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[SHIVOK SP211] August 26, 2015
Page1
CH 2
Motion along a Straight Line
I. Motion
A. Wefindmovingobjectsallaroundus.
B. Thestudyofmotioniscalled_________________________;itistheclassificationandcomparisonofmotions.
1. Examples:
a) TheEarthorbitsaroundtheSun
b) AroadwaymoveswithEarth’srotation
2. Inthischapter,wewillstudymotionthattakesplaceinastraightline.
3. Forcescausemotion.Wewillfindout,asaresultofapplicationofforce,iftheobjectsspeedup,slowdown,ormaintainthesamerate.(Forthischapter,wediscussonlythemotionitself,nottheforcesthatcauseit.)
4. Themovingobjectherewillbeconsideredasaparticle.Ifwedealwithastiff,extendedobject,wewillassumethatallparticlesonthebodymoveinthesamefashion.Wewillstudythemotionofaparticle,whichwillrepresenttheentirebody.
a) Aparticleiseither:
(1) Apoint‐likeobject(suchasanelectron)
(2) Oranobjectthatmovessuchthateachparttravelsinthesamedirectionatthesamerate(norotationorstretching)
5. Exampleofmotioninstraightline
[SHIVOK SP211] August 26, 2015
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II. Positionanddisplacement
A. Thelocationofanobjectisusuallygivenintermsofastandardreferencepoint,calledtheorigin.Thepositivedirectionistakentobethedirectionwherethecoordinatesareincreasing,andthenegativedirectionasthatwherethecoordinatesaredecreasing.
B. Achangeinthecoordinatesofthepositionofthebodydescribesthe_______________ofthebody:∆xisthechangeinx,(finalposition)–(initialposition)
C. Forexample,ifthex‐coordinateofabodychangesfromx1tox
2,thenthe
displacement,
D. Displacementisa________________________quantity.Thatis,aquantitythathasbothmagnitudeanddirectioninformation.
E. Anobject’sdisplacementisx=‐4mmeansthattheobjecthasmovedtowardsdecreasingx‐axisby4m.Thedirectionofmotion,here,istowarddecreasingx.
F. Theactualdistancecoveredisirrelevant
G. Exampleproblem:
1. Herearethreepairsofinitialandfinalpositions,respectively,alongthex‐axis.Whichpairsgiveanegativedisplacement?
(a) ‐3m, +5m ; (b) ‐3m, ‐7m ; (c) 7m, ‐3m
[SHIVOK SP211] August 26, 2015
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III. AverageVelocityandAverageSpeed
A. Acommonwaytodescribethemotionofanobjectistoshowagraphofthepositionasafunctionoftime.
B. Averagevelocity,orvavg,isdefinedasthedisplacementoverthetime
duration.
1. Averagevelocityhasunitsof(distance)/(time)
2. Meterspersecond,m/s
C. Theaveragevelocityhasthesamesignasthedisplacement
[SHIVOK SP211] August 26, 2015
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D. Themagnitudeoftheslopeofthex‐tgraphgivestheaveragevelocity
1. Here,theaveragevelocityis:
E. AverageSpeed
1. Averagespeedistheratioofthetotaldistancetraveledtothetotaltimeduration.Itisascalarquantity,anddoesnotcarryanysenseofdirection.
2. Averagespeedisalwayspositive(nodirection)
F. ExampleProblems:
1.
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4.
Note: Notice savg is not always the same as the magnitude of vavg.
IV. InstantaneousVelocityandSpeed
A. Theinstantaneousvelocityofaparticleataparticularinstantisthevelocityoftheparticleatthatinstant.
B. Obtainedfromaveragevelocitybyshrinking∆t;heretapproachesalimitingvalue:
1. v,theinstantaneousvelocity,istheslopeofthetangentoftheposition‐timegraphatthatparticularinstantoftime.
C. Velocityisavectorquantity(units(distance)/(time))andhaswithitanassociatedsenseofdirection.Thesignofthevelocityrepresentsitsdirection.
D. Speedisthemagnitudeof(instantaneous)velocity
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V. Averageandinstantaccelerations
A. Averageaccelerationisthechangeofvelocityoverthechangeoftime.
B. Assuch,
1. Herethevelocityisv1attimet
1,andthevelocityisv
2attimet
2.
