RaymondA.SerwayChrisVuille

ChapterTwoMotioninOneDimension

Dynamics

• Thebranchofphysicsinvolvingthemotionofanobjectandtherelationshipbetweenthatmotionandotherphysicsconcepts

• Kinematics isapartofdynamics– Inkinematics,youareinterestedinthedescriptionofmotion

– Not concernedwiththecauseofthemotion

Introduction

QuantitiesinMotion

• Anymotioninvolvesthreeconcepts– Displacement– Velocity– Acceleration

• Theseconceptscanbeusedtostudyobjectsinmotion

Introduction

BriefHistoryofMotion

• SumariaandEgypt–Mainlymotionofheavenlybodies

• Greeks– Alsotounderstandthemotionofheavenlybodies– Systematicanddetailedstudies– Geocentricmodel

Introduction

“Modern”IdeasofMotion

• Copernicus– Developedtheheliocentricsystem

• Galileo–Madeastronomicalobservationswithatelescope– Experimentalevidencefordescriptionofmotion– Quantitativestudyofmotion

Introduction

Position

• Definedintermsofaframeofreference– Achoiceofcoordinateaxes– Definesastartingpointformeasuringthemotion• Oranyotherquantity

– Onedimensional,sogenerallythex- ory-axis

Section2.1

Displacement

• Definedasthechangeinposition–• fstandsforfinalandistandsforinitial

– Unitsaremeters(m)inSI

Section2.1

DisplacementExamples• FromAtoB

– xi =30m– xf =52m– Dx=22m– Thedisplacementispositive,

indicatingthemotionwasinthepositivexdirection

• FromCtoF– xi =38m– xf =-53m– Dx=-91m– Thedisplacementisnegative,

indicatingthemotionwasinthenegativexdirection

Section2.1

Displacement,Graphical

Section2.1

VectorandScalarQuantities

• Vectorquantitiesneedbothmagnitude(size)anddirectiontocompletelydescribethem– Generallydenotedbyboldfacedtypeandanarrowovertheletter

– +or– signissufficientforthischapter• Scalarquantitiesarecompletelydescribedbymagnitudeonly

Section2.1

DisplacementIsn’tDistance

• Thedisplacementofanobjectisnotthesameasthedistanceittravels– Example:Throwaballstraightupandthencatchitatthesamepointyoureleasedit• Thedistanceistwicetheheight• Thedisplacementiszero

Section2.1

Speed

• Theaveragespeed ofanobjectisdefinedasthetotaldistancetraveleddividedbythetotaltimeelapsed

– Speedisascalarquantity

Section2.2

Speed,cont

• Averagespeedtotallyignoresanyvariationsintheobject’sactualmotionduringthetrip

• Thepathlengthandthetotaltimeareallthatisimportant– Bothwillbepositive,sospeedwillbepositive

• SIunitsarem/s

Section2.2

PathLengthvs.Distance

• Distancedependsonlyontheendpoints

– Thedistancedoesnotdependonwhathappensbetweentheendpoints

– Isthemagnitudeofthedisplacement

• Pathlengthwilldependontheactualroutetaken

Section2.2

Velocity

• Ittakestimeforanobjecttoundergoadisplacement• Theaveragevelocity israteatwhichthedisplacementoccurs

• Velocitycanbepositiveornegative– Dtisalwayspositive

• Averagespeedisnotthesameastheaveragevelocity

Section2.2

Velocitycontinued

• Directionwillbethesameasthedirectionofthedisplacement,+or- issufficientinone-dimensionalmotion

• Unitsofvelocityarem/s(SI)– Otherunitsmaybegiveninaproblem,butgenerallywillneedtobeconvertedtothese

– Inothersystems:• USCustomary:ft/s• cgs:cm/s

Section2.2

Speedvs.Velocity

• Carsonbothpathshavethesameaveragevelocitysincetheyhadthesamedisplacementinthesametimeinterval

• Thecaronthebluepathwillhaveagreateraveragespeedsincethepathlengthittraveledislarger

Section2.2

GraphicalInterpretationofVelocity

• Velocitycanbedeterminedfromaposition-timegraph

• Averagevelocityequalstheslopeofthelinejoiningtheinitialandfinalpointsonthegraph

• Anobjectmovingwithaconstantvelocitywillhaveagraphthatisastraightline

Section2.2

AverageVelocity,Constant

• Thestraightlineindicatesconstantvelocity

• Theslopeofthelineisthevalueoftheaveragevelocity

NotesonSlopes

• Thegeneralequationfortheslopeofanylineis

– Themeaningofaspecificslopewilldependonthephysicaldatabeinggraphed

• Slopecarriesunits

Section2.2

AverageVelocity,NonConstant

• Themotionisnon-constantvelocity

• Theaveragevelocityistheslopeofthestraightlinejoiningtheinitialandfinalpoints

Section2.2

InstantaneousVelocity

• Thelimitoftheaveragevelocityasthetimeintervalbecomesinfinitesimallyshort,orasthetimeintervalapproacheszero

• Theinstantaneousvelocityindicateswhatishappeningateverypointoftime– Themagnitudeoftheinstantaneousvelocityiswhatyoureadonacar’sspeedometer

Section2.2

InstantaneousVelocityonaGraph

• Theslopeofthelinetangenttothepositionvs.timegraphisdefinedtobetheinstantaneousvelocityatthattime– Theinstantaneousspeedisdefinedasthemagnitudeoftheinstantaneousvelocity

Section2.2

GraphicalInstantaneousVelocity

• Averagevelocitiesarethebluelines

• Thegreenline(tangent)istheinstantaneousvelocity

Section2.2

Acceleration

• Changingvelocitymeansanaccelerationispresent

• Accelerationistherateofchangeofthevelocity

• Unitsarem/s²(SI),cm/s²(cgs),andft/s²(USCust)

