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Mechanics:DescribingMotionandSolvingProblems(ReferenceChapter#1and#2Knight)

OverallAim:TousewhatyouknowaboutNewtonianMechanicstodescribewhatyouseeinwords,images,diagrams,andmathematicalrepresentations.

DONOW:Witnessthemotioninthedemoand/orvideoandtellmeeverythingyoucanfromthestartofthemotiontotheend

TheParticleModelandMotionDiagrams

LessonAim:Toreviewmotionterminologyandconceptsandapplytomotiondiagrams

TheParticleModelofMotion-Simplificationwherewetreatamovingobjectasifallofitsmasswereconcentratedatasinglepoint.Theparticle(akapoint)hasmassbutnosize,shape,ordistinctionbtwtopandbottomorleftandright.

• Goodforanalyzingtranslationalmotion(motionofanobjectalongatrajectory)butdoesn’tworkforobjectsundergoingarotation-exampleanalyzingthemotionofeachtoothonagear.

TheMotionDiagram

Usedotstorepresenttheparticleasitmovesinitstrajectoryand0,1,2,3,etc.tolocatethepositionoftheparticleatapointintimeorthe“framenumber”wherethetimeintervalbetweenpositionchangesisconstant

Q:Howisthemotionofthecarchangedifthemotiondiagramlookslikethis?

0 1 2 3

0 1 2 3

0 1 2 3

0

1

2

3

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TimeInterval(ChangeinTime),∆tistf-timeasurestheelapsedtimeasanobjectmovesfromaninitialposition,ritoafinalpositionrf.Thevalueof∆tisindependentofthespecificclockusedtomeasurethetimes.

Specificityoftermsisimportantforthiscourse:Changeintime,position,changeinposition,displacement,velocity,instantaneousvsaveragevelocityoracceleration,etc.

REVIEW:Scalarsarequantitiesthatcanbedescribedwithamagnitude(akanumber)onlywhilevectorsrequiremagnitudeanddirection!

Howisachangeinpositiondescribed?(asadisplacement)∆r(BOLDTEXTDENOTESVECTORQUANITYinthesenotes)

Displacement,∆risr1-r0,orthepositionvectoratlocation1minusthepositionvectoratlocation0.

NOTEONVARIABLES&SUBSCRIPTS:Variablesinphysicsrepresentaphysicalquantitylikedistant,speed,time,etc.,soyouwillusesymbolsotherthanx,y,orzthatyouhaveusedinmathclass.Inthiscasepositionisgiventhesymbol“d”.TheGreeksymboldelta,Δ,means...whichequals_________________-________________always!Ifasubscriptisused,thenumberindicatestheinstantandtellsyourightawaythatthatisanistantaneousvalue.Forexample,positionattime=0is0cmandisdenotedasd1andthepositionattime=3sis12cmanddenotedasd2.

Δd=df,i=df-di

So,ifIwantedtocalculatethechangeinpositionfromposition1toposition2,Iwouldwrite

d2,1=d2-d1=12cm-0cm=12cm

sinceinitialisposition1andfinalisposition2

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Lookatthediagrambelowanddistinguishbetweenthepositionvectorsandthedisplacementvector!

end?

QUESTION:Howdothepositionvectorschangeiftheoriginorreferencepointwasat(50ft,0)?Howaboutthedisplacementvector?

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REVIEW:VECTORADDITIONANDSUBTRACTION(HEADTOTAILMETHOD)

Vector addition

Vector subtraction

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Question:Howwouldyoufindthedisplacementvectoroftheparticleasitgoesfromitsinitialtofinalposition?

Describingthetimerateofchangeofdistancetraveled(speed)andtimerateofchangeoftheposition(velocity)

SCALER:

AverageSpeed,vavgis!"#$%&'( !"#$%&%'

!"#$ !"#$%&'( !"#$% !"#$%&'()Speedisnotavectorandisnotbasedon

displacementbutontheactualdistancetraveled!So,vavg=!"#$%&'( !"!"#$#%

∆ !

VECTOR:

AverageVelocity,vavg=∆!∆!or𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕

∆!and

InstantaneousVelocity,v=lim∆!→!∆𝒓∆!or!𝒓

!"iswhenyouarelookingatthevelocityatan

instantintime

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MotionDiagramswithVelocityVectorsThelengthofthevelocityvectorrepresentstheaveragespeedoftheobjectasitmovesbetweenthetwopoints.Example:TortoisevstheHareWhoismovingfaster?Howdoyouknow?

