CBSEClass11physics

ImportantQuestions

Chapter4

MotioninAPlane

1MarksQuestions

1.Whatis“Trajectoryofaprojectile?

Ans:Thepathfollowedbyaprojectileiscalledtrajectoryofprojectilee.g.parabola.

2.Aprojectileisfiredatanangleof30owiththehorizontalwithvelocity10m/s.Atwhat

anglewiththeverticalshoulditbefiredtogetmaximumrange?

Ans:Maximumrangeisobtainedatanangleof45o.

3. Whatisthevalueofangularspeedfor1revolution?

Ans:Foronecompleterevolution, intimeperiodt=T,

4. Giveanexampleofabodymovingwithuniformspeedbuthavingavariablevelocity

andanaccelerationwhichremainsconstantinmagnitudebutchangesindirection

Ans:Abodymovinginacircularpath.

5.Whatisthedirectionofcentripetalforcewhenparticleisfollowingacircularpath?

Ans:Thedirectionofthecentripetalforceistowardsthecentreofthecircle.

6.Twovectors areperpendiculartoeachother.Whatisthevalueof ?

Ans:Since

1

7.Whatwillbetheeffectonhorizontalrangeofaprojectilewhenitsinitialvelocityis

doubled,keepingtheangleofprojectionsame?

Ans:Fourtimestheinitialhorizontalrange.

8.Whatwillbetheeffectonmaximumheightofaprojectilewhenitsangleofprojection

ischangedfrom30oto60o,keepingthesameinitialvelocityofprojection?

Ans:Threetimestheinitialverticalheight.

9.Whatistheangularvelocityofthehourhandofaclock?

Ans: radianperhour.

10.Abodyismovingonacurvedpathwithaconstantspeed.Whatisthenatureofits

acceleration?

Ans:Accelerationmustbeperpendiculartothedirectionofmotionandiscalledcentripetal

acceleration.

11. State,foreachofthefollowingphysicalquantities,ifitisascalaroravector:

volume,mass,speed,acceleration,density,numberofmoles,velocity,angular

frequency,displacement,angularvelocity.

Ans.Scalar:Volume,mass,speed,density,numberofmoles,angularfrequency

Vector:Acceleration,velocity,displacement,angularvelocity

Ascalarquantityisspecifiedbyitsmagnitudeonly.Itdoesnothaveanydirectionassociated

withit.Volume,mass,speed,density,numberofmoles,andangularfrequencyaresomeof

thescalarphysicalquantities.

2

Avectorquantityisspecifiedbyitsmagnitudeaswellasthedirectionassociatedwithit.

Acceleration,velocity,displacement,andangularvelocitybelongtothiscategory.

12. Pickoutthetwoscalarquantitiesinthefollowinglist:

force,angularmomentum,work,current,linearmomentum,electricfield,average

velocity,magneticmoment,relativevelocity.

Ans.Workandcurrentarescalarquantities.

Workdoneisgivenbythedotproductofforceanddisplacement.Sincethedotproductof

twoquantitiesisalwaysascalar,workisascalarphysicalquantity.

Currentisdescribedonlybyitsmagnitude.Itsdirectionisnottakenintoaccount.Hence,itis

ascalarquantity.

13. Pickouttheonlyvectorquantityinthefollowinglist:

Temperature,pressure,impulse,time,power,totalpathlength,energy,gravitational

potential,coefficientoffriction,charge.

Ans.Impulse

Impulseisgivenbytheproductofforceandtime.Sinceforceisavectorquantity,itsproduct

withtime(ascalarquantity)givesavectorquantity.

3

CBSEClass11physics

ImportantQuestions

Chapter4

MotioninAPlane

2MarksQuestions

1.Whatistheanglebetweentwoforcesof2Nand3Nhavingresultantas4N?

Ans:Using weget

2.Whatistheangleofprojectionatwhichhorizontalrangeandmaximumheightare

equal?

Ans:Equating,

4

3.Provethatforelevationswhichexceedorfallshortof45obyequalamountsthe

rangesareequal?

Ans:Weknow

So,

R1=R2

4.Atwhatrangewillaradarsetshowafighterplaneflyingat3kmaboveitscentreand

atdistanceof4kmfromit?

Ans:Herestraightdistanceoftheobjectformtheradar=OB

5

5.Twoforces5and10kgwtareactingwithaninclinationof120obetweenthem.What

istheanglewhichtheresultantmakeswith10kgwt?

Ans:

6

6.Astoneisthrownverticallyupwardsandthenitreturnstothethrower.Isita

projectile?Explain?

