareof:
1.Theyareonlyforsituationswheretheaccelerationisconstant.
2.Thewaywehavewrittenthemisreallyjustfor1-Dmotion.
Missing
Variable
Goodfor
finding
x
a,t,v
a
x,t,v
at2
v
2
x,a,t
Equation
v = v0 + at
x=
(v + v0 )t
2
x = x0 + v0 t +
v2 = v20 + 2ax
t
a,x,v
Solving Problems
1.Diagram:drawapicture
2.Characters:Considertheproblemastory.Whoarethecharacters?
3.Find:clearlylistsymbolicallywhatwe'relookingfor.
4.Solve:statethebasicideabehindsolution,inafewwords
(physicalprinciplesused,etc.)
5.Assess:doesanswermakesense?
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Example Problem #4:
Ataxiissittingataredlight.Thelightturnsgreenandthetaxi
acceleratesat2.5m/s2for3seconds.Howfardoesittravelduringthis
time?
Example Problem #5:
Aparticleisatrest.Whataccelerationvalueshouldwegiveitsothatit
willbe2metersawayfromitsstartingpositionafter0.4seconds?
Example Problem #6:
Asubwaytrainacceleratesstartingatx=200muniformlyuntilit
reachesx=350m,atauniformaccelerationvalueof0.5m/s2.
a.Ifithadaninitialvelocityof0m/s,whatwillthedurationofthis
accelerationbe?
b.Ifithadaninitialvelocityof8m/s,whatwillthedurationofthis
accelerationbe?
Example Problem #7:
Ifx(t) = 4 − 27t + t3 ,findv(t)anda(t).Also,findthetimewhenthe
velocityiszero.
Example Problem #8:
Ahummingbirdjustnoticedabrightredflower.Sheacceleratesina
straightlinetowardstheflower,from1.0m/sto8.5m/satarateof3.0
m/s2 .Howfardoesshetraveltoreachthefinalvelocity?
Example Problem #9:
TraianVuia,aRomanianInventor,wantedto
reach17m/sinordertotakeoffinhisflying
machine.Hisplanecouldaccelerateat2m/s2.The
onlyrunwayhehadaccesstowas80meterslong.
Willhereachthenecessaryspeed?
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Plotting
0s
1s
0
2s
3s 4s 5s 6s 7s
10
20
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
8s
Let'slookatthe
motionofahoney
badger.
9s
30
40
50
+x (ft)
Aftereachsecond,
wenotewherethe
honeybadgeris
alongthexaxis.
50
distance [ft]
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
time [s]
Two plots
+x
+x
4
1.4
3
1.0
velocity [m/s]
position [m]
5
2
1
0
t
-1
0.8
0.4
0.0
t
-0.4
-0.8
-2
-1.0
-3
-4
-5
-1.4
0
0
1
2
3
4 5 6 7
times [s]
8
1
2
3
9 10
4 5 6 7
times [s]
8
9 10
Quick Question 7
Hereisthepositionplotforacarintraffic.Whichofthe
followingwouldbethecorrespondingvelocitygraph?
+x
+v
0
t
+v
0
t
a
+v
0
t
b
+v
0
t
c
0
t
d
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Quick Question 8
Whichofthefollowingvelocityvs.timegraphsrepresentsanobjectwithanegativeconstant
acceleration?
v
0
A
v
t(s)
0
B
v
t(s)
0
C
v
t(s)
0
D
t(s)
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Derivekinematicsusingcalculus.
Wecanderivenearlyallofkinematics(forcasseswithconstantacceleration)byconsideringtherelationships
betweenderivativesandintegrals.Let'sbeginwiththedefinitionofacceleration:
a=
Δv
Δt
Ifwemakethe ΔvandΔvinfinitesimallysmall,dvanddt,thenwecanrewritethisas:
a=
dv
⇒ dv = a dt
dt
Now,wecantaketheindefiniteintegralofbothsides:
∫ dv = ∫ a dt
Since aisassumedtobeconstant,wecanremovefromtheintegrand.Performingtheindefiniteintegrals:
v = at + C1
whereC1 istheconstantofintegration.Todeterminetheconstant C ,considertheequationwhen t = 0.Thisisthe
'initialcondition',thusthevelocityatthispointwillbetheinitialvelocity:v0 .Wethereforeobtain:
v = v0 + at
byconsideringjustthedefinitionofaccelerationandtheconceptofintegration.