Question:CanyoutelliftheHareorTortoiseisacceleratingormovingataconstantvelocity

Describingthetimerateofchangeofthevelocityvector

AverageAcceleration,aavg=∆𝒗∆!or𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚

∆!and

InstantaneousAcceleration,a=lim∆!→!∆𝒗∆!or!𝒗

!"iswhenyouarelookingattheaccelerationataninstantin

time

Anobject’saverageaccelerationvectorpointsinthesamedirectionasthedirectionof∆v

Note:Anobject’saveragevelocityvectorpointsinthesamedirectionofthedisplacementvectorandthisisthedirectionofmotion

Whatwouldthemotiondiagramlooklikeforaballrollingdownanincline?(Justsketchit)

Note:Whendrawingacompletemotiondiagramwithvelocityvectors,youaredeterminingtheaverageacceleration!(Justas∆rwillgiveyouVavg)

Contextisimportantthesubscriptavgisoftendroppedwhendescribingaverageaccelerationorvelocity,soitisimportanttounderstandtheproblemyouareworkingwith

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TheCompleteMotionDiagram-FindingtheAccelerationVector-justuseheadtotailmethod

LEARNINGCHECK:Describeaphysicalscenarioforeachofthefollowingtwomotiondiagrams–what’sgoingonandhowdoyouknow?

1

2

Motiondiagramsforthiscoursepurposesarejustatooltohelpyouvisualizethemotion.Iwillnotaskyoutodrawthesetoscaleandwon’taskyoudrawtheseonanexam,butsketchingthemcanbeveryhelpfulandyoushouldbeabletointerpretthem.

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Determiningsignsforposition,velocity,andacceleration(Traditionalconventions+xistotheright,+yisup,and+zisoutofthepage)

Assessment for Understanding: Two balls are on tracks A and B. Ball A is released from rest and rolls down an incline while ball B rolls horizontally at constant speed. Ball B overtakes ball A near the beginning, as the motion diagram shows, but later ball A overtakes ball B. Identify the time or times (if any) at which the two balls have the same speed.

WHY?EXPLAINTHIS!!!

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PositionTimePlots:Allowyoutotrackthepositionofaparticleasafunctionoftime(YouneedtoknowhowtodrawandinterprettheseforHW,Quizzes,Tests,etc)

Examplefora1-Dmotionofastudentwalking

LearningCheck:Whatdoes1-Dmean?

Howdothetwopositiontimeplotsdifferintermsofinformation?

meters meters

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PuttingEverythingTogethertoSolveProblems:

YOUWILLNEEDGOODPHYSICALINSIGHT,THEABILITYTOEXPRESSYOURSELFINWORDSANDEQUATIONS+ABILITYTOSOLVEPROBLEMSMATHEMATICALLY!

Aim:Tobeabletoplotposition,velocity,andaccelerationvstimegraphs

VisualizationoftheSituationExample-Motiontype,Forces,Torques,Constraints,etc.

IFYOUCAN’T“SEE”THEPROBLEM,YOUWON’TBEABLETOSOLVEIT!

*THISTAKESALOTOFEXPERIENCEANDPRACTICETODOWELL

• Identifyknowns&unknowns(couldbepositions,velocities,andaccelerationsasafunctionoftimeorforces,torques,physicaldataonthebody(beyondparticlemodel)

• Representthesituationusingsketches,diagrams,graphs,etc.

Understandhowtoapproachproblemphysically-applyamodel(Ex:Newton’sLaws,ConservationofEnergy,Gauss’sLaw,Coulomb’sLaw,etc))tosetupequations

Mathematicalsolutioninvariablestoarriveatarelationship&/ornumericalvaluewithinagivenuncertainty(sigfigs)

THEPHYSICS

THEMATH

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IDENTIFYINGTHECOMPONENTSOFTHEPROBLEMSOLVINGPROCESSASSHOWNINTHISPICTORALREPRESENTATIONOFTHEFOLLOWINGPROBLEM-GUESSSSYSTEMSTILLWORKS!

THISPROBLEMISANEXAMPLEFORHOWPROBLEMSWILLGETABITMORECOMPLEX:

Asmallrocketislaunchedverticallywithanaccelerationof30m/s2suchthatitrunsoutoffuelafter30seconds.Whatisitsmaximumaltitude?

WORKINPAIRSTOPUTTOGETHERYOURBESTPRESENTATIONOFASOLUTION-DOtheBESTYOUCAN!