Ans:Astonecannotbeconsideredasaprojectilebecauseaprojectilemusthavetwo

perpendicularcomponentsofvelocitiesbutinthiscaseastonehasvelocityinonedirection

whilegoinguporcomingdownwards.

7.Whichisgreatertheangularvelocityofthehourhandofawatchorangularvelocity

ofeartharounditsownaxis?

Ans:Inhourhandofawatch(T)=12h

ForrotationofearthT=24h

8.Whydoesthedirectionofmotionofaprojectilebecomehorizontalatthehighest

pointofitstrajectory?

Ans:Atthehighestpointverticalcomponentofvelocitybecomeszerothusdirectionof

motionofprojectilebecomeshorizontal.

9.Avector hasmagnitude2andanothervector havemagnitude3andis

perpendiculartoeachother.Byvectordiagramfindthemagnitudeof and

7

showitsdirectioninthediagram.

Ans:Here

10.Findaunitvectorparalleltotheresultantofthevectors

Ans:Weknow

8

11.Astonetiedattheendofstringiswhirledinacircle.Ifthestringbreaks,thestone

fliesawaytangentially.Why?

Ans:Whenastoneismovingaroundacircularpath,itsvelocityactstangenttothecircle.

Whenthestringbreaks,thecentripetalforcewillnotact.Duetoinertia,thestonecontinues

tomovealongthetangenttocircularpath,andfliesofftangentiallytothecircularpath.

12. Whatarethetwoanglesofprojectionofaprojectileprojectedwithvelocity30m/s,

sothatthehorizontalrangeis45m.Take,g=10m/s2.

Ans:

13.Thebladesofanaeroplanepropellerarerotatingattherateof600revolutionsper

minute.Calculateitsangularvelocity.

Ans:

9

14.Whatisauniformcircularmotion?Explainthetermstimeperiod,frequencyand

angularvelocity.Establishrelationbetweenthem.

Ans:Whenanobjectmovesinacircularpathwithconstantspeedthenthemotioniscalled

uniformcircularmotion

Timeperiod–Thetimetakenbytheobjecttocompleteonerevolution

Frequency–Thetotalnumberofrevolutionsinonesecondiscalledthefrequency.

Angularvelocity–Itisdefinedasthetimerateofchangeofangulardisplacement.

15.Abodyofmassmisthrownwithvelocity atangleof30otothehorizontaland

anotherbodyBofthesamemassisthrownwithvelocity atanangleof60otothe

horizontal.FindtheratioofthehorizontalrangeandmaximumheightofAandB?

Ans:(1)When

When

10

(2)When

When

16. Readeachstatementbelowcarefullyandstatewithreasons,ifitistrueorfalse:

(a) Themagnitudeofavectorisalwaysascalar,(b)eachcomponentofavectoris

alwaysascalar,(c)thetotalpathlengthisalwaysequaltothemagnitudeofthe

displacementvectorofaparticle.(d)theaveragespeedofaparticle(definedastotal

pathlengthdividedbythetimetakentocoverthepath)iseithergreaterorequaltothe

magnitudeofaveragevelocityoftheparticleoverthesameintervaloftime,(e)Three

vectorsnotlyinginaplanecanneveradduptogiveanullvector.

Ans.

(a) True

(b) False

(c) False

(d) True

11

(e) True

Explanation:

(a) Themagnitudeofavectorisanumber.Hence,itisascalar.

(b) Eachcomponentofavectorisalsoavector.

(c) Totalpathlengthisascalarquantity,whereasdisplacementisavectorquantity.Hence,

thetotalpathlengthisalwaysgreaterthanthemagnitudeofdisplacement.Itbecomesequal

tothemagnitudeofdisplacementonlywhenaparticleismovinginastraightline.

(d) Itisbecauseofthefactthatthetotalpathlengthisalwaysgreaterthanorequaltothe

magnitudeofdisplacementofaparticle.

(e) Threevectors,whichdonotlieinaplane,cannotberepresentedbythesidesofatriangle

takeninthesameorder.

17. Statewithreasons,whetherthefollowingalgebraicoperationswithscalarand

vectorphysicalquantitiesaremeaningful:

(a) addinganytwoscalars,(b)addingascalartoavectorofthesamedimensions,(c)

multiplyinganyvectorbyanyscalar,(d)multiplyinganytwoscalars,(e)addingany

twovectors,(f)addingacomponentofavectortothesamevector.