C. Theinstantaneousaccelerationisdefinedas:
1. Slopeoftangentlineofthevelocityvs.timegraph
D. Intermsofthepositionfunction,theaccelerationcanbedefinedas:
E. TheSIunitsforaccelerationare________________.
F. Ifaparticlehasthesamesignforvelocityandacceleration,thenthatparticleisspeedingup.
G. Conversely,ifaparticlehasoppositesignsforthevelocityandacceleration,thentheparticleisslowingdown.
H. Ourbodiesoftenreacttoaccelerationsbutnottovelocities.Afastcaroftendoesnotbothertherider,butasuddenbrakeisfeltstronglybytherider.Thisiscommoninamusementcarrides,wheretherideschangevelocitiesquicklytothrilltheriders.
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I. ThemagnitudeofaccelerationfallingneartheEarth’ssurfaceis
9.8m/s2,andisoftenreferredtoasg.
1. Onarollercoaster,youmightexperiencebriefaccelerationsofupto3g,whichis(3)*(9.8m/s2),orabout29m/s2,whichisathrill.Anun‐trainedindividualcanblackoutbetween4and6g,particularlyifthisispulledsuddenly.
J. Sampleproblems:
1. Overashortintervalneartimet=0thecoordinateofanautomobileinmetersisgivenbyx(t)=27t–4.0t3,wheretisinseconds.Attheendof1.0stheaccelerationoftheautois:A)27m/s2B)4.0m/s2C)–4.0m/s2D)–12m/s2E)–24m/s2
2. Overashortinterval,startingattimet=0,thecoordinateofanautomobileinmetersisgivenbyx(t)=27t–4.0t3,wheretisinseconds.Themagnitudesoftheinitial(att=0)velocityandaccelerationoftheautorespectivelyare:A) 0; 12 m/s2
B) 0; 24 m/s2
C) 27 m/s; 0 D) 27 m/s; 12 m/s2
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VI. Constantacceleration
A. Whentheaccelerationisconstant,itsaverageandinstantaneousvaluesarethesame…so:
means that Eqn (2‐11) 1. Here,velocityatt=0isv
o.
B. Similarly,,whichmeansthat
C. Alsovavg=½(vo+v)=>vavg=½vo+½vvavg=vo+½at
D. ThusX=Xo+(vo+½at)tEqn(2‐15)
E. EliminatingtfromtheaboveboxedEquationsgivesus
Eqn(2‐16)
F. Eliminatingafromequations2‐11and2‐15givesusoreliminatingvogivesusEqn’s2‐17and2‐18listedinthetablebelow.Proofscanbeshowninmyofficeifyouareinterested.
Can only be used for CONSTANT
Acceleration!
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G. Thefirsttwoequationscanbeobtainedbyintegratingaconstantaccelerationsoneverneedtobememorized.
1. Theotheradditionalusefulformsallcomefromalgebrarearrangementsofthefirsttwoequationsdependingontheknownandmissingvalues.
a) NowunderVF=VO+at.Squarebothsides.YougetVF2=(VO+at)(VO+at)whichusingthefoilmethod=
b) VO2+2aVOt+a2t2.NowifIfactoroutthe2anoticeIgetVF2=VO2+2a(VOt+(1/2)at2).Solookaboutisn’tthepartinparenthesisjustDX.Sodon’tyouhaveVF2=VO2+2a(DX).Hmmmnomemorizationrequiredhereeither.Equation2‐16learned.Threeof5done.
c) Nowlookatthefirstequationifwearenotgivena,wecanrearrangethefirstequationforathensubstitutethatequationintothesecondequationlikethis.
d) a=(Vf‐Vo)/tXf‐Xo=VOt+(1/2)t2(Vf‐Vo)/t.Thenoneofthet’scancelsoIhaveVot+(1/2)Vft‐(1/2)Vot.SoifIfactorout½IgetXf‐Xo=½(Vo+Vf)t.Fourthequationlearned.Nothingmemorized.
e) FINALLYifIgobacktothefirstequationandassumetheydidnotgivemeVo,IrearrangeforVo=Vf‐at.
f) SubstitutethatintothefourthequationandIgetXf‐Xo=½(Vf‐at+Vf)t,thusfactorout…Xf‐Xo=Vft‐(1/2)at2.Fifthequationlearned.Nothingmemorized.
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H. SampleProblems:
1. Howfardoesacartravelin6sifitsinitialvelocityis2m/sanditsaccelerationis2m/s2intheforwarddirection?
A) 12 m B) 14 m C) 24 m D) 36 m E) 48 m
2. Adragracingcarstartsfromrestatt=0andmovesalongastraightlinewithvelocitygivenbyv=bt2,wherebisaconstant.Theexpressionforthedistancetraveledbythiscarfromitspositionatt=0is?