Wecanlikewiseconsiderthedefinitionofinstantaneousvelocity:
v=
dx
dt
Asimilaroperationleadsto:
∫ dx = ∫ v dt
Now,wecannotremovevfromthisintegrandsinceitisnotaconstantvalue.However,wejustfiguredouta
relationbetweenvelocityandtimeabove,so:
∫ dx = ∫ (v0 + at) dt
Inthiscase,v0 andaarebothconstants.Sotheindefiniteintegralcanbesolved:
1
x = v0 t + at2 + C2
2
Again,wehaveaconstantofintegrationtosolvefor:C2 .Let'sagainconsider t = 0,i.e.theinitialcondition.When
t = 0,theobjectwillbelocatedattheinitial xposition, x0 .ThusC2 = x0 .Finally,wehaveanequationforxasa
functionoftimegivenalltheinitialconditionsofpositionandvelocity:
1
x = x0 + v0 t + at2
2
Thisisourfundamentalquadraticequationthatdescribesthemotionofaparticleundergoingtranslationwith
constantacceleration.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
v(t)
x(t)
x(t)
0
0
0
t
v = v0 + at
t
t
x = vt
x = v0 t +
at2
2
velocityasafunctionof positionasafunctionof
positionasafunctionof
time:v(t)
timex(t).
time
Accelerationisconstant (vel.constant,accel=0)
Example Problem #10:
Aturtleandarabbitaretohavearace.Theturtle’saveragespeedis0.9
m/s.Therabbit’saveragespeedis9m/s.Thedistancefromthestarting
linetothefinishlineis1500m.Therabbitdecidestolettheturtlerun
beforehestartsrunningtogivetheturtleaheadstart.What,
approximately,isthemaximumtimetherabbitcanwaitbeforestarting
torunandstillwintherace?
Example Problem #11:
Acarandamotorcycleareatx0 = 0 att = 0 .Thecarmovesata
constantvelocityv0 .Themotorcyclestartsatrestandaccelerateswith
constantaccelerationa.
a.Findthetwheretheymeet.
b.Findthepositionxwheretheymeet.
c.Findthevelocityofthemotorcyclewhentheymeet.
Thisproblemisaskingustodescribethekinematicsofthesituationinthemostgeneraltermspossible.There
arenonumbersgiven,sowemustdoeverythingusingsymbolicalgebra.First,let'smakesureweunderstand
thesetup.Therearetwovehicles:acarandamotorcycle.Theycanbeconsideredparticlesmeaningtheyare
pointlike.Theactionstartsatt=0.Atthistime,bothvehiclesarelocatedattheorigin.Themotorcycleis
stationary,butthecarhasavelocity,v0 .(*v0 isjustasymbolthatcouldbeanumber,like10m/sor34.3mph.
Butweleaveitasasymbolsothatwecansolvethisprobleminageneralway,applicabletoanycar!)Nowthe
carwillmovefartherthanthemotorcycleatfirst.However,themotorcyclewillcatchupandovertakethecar
becauseitisaccelerating.
a)Findoutwhen,i.e.atwhattime,theyareatthesameposition.So,weneedfunctionsthattelluswhereeach
vehicleislocatedatagiventime.Wecanstartwiththebasickinematicequationofmotion:
1
x = x0 + v0 t + at2
2
Forthecar,sincethereisnoacceleration,a = 0,andx0 = 0,thisequationsimplifiesto:
xcar = v0 t
Forthemotorcycle,ithasnointitialvelocity,v0 = 0,butitdoeshasanaccelerationa.Italsostartsfromthe
origin:
moto
1
= a2
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
1
xmoto = at2
2
Thequestionaskwhentheobjectsmeet?Thatis,whenarethexvaluesthesame.So,wecanjustsetthetwo
equationsequaltoeachother.
xcar = xmoto
1
v0 t = at2
2
andsolvethisfort.
t=
2v0
a
.Nowwehaveanequationfortthatwecanusegivenanyaccelerationandinitialvelocity.
b)Wheredoesthisoccur?Wecanusethetimeexpressioninoneofthepreviouspositionequations.