GIVENS&UNKNOWNS

SimplifyingAssumption/(s)

PictorialRepresentation

1. TypeofPictorialRepresentation:2. Knowns:3. Unknowns:4. CoordinateSystem:5. SymbolUse:

EQUATION/SOLVE/SUBSTITUTION

IDPROBLEMTYPE(KINEMATICS/NEWTON’SLAWS/CONERVATIONLAWS-ENERGYORMOMENTUM)

MATHEMATICALREPRESENTATION

DOESANSWER/SOLUTIONMAKESENSE?

NOWYOUSOLVEANDMAKESUREYOURANSWERMAKESSENSE&YOUEXPRESSYOURANSWERUSING3SIGNIFICANTFIGURES

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Anotherlookattheballfallingwhendroppedfromrestproblemusingitsmotiondiagram:Sketchtheposition-timeplotandvelocitytimeplotforthetwomotiondiagrambelow.Follow-up:Wheredoesthelargestaccelerationoccur?

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DESCRIBETHEMOTIONDEPICTEDASCOMPLETELYASPOSSIBLEFORTHEPOSITION-TIMEPLOTSHOWNBELOWINCORPORATINGTHEVALUESGIVEN.ASSUMETHATTHEINITIALVELOCITYIS1.0km/mintotheleftatx=10km.WHATWOULDTHEVELOCITY-TIMEPLOTLOOKLIKE?PLOTandLABEL-DONOTSKETCH!

Whycanyouuseaveragevelocitytocalculateallvaluesalongthepositiontimecurve?

Theterm“plot”meansuserealvaluesandtoscaleforAP

VERBALDESCRIPTIONUSINGVALUES,ANDDIRECTIONOFMOTION:

Forthep-tcurve:

Apositivepositionmeans…

Anegativepositionmeans…

Azeroslopemeans…

Apositiveslopemeans…

Anegativeslopemeans…

Thesteepertheslopethe…

FortheV-tcurve:

Apositivevelocitymeans…

Anegativevelocitymeans…

Azeroslopemeans…

Apositiveslopemeans…

Anegativeslopemeans…

Thesteepertheslopethe….

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Learning Check 1: Match the Displacement- time curve with the proper Velocity- time curve

Learning Check 2: Remember what you learned last year Match the Velocity Curve with its Displacement Curve and describe the 1-D motion (stories are fine)

(C4 CHAPTER #2) Learning Check 3: Below are two displacement- time graphs for the motion of objects A and B as they move along the same axis. What can you tell me about the motions depicted in each? Do objects A and B ever have the same speed? If so, at what time or times?

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Aim:Howarethevarious1-Dmotionsdescribedinwords,math,diagrams,andplots?

Possibilitiesfor1-DMotionallmovingNOMO(Nochangeinposition)

ConstantVelocity(V=const;a=0)

ConstantAcceleration(a=const)*IncludesFreeFall

Non-ConstantAcceleration(aincreasesordecreaseswithtimeorposition)

Bestfriendsfromlastyear

NEW

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ConstantVelocityMotion(UniformMotion)-Straight-linemotioninwhichequaldisplacementsoccurduringany

successiveequaltimeintervalsAverageVelocity,vavg=∆𝒓∆!or𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕

∆!ifworkingalongthexaxis,vavg=

∆𝒙∆!orifalong

theyaxis,vavg=∆𝒚∆!.Sincevelocitydoesn’tchangetheaveragevalueistheinstantaneousvalue.

AcceleratedMotion(WhenVelocityisnotConstant)-InstantaneousVelocityandtheDerivativeofPositionwithRespecttoTime

TryingtofindV2forthisverticalacceleration

Thelimitingcase:InstantaneousVelocity,v=lim∆!→!∆𝒚∆!or!𝒚

!"

*Thisyearitwillbeimportanttoconsiderinitialpositionsandconditions!

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Theinstantaneousvelocityataninstantistheslopeatapointoftheposition-timecurve.

Equations:ConstantAccelerationandUniform(ConstantVelocityMotion)in1-D

*THESEEQUATIONSAPPLYONLYFORUNIFORM(AKACONSTANT)ACCELERATION

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PRACTICE-MORECOMPLEX(FROMMITOPENCOURSEWARE)SOLVEONANOTHERSHEETOFPAPERAbusleavesastopatMITandacceleratesataconstantratefor5seconds.Duringthistimethebustraveled25meters.Thenthebustraveledataconstantspeedfor15seconds.Thenthedrivernoticedaredlight18metersaheadandslamsonthebrakes.Assumethebusdeceleratesataconstantrateandcomestoastopsometimelaterjustatthelight.

a) Whatwastheinitialaccelerationofthebus?

b)Whatwasthevelocityatthebusafter5seconds?

c)Whatwasthebrakingaccelerationofthebus?Isitpositiveornegative?

d)Howlongdidthebusbrake?

e)Whatwasthedistancefromthebusstoptothelight?

f)Makeagraphofthepositionvs.timefortheentiretrip.

g)Makeagraphofthevelocityvs.timefortheentiretrip.

h)Makeagraphoftheaccelerationvs.timefortheentiretrip.