Ans.(a)Meaningful

(b) NotMeaningful

(c) Meaningful

(d) Meaningful

(e) Meaningful

(f) Meaningful

Explanation:

12

(a)Theadditionoftwoscalarquantitiesismeaningfulonlyiftheybothrepresentthesame

physicalquantity.

(b)Theadditionofavectorquantitywithascalarquantityisnotmeaningful.

(c) Ascalarcanbemultipliedwithavector.Forexample,forceismultipliedwithtimeto

giveimpulse.

(d) Ascalar,irrespectiveofthephysicalquantityitrepresents,canbemultipliedwith

anotherscalarhavingthesameordifferentdimensions.

(e) Theadditionoftwovectorquantitiesismeaningfulonlyiftheybothrepresentthesame

physicalquantity.

(f) Acomponentofavectorcanbeaddedtothesamevectorastheybothhavethesame

dimensions.

18. Threegirlsskatingonacircularicegroundofradius200mstartfromapointPon

theedgeofthegroundandreachapointQdiametricallyoppositetoPfollowing

differentpathsasshowninFig.4.20.Whatisthemagnitudeofthedisplacementvector

foreach?Forwhichgirlisthisequaltotheactuallengthofthepathskated?

Ans.

Displacementisgivenbytheminimumdistancebetweentheinitialandfinalpositionsofa

particle.Inthegivencase,allthegirlsstartfrompointPandreachpointQ.Themagnitudes

oftheirdisplacementswillbeequaltothediameteroftheground.

Radiusoftheground=200m

13

Diameteroftheground= =400m

Hence,themagnitudeofthedisplacementforeachgirlis400m.Thisisequaltotheactual

lengthofthepathskatedbygirlB.

19. Amancanswimwithaspeedof4.0km/hinstillwater.Howlongdoeshetaketo

crossariver1.0kmwideiftheriverflowssteadilyat3.0km/handhemakeshisstrokes

normaltotherivercurrent?Howfardowntheriverdoeshegowhenhereachesthe

otherbank?

Ans.Speedoftheman, =4km/h

Widthoftheriver=1km

Timetakentocrosstheriver

Speedoftheriver, =3km/h

Distancecoveredwithflowoftheriver=

20. Astonetiedtotheendofastring80cmlongiswhirledinahorizontalcirclewitha

constantspeed.Ifthestonemakes14revolutionsin25s,whatisthemagnitudeand

directionofaccelerationofthestone?

Ans.Lengthofthestring,l=80cm=0.8m

Numberofrevolutions=14

14

Timetaken=25s

Frequency,

Angularfrequency,

Centripetalacceleration,

Thedirectionofcentripetalaccelerationisalwaysdirectedalongthestring,towardthe

centre,atallpoints.

21. Anaircraftexecutesahorizontalloopofradius1.00kmwithasteadyspeedof900

km/h.Compareitscentripetalaccelerationwiththeaccelerationduetogravity.

Ans.

Radiusoftheloop,r=1km=1000m

Speedoftheaircraft,v=900km/h

Centripetalacceleration,

Accelerationduetogravity,g=

15

22. Readeachstatementbelowcarefullyandstate,withreasons,ifitistrueorfalse:

(a) Thenetaccelerationofaparticleincircularmotionisalwaysalongtheradiusof

thecircletowardsthecentre

(b) Thevelocityvectorofaparticleatapointisalwaysalongthetangenttothepathof

theparticleatthatpoint

(c) Theaccelerationvectorofaparticleinuniformcircularmotionaveragedoverone

cycleisanullvector

Ans.

(a) False

Thenetaccelerationofaparticleincircularmotionisnotalwaysdirectedalongtheradiusof

thecircletowardthecentre.Ithappensonlyinthecaseofuniformcircularmotion.

(b) True

Atapointonacircularpath,aparticleappearstomovetangentiallytothecircularpath.

Hence,thevelocityvectoroftheparticleisalwaysalongthetangentatapoint.

(c) True

Inuniformcircularmotion(UCM),thedirectionoftheaccelerationvectorpointstowardthe

centreofthecircle.However,itconstantlychangeswithtime.Theaverageofthesevectors

overonecycleisanullvector.

23. Acricketercanthrowaballtoamaximumhorizontaldistanceof100m.Howmuch

highabovethegroundcanthecricketerthrowthesameball?

Ans.

16

Maximumhorizontaldistance,R=100m

Thecricketerwillonlybeabletothrowtheballtothemaximumhorizontaldistancewhen

theangleofprojectionis45°,i.e.,θ=45°.