xcar = v0
v20
2v0
=2
a
a
Itshouldalsobethesameifweputinthetimeinthemotorcycle'spositionequation:
2
v0
1
1 2v0 2
xmoto = at2 = a(
) =2
a
a
2
2
c)Whatisthespeedofthemotorcycle?Wefirstneedtofindanequationforspeedofthemotorcycles.Let'sthe
relationshipbetweenpositionandvelocity:
v=
So,whentimeist =
dx
= at
dt
2v0
,thespeedofthemotorcyclewillbe:
a
v = at = a (
2v0
) = 2v0
a
Noticehowtheaccelerationtermisgone.Thespeedofthemotorcyclewhenthetwoobjectmeetisindependent
ofitsacceleration.That'saninterestingbitofinformationthatwouldhavebeenlostifwedidthisproblemusing
numbersinsteadofletters.
Thisplotshowsgraphicallythesituation.Wecancomparetheslopesthatthe
intersectionandseethattheslopeofthemotorcycleisroughlytwicethatofthe
car.
x
motorcycle
car
meeting time
t(s)
Page 17
PHY 207 - 1d-kinematics - J. Hedberg - 2016
Quick Question 9
Belowisthegraphofanobjectmovingalongthexaxis
Duringwhichsection(s)doestheobjecthaveaconstantvelocity?
Quick Question 10
Duringwhichsection(s)istheobjectspeedingup?
Quick Question 11
Duringwhichsection(s)istheobjectstandingstill?
Quick Question 12
Duringwhichsection(s)istheobjectmovingtotheleft?(assumeleftisnegativexdirection.)
Free Fall
Afreelyfallingobjectisanyobjectmovingfreelyundertheinfluenceof
gravityalone.
Objectcouldbe:
1.Dropped=releasedfromrest
2.Throwndownward
3.Thrownupward
Itdoesnotdependupontheinitialmotionoftheobject.
1.Theaccelerationofanobjectinfreefallisdirecteddownward
(negativedirection),regardlessoftheinitialmotion.
2.Themagnitudeoffreefallaccelerationis 9.8m/ s2 = g .
3.Wecanneglectairresistance.
4.We'llchooseouryaxistobepositiveupward.
5.ConsidermotionnearEarth’ssurfacefornow.
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Kinematicequationinthecaseoffreefall:
1. v = v0 − gt
2. y = ¯v̄¯t = 12 (v0 + v)t
3. y = y0 + v0 t − 12 gt2
4. v2 = v20 − 2gy
Theyarethesame.Wejustreplacedx → yanda → −g.
Quick Question 13
C
B
D
A
E
Anarrowislaunchedverticallyupward.Itmovesstraightuptoamaximumheight,then
fallstotheground.Thetrajectoryofthearrowisshown.Atwhichpointofthetrajectory
isthearrow’saccelerationthegreatest?Ignoreairresistance;theonlyforceactingis
gravity.
a)pointA
b)pointB
c)pointC
d)pointD
e)pointE
f)Noneofthesebecauseitisthesameeverywhere.
Example Problem #12:
Anobjectisthrownupwardat20m/s:
a.Howlongwillittaketoreachthetop
b.Howhighisthetop?
c.Howlongtoreachthebottom?
d.Howfastwillitbegoingwhenitreachesthebottom?
Quick Question 14
Anarrowislaunchedverticallyupward.Itmovesstraightuptoamaximumheight,thenfallstothe
ground.Whichgraphbestrepresentstheverticalvelocityofthearrowasafunctionoftime?Ignoreair
resistance;thearrowisinfreefall!.
+v
+v
0
t
A
+v
0
t
B
+v
0
t
C
+v
0
t
D
0
t
E
Example Problem #13:
Ifanobjectisthrownupwardfromaheighty0 withaspeedv0 ,when
willithittheground?
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PHY 207 - 1d-kinematics - J. Hedberg - 2016
Example Problem #14:
Drop a wrench
Aworkerdropsawrenchdowntheelevatorshaftofatallbuilding.
a.Whereisthewrench1.5secondslater?
b.Howfastisthewrenchfallingatthattime?
Example Problem #15:
Arockisthrownupwardwithavelocityof49m/sfromapoint15m
abovetheground.
a.Whendoestherockreachitsmaximumheight?
b.Whatisthemaximumheightreached?
c.Whendoestherockhittheground?
Example Problem #16:
Drawposition,velocity,andaccelerationgraphsasafunctionsoftime,
foranobjectthatisletgofromrestoffthesideofacliff.
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