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FreeFall-Motionwheretheonlyforceactingonanobjectistheforceduetogravity,FgRememberGalileoandthegravitydiluterakatheinclinedplaneWhenworkingonmotionslidingupordowninclines,youjustplaceyourcoordinateaxessothattheyalignwiththemotion.Thex-axisparalleltotheplaneandtheyaxisperpendicular!Thesigndependsonyou-Isuptheplaneordowntheplaneisconsideredpositive?Youdecide!Justbesticktoyourcoordinatesystemandsignconventions.

When the angle of the incline= 90 degrees, then you have free fall!

Use the angle of incline to slow down the motion so you can measure it!

MotiononanInclinedPlane

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FreeFallPracticeProblem:

Twostonesarereleasedfromrestatacertainheight,oneaftertheother.

a)Willthedifferencebetweentheirspeedsincrease,decrease,orstaythesame?

b)Willtheirseparationdistanceincrease,decrease,orstaythesame?

c)Willthetimeintervalbetweentheinstantsatwhichtheyhitthegroundbesmallerthan,equalto,orlargerthanthetimeintervalbetweentheinstantsoftheirrelease?d)Plotthespeedvs.timeforbothballsinthesameplot.e)Plotthepositionvs.timeofthetwoballsinthesameplot.(DOTHISONGRAPHPAPERANDMARKALLSIGNIFICANTPOINTSOFINTEREST)

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ANOLDQUIZPROBLEM:Trythisone!

Directions:Youmayusereferencetablesandcalculators.Showallworkandenjoy.Youhave15minutes.

Arubberballisthrownupfromthegroundwithspeedvo.Simultaneously,asecondrubberballatheighthdirectlyabovetheballisreleasedfromrest.A. Atwhatheightabovethegrounddotheballscollide?Expressyouranswerintermsoffundamental

constants(inthiscaseg)andthevariablesgiven.

B. Whatisthemaximumvalueofhforwhichacollisionoccursbeforethefirstballfallsbacktotheground?

C. Forwhatvalueofhdoesthecollisionoccurwhenthefirstballisatitshighestheight?

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Thisyear,accelerationsmaynotalwaysbeconstantandyour“oldfriends”equationsonlyapplytoconstantacceleration.So,thisisonereasonyouwillneedcalculus!Now,youcanalsohaveaninstantaneousaccelerationthatvaries.

Inotherwords,youtakethederivativeofthevelocitywithrespecttotimetofindtheinstantaneousacceleration.Tofindinstantaneousvelocity,youtakethederivateofthedisplacementwithrespecttotime.

Instantaneous acceleration as at a specific instant of time t is given by the

derivative of the velocity

Application: Rank in order, from largest to smallest, the accelerations aA– a

C at points A – C.

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TableofCommonDerivatives

Youneedtoknowtheseandbeabletoapplytofindrateofchangeoffunctions

1. 0.)( =constdxd

2. 1)( =xdxd

3. 1)( −= nn nxxdxd

4. [ ] [ ])()( xfdxdcxcf

dxd

=

5. [ ] [ ] [ ])()()()( xgdxdxf

dxdxgxf

dxd

+=+

6. [ ] [ ] [ ])()()()( xgdxdxf

dxdxgxf

dxd

−=−

7. [ ] [ ] [ ])()()()()()( xfdxdxgxg

dxdxfxgxf

dxd

+= (The Product Rule)

8. [ ] [ ]

[ ]2)(

)()()()(

)()(

xg

xgdxdxfxf

dxdxg

xgxf

dxd −

=⎥⎦

⎤⎢⎣

⎡ (The Quotient Rule)

9. xxdxd cos)(sin =

10. xxdxd sin)(cos −=

11. xxdxd 2sec)(tan =

12. x

xdxd 1)(ln =

13. xx eedxd

=

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Integrals are anti-derivatives- they are used to calculate the area under a curve *If acceleration is constant, then you can use simple geometry to figure out the area under the curve- just be careful to note initial conditions!

Application Check:

WILLTALKABOUTSOON!