Thehorizontalrangeforaprojectionvelocityv,isgivenbytherelation:

Theballwillachievethemaximumheightwhenitisthrownverticallyupward.Forsuch

motion,thefinalvelocityviszeroatthemaximumheightH.

Acceleration,a= Usingthethirdequationofmotion:

24. Thepositionofaparticleisgivenby

Wheretisinsecondsandthecoefficientshavetheproperunitsforrtobeinmetres.

(a) Findthevandaoftheparticle?

(b) Whatisthemagnitudeanddirectionofvelocityoftheparticleatt=2.0s?

Ans.

17

(a)

Thepositionoftheparticleisgivenby:

Velocity ,oftheparticleisgivenas:

Acceleration ,oftheparticleisgivenas:

(b) 8.54m/s,69.45°belowthex-axis

Wehavevelocitysector,

At

Themagnitudeofvelocityisgivenby:

18

Thenegativesignindicatesthatthedirectionofvelocityisbelowthex-axis.

25:Foranyarbitrarymotioninspace,whichofthefollowingrelationsaretrue:

(a)

(b)

(c)

(d)

(e)

(The‘average’standsforaverageofthequantityoverthetimeintervalt1tot2)

Ans.(b)and(e)

(a)Itisgiventhatthemotionoftheparticleisarbitrary.Therefore,theaveragevelocityof

theparticlecannotbegivenbythisequation.

(b)Thearbitrarymotionoftheparticlecanberepresentedbythisequation.

(c)Themotionoftheparticleisarbitrary.Theaccelerationoftheparticlemayalsobenon-

uniform.Hence,thisequationcannotrepresentthemotionoftheparticleinspace.

(d)Themotionoftheparticleisarbitrary;accelerationoftheparticlemayalsobenon-

uniform.Hence,thisequationcannotrepresentthemotionofparticleinspace.

19

(e)Thearbitrarymotionoftheparticlecanberepresentedbythisequation.

26. Readeachstatementbelowcarefullyandstate,withreasonsandexamples,ifitis

trueorfalse:

Ascalarquantityisonethat

(a) isconservedinaprocess

(b) cannevertakenegativevalues

c) mustbedimensionless

(d) doesnotvaryfromonepointtoanotherinspace

(e) hasthesamevalueforobserverswithdifferentorientationsofaxes

Ans.(a)False

Despitebeingascalarquantity,energyisnotconservedininelasticcollisions.

(b) False

Despitebeingascalarquantity,temperaturecantakenegativevalues.

(c) False

Totalpathlengthisascalarquantity.Yetithasthedimensionoflength.

(d) False

Ascalarquantitysuchasgravitationalpotentialcanvaryfromonepointtoanotherinspace.

(e) True

Thevalueofascalardoesnotvaryforobserverswithdifferentorientationsofaxes.

27. (a)Avectorhasmagnitudeanddirection.Doesithavealocationinspace?(b)Canit

varywithtime?(c)Willtwoequalvectorsaandbatdifferentlocationsinspace

necessarilyhaveidenticalphysicaleffects?Giveexamplesinsupportofyouranswer.

Ans.(a)No;(b)Yes;(c)No

Generallyspeaking,avectorhasnodefinitelocationsinspace.Thisisbecauseavector

remainsinvariantwhendisplacedinsuchawaythatitsmagnitudeanddirectionremainthe

same.However,apositionvectorhasadefinitelocationinspace.

20

Avectorcanvarywithtime.Forexample,thedisplacementvectorofaparticlemovingwith

acertainvelocityvarieswithtime.

Twoequalvectorslocatedatdifferentlocationsinspaceneednotproducethesamephysical

effect.Forexample,twoequalforcesactingonanobjectatdifferentpointscancausethe

bodytorotate,buttheircombinationcannotproduceanequalturningeffect.

28. (a)Avectorhasbothmagnitudeanddirection.Doesitmeanthatanythingthathas

magnitudeanddirectionisnecessarilyavector?(b)Therotationofabodycanbe

specifiedbythedirectionoftheaxisofrotation,andtheangleofrotationaboutthe

axis.Doesthatmakeanyrotationavector?

Ans.(a)No;(b)No

Aphysicalquantityhavingbothmagnitudeanddirectionneednotbeconsideredavector.

Forexample,despitehavingmagnitudeanddirection,currentisascalarquantity.The

essentialrequirementforaphysicalquantitytobeconsideredavectoristhatitshould

followthelawofvectoraddition.

Generallyspeaking,therotationofabodyaboutanaxisisnotavectorquantityasitdoesnot

followthelawofvectoraddition.However,arotationbyacertainsmallanglefollowsthe

lawofvectoradditionandisthereforeconsideredavector.