3. Thefollowingequationsgivethepositionofx(t)ofaparticleinfoursituations.Forwhichofthesescenariosdothe“big5equations”apply?
a) ( ) 3 4x t t
b) 3 2( ) 5 4 6x t t t
c) 2
2 4( )x t
t t
d) 2( ) 5 3x t t
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VII. Free‐FallAcceleration
A. InthiscaseobjectsclosetotheEarth’ssurfacefalltowardstheEarth’ssurfacewithnoexternalforcesactingonthemexceptfortheirweight.
B. Usetheconstantaccelerationmodelwith“a”replacedby“‐g”,whereg=9.8m/s2formotionclosetotheEarth’ssurface.NOTICEthatgdoesnotequal‐9.8m/s2BUTa=‐g!
1. Interestingwebsitewhichwillgiveyoutheactualgvalueforyourgeographiclocationis:http://www.physicsclassroom.com/class/circles/u6l3e.cfm.
a) ThevalueofgforAnnapolis,MDis9.80171m/s2.
b) Interestingly,thevalueofgforHonnolulu,HIis9.78452m/s2.
C. Inaddition,itisimportanttonotetheaccelerationisnotalwaysaconstant=9.80m/s2(thisisonlytrueinFree‐fallacceleration).
D. Invacuum,afeatherandanapplewillfallatthesamerate.Youcanseethisforyourselfat:
http://www.youtube.com/watch?v=_XJcZ‐KoL9o
E. FreefallQuantitative:
1. ChoosepositiveYasupwards(a=‐g.AgainNotegisnon‐negative!)
Vx =
X =
V2x =
Vy =
Y =
V2y =
Monday’s Lesson Review
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VIII. Exampleproblems:
A. Boyontopofbuildingreleasesfromrestarock,whichstrikestheground3.5seclatter.Howtallisthebuilding?Whatisthevelocityatthebottompicosecondsbeforeitstruck?
Solution: Y =
h =
V=
B. Abatterhitstheball40mverticallyupward.Whatisaccelerationbefore,at,andafterballreachesthemaximumheight?Whatisvelocityatthemaximumheight?Whatistheinitialandfinalvelocity?Howlongistheballinflight?
Solution:
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C. Astoneisthrownverticallyupwardwithaninitialspeedof19.5m/s.Itwillrisetoamaximumheightof:A) 4.9 m B) 9.8 m C) 19.4 m D) 38.8 m E) none of these
Solution:
D. Aprojectileisshotverticallyupwardwithagiveninitialvelocity.Itreachesamaximumheightof100m.If,onasecondshot,theinitialvelocityisdoubledthentheprojectilewillreachamaximumheightof:A) 70.7 m B) 141.4 m C) 200 m D) 241 m E) 400 m
Solution:
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E. Anelevatorismovingupwardwithconstantacceleration.Thedashedcurveshowsthepositionyoftheceilingoftheelevatorasafunctionofthetimet.Attheinstantindicatedbythedot,aboltbreakslooseanddropsfromtheceiling.Whichcurvebestrepresentsthepositionoftheboltasafunctionoftime?
A) A B) B C) C D) D E) E
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IX. GraphicalIntegrationinMotionAnalysis
A. Integratingacceleration:
1. Givenagraphofanobject'saccelerationaversustimet,wecanintegratetofindvelocity.
a) TheFundamentalTheoremofCalculusgives:
2. FromyourknowledgeoftheendofCalcIandfromCalcII,youshouldrememberthatiftheaxesofthegrapharetheitemsbehindtheintegralsymbol,thatanothergroupofwordsforintegralis________________________________.
a) Thusthedefiniteintegralontherightcanbeevaluatedfromagraph:
B. Integratingvelocity:
1. Givenagraphofanobject'svelocityvversustimet,wecanintegratetofindposition.
a) TheFundamentalTheoremofCalculusgives:
b) Similarlyasabove,thedefiniteintegralontherightcanbeevaluatedfromagraph:
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C. Example:
1. AparticularsalamanderofthegenusHydromantescapturespreybylaunchingitstongueasaprojectile:Theskeletalpartofthetongueisshotforward,unfoldingtherestofthetongue,untiltheouterportionlandsontheprey,stickingtoit.TheFigurebelowshowstheaccelerationmagnitudeaversustimetfortheaccelerationphaseofthelaunchinatypicalsituation.
Theindicatedaccelerationsare 22 400 /a m s and 2
1 100 /a m s .Whatisthe
outwardspeedofthetongueattheendoftheaccelerationphase?