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:
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(d) Lettwovectors and berepresentedbytheadjacentsidesofaparallelogramPORS,
asshowninthegivenfigure.
Thefollowingrelationscanbewrittenforthegivenparallelogram.
ThequantityontheLHSisalwayspositiveandthatontheRHScanbepositiveornegative.
Tomakebothquantitiespositive,wetakemodulusonbothsidesas:
Ifthetwovectorsactinastraightlinebutintheoppositedirections,thenwecanwrite:
Combiningequations(iv)and(v),weget:
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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
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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,
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Accelerationoftheparticle
Also,
But,
Integratingbothsides:
Where,
=Velocityvectoroftheparticleatt=0
=Velocityvectoroftheparticleattimet
But,
Integratingtheequationswiththeconditions:att=0;r=0andatt=t;r=r
Sincethemotionoftheparticleisconfinedtothex-yplane,onequatingthecoefficientsof
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5. areunitvectorsalongx-andy-axisrespectively.Whatisthemagnitudeand
directionofthevectors ,and ?Whatarethecomponentsofavector
alongthedirectionsof and ?[Youmayusegraphicalmethod]
Ans.
Consideravector ,givenas:
Oncomparingthecomponentsonbothsides,weget:
Hence,themagnitudeofthevector is .
Let betheanglemadebythevector ,withthex-axis,asshowninthefollowingfigure.
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Hence,thevector makesanangleof withthex-axis.
Hence,themagnitudeofthevector is .
Let betheanglemadebythevector ,withthex-axis,asshowninthefollowingfigure.
Hence,thevector makesanangleof
Itisgiventhat:
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Oncomparingthecoefficientsof ,wehave:
Let makeanangle withthex-axis,asshowninthefollowingfigure.
Anglebetweenthevectors
Componentofvector ,alongthedirectionof ,makinganangle
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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 .
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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.
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