29. Canyouassociatevectorswith(a)thelengthofawirebentintoaloop,(b)aplane

area,(c)asphere?Explain.

Ans.(a)No;(b)Yes;(c)No

(a) Onecannotassociateavectorwiththelengthofawirebentintoaloop.

(b) Onecanassociateanareavectorwithaplanearea.Thedirectionofthisvectorisnormal,

inwardoroutwardtotheplanearea.

(c) Onecannotassociateavectorwiththevolumeofasphere.However,anareavectorcan

beassociatedwiththeareaofasphere.

21

CBSEClass11physics

ImportantQuestions

Chapter4

MotioninAPlane

3MarksQuestions

1.Deriveexpressionsforvelocityandaccelerationforuniformcircularmotion.

ORDeriveexpressionforlinearaccelerationinuniformcircularmotion.

Ans:(1)IF

Andangularvelocity

Using

Substitutingin(1)

(2) Since

22

2.Deriveanequationforthepathofaprojectilefiredparalleltohorizontal.

Ans:Letaprojectilehavinginitialuniformhorizontalvelocityubeundertheinfluenceof

gravity,thenatanyinstanttatpositionPthehorizontalandvertical.

Forhorizontalmotion

Forverticalmotion

Weget

23

Or

Usingequation(1)and(2)

3.(a)Definetimeofflightandhorizontalrange?

(b) FromacertainheightabovethegroundastoneAisdroppedgently.Simultaneously

anotherstoneBisfiredhorizontally.Whichofthetwostoneswillarriveontheground

earlier?

Ans:(a)Timeofflight–Thetimetakenbytheprojectiletocompleteitstrajectoryiscalled

timeofflight.

HorizontalRange–Themaximumhorizontaldistancecoveredbytheprojectileformthe

footofthetowertothepointwhereprojectilehitsthegroundiscalledhorizontalrange.

(b) Boththestoneswillreachthegroundsimultaneouslybecausetheinitialverticalvelocity

inbothcasesiszeroandbotharefallingwithsameaccelerationequaltoaccelerationdueto

gravity.

4.Atwhatpointofprojectilemotion(i)potentialenergymaximum(ii)Kineticenergy

maximum(iii)totalmechanicalenergyismaximum

Ans:(1)P.E.Willbemaximumatthehighestpoint

(P.E.)highestpoint=mgH

(2)K.Ewillbeminimumatthehighestpoint

24

(Verticalcomponentofvelocityiszero)

(3)Totalmechanicalenergy

5. AcycliststartsfromthecentreOofacircularparkofradius1km,reachestheedge

Pofthepark,thencyclesalongthecircumference,andreturnstothecentrealongQO

asshowninFig.4.21.Iftheroundtriptakes10min,whatisthe(a)netdisplacement,(b)

averagevelocity,and(c)averagespeedofthecyclist?

Ans.(a)Displacementisgivenbytheminimumdistancebetweentheinitialandfinal

positionsofabody.Inthegivencase,thecyclistcomestothestartingpointaftercyclingfor

10minutes.Hence,hisnetdisplacementiszero.

25

(b) Averagevelocityisgivenbytherelation:

Averagevelocity

Sincethenetdisplacementofthecyclistiszero,hisaveragevelocitywillalsobezero.

(c) Averagespeedofthecyclistisgivenbytherelation:

Averagespeed

Totalpathlength=OP+PQ+QO

Timetaken=10min

∴Averagespeed

6. Apassengerarrivinginanewtownwishestogofromthestationtoahotellocated10

kmawayonastraightroadfromthestation.Adishonestcabmantakeshimalonga

circuitouspath23kmlongandreachesthehotelin28min.Whatis(a)theaverage

speedofthetaxi,(b)themagnitudeofaveragevelocity?Arethetwoequal?

Ans.(a)Totaldistancetravelled=23km

26

Totaltimetaken=28min

∴Averagespeedofthetaxi

(b) Distancebetweenthehotelandthestation=10km=Displacementofthecar

∴Averagevelocity

Therefore,thetwophysicalquantities(averagespeedandaveragevelocity)arenotequal.

7. Rainisfallingverticallywithaspeedof .Awomanridesabicyclewitha

speedof10 inthenorthtosouthdirection.Whatisthedirectioninwhichshe

shouldholdherumbrella?

Ans.Thedescribedsituationisshowninthegivenfigure.