Section2.3

AverageAcceleration

• Vectorquantity• Whentheobject’svelocityandaccelerationareinthesamedirection(eitherpositiveornegative),thenthespeedoftheobjectincreaseswithtime

• Whentheobject’svelocityandaccelerationareintheoppositedirections,thespeedoftheobjectdecreaseswithtime

Section2.3

NegativeAcceleration

• Anegativeaccelerationdoesnotnecessarilymeantheobjectisslowingdown

• Iftheaccelerationandvelocityarebothnegative,theobjectisspeedingup

• “Deceleration”meansadecreaseinspeed,notanegativeacceleration

Section2.3

InstantaneousandUniformAcceleration

• Thelimitoftheaverageaccelerationasthetimeintervalgoestozero

• Whentheinstantaneousaccelerationsarealwaysthesame,theaccelerationwillbeuniform– Theinstantaneousaccelerationswillallbeequaltotheaverageacceleration

Section2.3

GraphicalInterpretationofAcceleration

• Averageaccelerationistheslopeofthelineconnectingtheinitialandfinalvelocitiesonavelocityvs.timegraph

• Instantaneousaccelerationistheslopeofthetangenttothecurveofthevelocity-timegraph

Section2.3

AverageAcceleration– GraphicalExample

Section2.3

RelationshipBetweenAccelerationandVelocity

• Uniformvelocity(shownbyredarrowsmaintainingthesamesize)

• Accelerationequalszero

Section2.4

RelationshipBetweenVelocityandAcceleration

• Velocityandaccelerationareinthesamedirection• Accelerationisuniform(violetarrowsmaintainthesamelength)

• Velocityisincreasing(redarrowsaregettinglonger)• Positivevelocityandpositiveacceleration

Section2.4

• Accelerationandvelocityareinoppositedirections• Accelerationisuniform(violetarrowsmaintainthesame

length)• Velocityisdecreasing(redarrowsaregettingshorter)• Velocityispositiveandaccelerationisnegative

Section2.4

RelationshipBetweenVelocityandAcceleration

MotionDiagramSummary

Section2.4

EquationsforConstantAcceleration

• Theseequationsareusedinsituationswithuniformacceleration

Section2.5

Notesontheequations

• Givesdisplacementasafunctionofvelocityandtime

• Usewhenyoudon’tknowandaren’taskedfortheacceleration

Section2.5

Notesontheequations

• Showsvelocityasafunctionofaccelerationandtime

• Usewhenyoudon’tknowandaren’taskedtofindthedisplacement

Section2.5

GraphicalInterpretationoftheEquation

Section2.5

Notesontheequations

• Givesdisplacementasafunctionoftime,velocityandacceleration

• Usewhenyoudon’tknowandaren’taskedtofindthefinalvelocity

• Theareaunderthegraphofvvs.tforanyobjectisequaltothedisplacementoftheobject

Section2.5

Notesontheequations

• Givesvelocityasafunctionofaccelerationanddisplacement

• Usewhenyoudon’tknowandaren’taskedforthetime

Section2.5

Problem-SolvingHints

• Readtheproblem• Drawadiagram– Chooseacoordinatesystem– Labelinitialandfinalpoints– Indicateapositivedirectionforvelocitiesandaccelerations

• Labelallquantities,besurealltheunitsareconsistent– Convertifnecessary

• Choosetheappropriatekinematicequation

Section2.5

Problem-SolvingHints,cont

• Solvefortheunknowns– Youmayhavetosolvetwoequationsfortwounknowns

• Checkyourresults– Estimateandcompare– Checkunits

Section2.5

GalileoGalilei

• 1564- 1642• Galileoformulatedthelawsthatgovernthemotionofobjectsinfreefall

• Alsolookedat:– Inclinedplanes– Relativemotion– Thermometers– Pendulum

Section2.6

FreeFall

• Afreelyfallingobjectisanyobjectmovingfreelyundertheinfluenceofgravityalone– Freefalldoesnotdependontheobject’soriginalmotion

• Allobjectsfallingneartheearth’ssurfacefallwithaconstantacceleration

• Theaccelerationiscalledtheaccelerationduetogravity,andindicatedbyg

Section2.6

AccelerationduetoGravity

• Symbolizedbyg• g =9.80m/s²– Whenestimating,useg » 10m/s2

• g isalwaysdirecteddownward– Towardthecenteroftheearth

• Ignoringairresistanceandassumingg doesn’tvarywithaltitudeovershortverticaldistances,freefallisconstantlyacceleratedmotion

Section2.6

FreeFall– anobjectdropped

• Initialvelocityiszero• Letupbepositive– Conventional

• Usethekinematicequations– Generallyuseyinsteadofxsincevertical

• Accelerationisg =-9.80m/s2

vo=0

a=g

Section2.6

FreeFall– anobjectthrowndownward

• a=g =-9.80m/s2

• Initialvelocity¹ 0– Withupwardbeingpositive,initialvelocitywillbenegative

Section2.6

FreeFall– objectthrownupward

• Initialvelocityisupward,sopositive

• Theinstantaneousvelocityatthemaximumheightiszero

• a=g=-9.80m/s2 everywhereinthemotion

v = 0

Section2.6

Actuallystraightbackdown

Thrownupward,cont.

• Themotionmaybesymmetrical– Thentup =tdown– Thenv=-vo

• Themotionmaynotbesymmetrical– Breakthemotionintovariousparts• Generallyupanddown

Section2.6

Non-symmetricalFreeFallExample

• Needtodividethemotionintosegments

• Possibilitiesinclude– Upwardanddownward

portions– Thesymmetricalportionback

tothereleasepointandthenthenon-symmetricalportion

Section2.6

CombinationMotions

Section2.6