Here,

=Velocityofthecyclist

27

=Velocityoffallingrain

Inordertoprotectherselffromtherain,thewomanmustholdherumbrellainthedirection

oftherelativevelocity(v)oftherainwithrespecttothewoman.

Hence,thewomanmustholdtheumbrellatowardthesouth,atanangleofnearly18°with

thevertical.

8. Theceilingofalonghallis25mhigh.Whatisthemaximumhorizontaldistancethat

aballthrownwithaspeedof cangowithouthittingtheceilingofthehall?

Ans.Speedoftheball,u=40m/s

Maximumheight,h=25m

Inprojectilemotion,themaximumheightreachedbyabodyprojectedatanangleθ,isgiven

bytherelation:

=0.30625

sinθ=0.5534

∴θ= (0.5534)=33.60°

28

Horizontalrange,R

9. Anaircraftisflyingataheightof3400mabovetheground.Iftheanglesubtendedat

agroundobservationpointbytheaircraftpositions10.0sapartis30°,whatisthespeed

oftheaircraft?

Ans.Thepositionsoftheobserverandtheaircraftareshowninthegivenfigure.

Heightoftheaircraftfromground,OR=3400m

Anglesubtendedbetweenthepositions,∠POQ=30Time=10s

InΔPRO:

ΔPROissimilartoΔRQO.

∴PR=RQPQ=PR+RQ

29

=2PR=2×3400tan15°

=6800×0.268=1822.4m

∴Speedoftheaircraft

10. Abulletfiredatanangleof30°withthehorizontalhitstheground3.0kmaway.By

adjustingitsangleofprojection,canonehopetohitatarget5.0kmaway?Assumethe

muzzlespeedtothefixed,andneglectairresistance.

Ans.No

Range,R=3km

Angleofprojection, =30°

Accelerationduetogravity,g=

Horizontalrangefortheprojectionvelocity ,isgivenbytherelation:

Themaximumrange( isachievedbythebulletwhenitisfiredatanangleof45°with

thehorizontal,thatis,

Oncomparingequations(1)and(2),weget:

Hence,thebulletwillnothitatarget5kmaway.

30

CBSEClass11physics

ImportantQuestions

Chapter4

MotioninAPlane

4MarksQuestions

1. Givena+b+c+d=0,whichofthefollowingstatementsarecorrect:

(a) a,b,c,anddmusteachbeanullvector,

(b) Themagnitudeof(a+c)equalsthemagnitudeof(b+d),

(c) Themagnitudeofacanneverbegreaterthanthesumofthemagnitudesofb,c,and

d,

(d) b+cmustlieintheplaneofaanddifaanddarenotcollinear,andintheline

Ans.(a)Incorrect

Inordertomakea+b+c+d=0,itisnotnecessarytohaveallthefourgivenvectorstobe

nullvectors.Therearemanyothercombinationswhichcangivethesumzero.

(b) Correct

a+b+c+d=0

a+c=-(b+d)

Takingmodulusonboththesides,weget:

Hence,themagnitudeof(a+c)isthesameasthemagnitudeof(b+d).

(c) Correct

a+b+c+d=0

a=(b+c+d)

Takingmodulusbothsides,weget:

31

|a|=|b+c+d|

…(i)

Equation(i)showsthatthemagnitudeofaisequaltoorlessthanthesumofthemagnitudes

ofb,c,andd.

Hence,themagnitudeofvectoracanneverbegreaterthanthesumofthemagnitudesofb,

c,andd.

(d) Correct

Fora+b+c+d=0

a+(b+c)+d=0

Theresultantsumofthethreevectorsa,(b+c),anddcanbezeroonlyif(b+c)lieinaplane

containingaandd,assumingthatthesethreevectorsarerepresentedbythethreesidesofa

triangle.

Ifaanddarecollinear,thenitimpliesthatthevector(b+c)isinthelineofaandd.This

implicationholdsonlythenthevectorsumofallthevectorswillbezero.

2. Inaharbour,windisblowingatthespeedof72km/handtheflagonthemastofa

boatanchoredintheharbourfluttersalongtheN-Edirection.Iftheboatstartsmoving

ataspeedof51km/htothenorth,whatisthedirectionoftheflagonthemastofthe

boat?

Ans.Velocityoftheboat, =51km/h

Velocityofthewind, =72km/h

Theflagisflutteringinthenorth-eastdirection.Itshowsthatthewindisblowingtowardthe

north-eastdirection.Whentheshipbeginssailingtowardthenorth,theflagwillmovealong

thedirectionoftherelativevelocity( )ofthewindwithrespecttotheboat.

Theanglebetween =90°+45°

32

Anglewithrespecttotheeastdirection= =0.11°

Hence,theflagwillflutteralmostdueeast.

3. Afighterplaneflyinghorizontallyatanaltitudeof1.5kmwithspeed720km/hpasses

directlyoverheadananti-aircraftgun.Atwhatanglefromtheverticalshouldthegun

befiredfortheshellwithmuzzlespeed tohittheplane?Atwhatminimum

altitudeshouldthepilotflytheplanetoavoidbeinghit?(Takeg= ).

Ans.Heightofthefighterplane=1.5km=1500m

Speedofthefighterplane,v=720km/h=200m/s

Letθbetheanglewiththeverticalsothattheshellhitstheplane.Thesituationisshownin

thegivenfigure.

Muzzlevelocityofthegun,u=600m/s

Timetakenbytheshelltohittheplane=t

Horizontaldistancetravelledbytheshell=

Distancetravelledbytheplane=vt

33

Theshellhitstheplane.Hence,thesetwodistancesmustbeequal.

=vt

Inordertoavoidbeinghitbytheshell,thepilotmustflytheplaneatanaltitude(H)higher

thanthemaximumheightachievedbytheshell.

34

CBSEClass11physics

ImportantQuestions

Chapter4

MotioninAPlane

5MarksQuestions

1.(a)Whatistheanglebetween if denotetheadjacentsidesofa

parallelogramdrawnformapointandtheareaoftheparallelogramis ?

(b) Stateandprovetriangularlawofvectoraddition?

Ans:(a)Areaofaparallelogram=

Areaofparallelogram=ABSinθ( Applyingcrossproduct)

Given,areaofparallelogram=

So,

35

(b) Triangularlawofvectoradditionstatesthatiftwovectorscanberepresentedbothin

magnitudeanddirectionbythesidesofatriangletakeninorderthentheirresultantisgiven

bythethirdsideofthetriangletakeninoppositeorder.

Proofàin ADC

2. Establishthefollowingvectorinequalitiesgeometricallyorotherwise:

(a)

36

(b)

(c)

(d)

Whendoestheequalitysignaboveapply?

Ans.(a)Lettwovectors and berepresentedbytheadjacentsidesofaparallelogram

OMNP,asshowninthegivenfigure.

Here,wecanwrite:

Inatriangle,eachsideissmallerthanthesumoftheothertwosides.

Therefore,inΔOMN,wehave:

ON<(OM+MN)

Ifthetwovectors and actalongastraightlineinthesamedirection,thenwecanwrite:

37

Combiningequations(iv)and(v),weget:

(b) Lettwovectors and berepresentedbytheadjacentsidesofaparallelogramOMNP,

asshowninthegivenfigure.

Here,wehave:

Inatriangle,eachsideissmallerthanthesumoftheothertwosides.

Therefore,inΔOMN,wehave:

…(iv)

Ifthetwovectors nd actalongastraightlineinthesamedirection,thenwecanwrite:

38

…(v)

Combiningequations(iv)and(v),weget:

(c) Lettwovectors and berepresentedbytheadjacentsidesofaparallelogramPORS,as

showninthegivenfigure.

Herewehave:

Inatriangle,eachsideissmallerthanthesumoftheothertwosides.Therefore,in we

have:

OS<OP+PS

Ifthetwovectorsactinastraightlinebutinoppositedirections,thenwecanwrite:

Combiningequations(iii)and(iv)weget:

39

(d) Lettwovectors and berepresentedbytheadjacentsidesofaparallelogramPORS,

asshowninthegivenfigure.

Thefollowingrelationscanbewrittenforthegivenparallelogram.

ThequantityontheLHSisalwayspositiveandthatontheRHScanbepositiveornegative.

Tomakebothquantitiespositive,wetakemodulusonbothsidesas:

Ifthetwovectorsactinastraightlinebutintheoppositedirections,thenwecanwrite:

Combiningequations(iv)and(v),weget:

40

3. Onanopenground,amotoristfollowsatrackthatturnstohisleftbyanangleof60°

afterevery500m.Startingfromagiventurn,specifythedisplacementofthemotorist

atthethird,sixthandeighthturn.Comparethemagnitudeofthedisplacementwith

thetotalpathlengthcoveredbythemotoristineachcase.

Ans.

Thepathfollowedbythemotoristisaregularhexagonwithside500m,asshowninthe

givenfigure

LetthemotoriststartfrompointP.

ThemotoristtakesthethirdturnatS.

∴Magnitudeofdisplacement=PS=PV+VS=500+500=1000m

Totalpathlength=PQ+QR+RS=500+500+500=1500m

ThemotoristtakesthesixthturnatpointP,whichisthestartingpoint.

∴Magnitudeofdisplacement=0

Totalpathlength=PQ+QR+RS+ST+TU+UP

=500+500+500+500+500+500=3000m

ThemotoristtakestheeightturnatpointR

∴Magnitudeofdisplacement=PR

41

Therefore,themagnitudeofdisplacementis866.03matanangleof30°withPR.

Totalpathlength=Circumferenceofthehexagon+PQ+QR

=6×500+500+500=4000m

Themagnitudeofdisplacementandthetotalpathlengthcorrespondingtotherequiredturns

isshowninthegiventable

Turn Magnitudeofdisplacement(m) Totalpathlength(m)

Third 1000 1500

Sixth 0 3000

Eighth 866.03;30° 4000

4. Aparticlestartsfromtheoriginatt=0swithavelocityof andmovesin

thex-yplanewithaconstantaccelerationof .

(a) Atwhattimeisthex-coordinateoftheparticle16m?Whatisthey-coordinateofthe

particleatthattime?

(b) Whatisthespeedoftheparticleatthetime?

Ans.

Velocityoftheparticle,

42

Accelerationoftheparticle

Also,

But,

Integratingbothsides:

Where,

=Velocityvectoroftheparticleatt=0

=Velocityvectoroftheparticleattimet

But,

Integratingtheequationswiththeconditions:att=0;r=0andatt=t;r=r

Sincethemotionoftheparticleisconfinedtothex-yplane,onequatingthecoefficientsof

43

,weget:

(a)Whenx=16m:

∴y=10×2+ =24m

(b) Velocityoftheparticleisgivenby:

44

5. areunitvectorsalongx-andy-axisrespectively.Whatisthemagnitudeand

directionofthevectors ,and ?Whatarethecomponentsofavector

alongthedirectionsof and ?[Youmayusegraphicalmethod]

Ans.

Consideravector ,givenas:

Oncomparingthecomponentsonbothsides,weget:

Hence,themagnitudeofthevector is .

Let betheanglemadebythevector ,withthex-axis,asshowninthefollowingfigure.

45

Hence,thevector makesanangleof withthex-axis.

Hence,themagnitudeofthevector is .

Let betheanglemadebythevector ,withthex-axis,asshowninthefollowingfigure.

Hence,thevector makesanangleof

Itisgiventhat:

46

Oncomparingthecoefficientsof ,wehave:

Let makeanangle withthex-axis,asshowninthefollowingfigure.

Anglebetweenthevectors

Componentofvector ,alongthedirectionof ,makinganangle

47

Let betheanglebetweenthevectors and .

Componentofvector ,alongthedirectionof ,makinganangle

48

6. Acyclistisridingwithaspeedof27km/h.Asheapproachesacircularturnonthe

roadofradius80m,heappliesbrakesandreduceshisspeedattheconstantrateof0.50

m/severysecond.Whatisthemagnitudeanddirectionofthenetaccelerationofthe

cyclistonthecircularturn?

Ans. 0.86m/s2;54.46°withthedirectionofvelocity

Speedofthecyclist,

Radiusofthecircularturn,r=80m

Centripetalaccelerationisgivenas:

Thesituationisshowninthegivenfigure:

SupposethecyclistbeginscyclingfrompointPandmovestowardpointQ.AtpointQ,he

appliesthebreaksanddeceleratesthespeedofthebicycleby .

49

ThisaccelerationisalongthetangentatQandoppositetothedirectionofmotionofthe

cyclist.

Sincetheanglebetween is90°,theresultantaccelerationaisgivenby:

Where istheangleoftheresultantwiththedirectionofvelocity

7. (a)Showthatforaprojectiletheanglebetweenthevelocityandthex-axisasa

functionoftimeisgivenby

(b) Showthattheprojectionangle foraprojectilelaunchedfromtheoriginisgiven

by

Wherethesymbolshavetheirusualmeaning.

Ans.(a)Let and respectivelybetheinitialcomponentsofthevelocityofthe

projectilealonghorizontal(x)andvertical(y)directions.

Let and respectivelybethehorizontalandverticalcomponentsofvelocityatapoint

P.

50

TimetakenbytheprojectiletoreachpointP=t

Applyingthefirstequationofmotionalongtheverticalandhorizontaldirections,weget:

(b)Maximumverticalheight,

Horizontalrange,

Solvingequations(i)and(ii),weget:

51