by studying this lesson you will be able to...f 43 by studying this lesson you will be able to •...
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By studying this lesson you will be able to •solveproblemsrelatedtodistance,timeandspeed •representinformationrelatedtodistanceandtimegraphically •solveproblemsrelatedtoliquidvolumes,timeandrate.
22.1 Speed
10 mA B
Letusassumethatabatteryoperatedtoycartakes5secondstotravelfrompointA topointBwhichis10maway.
Thenthedistancethatthecarhastravelledduring5secondsis10m.Ifthedistancethat thecarmoves forwardduringeach second is the same from themoment itstarts,thenthedistanceittravelsduringeachsecondis metres,thatis,2metres.Accordingly, as the car moves forward fromA, the rate at which the distancechangeswithrespecttotimeis2metrespersecond.WecandefinethisvalueasthespeedwithwhichthecartravelsfromAtoB.
Ifthedistancetravelledbyanobjectinmotionisaconstantperunitoftime,thentheobjectissaidtobetravellingwithuniformspeed.Further,thespeedoftheobjectisthenthedistancetravelledperunitoftime.Fromthispointon,onlyobjectswhichtravelwithuniformspeedwillbeconsideredinthislesson.
However, in reality, vehicles that travel on themain road are usually unable tomaintainauniformspeedthroughoutthewholejourneyduetothetrafficontheroadandvariousotherreasons.Theinstrumentcalledthespeedometergivesthespeedofavehicleatanygiveninstance.
Thespeeddenotedbythespeedometerinthefigurecanbewrittenas80 kmph.Itcanalsobewrittenas 80 km/h oras80 kmh–1.
Rate22
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Asyoutravelalongamainroad,youmayobserveroadsignswith40kmphand60kmphwrittenonthemtoindicatespeedlimits.Trytorecallthatheavyvehiclessuchaslorriescarryaboardatthebackwith40kmphwrittenonit.
For an object that is moving with uniform speed, the relationship between thethreequantities,namelythedistancetravelled,thetimetakenandthespeedcanbewrittenasfollows.
Speed= DistancetravelledTimetaken
Thisrelationshipcanalsobewritteninthefollowingsimpleform(withoutfractions).
Distance =Speed « Time
Example 1
Afeatherfloatingonairwithuniformspeed,drifts100min20seconds.Calculatethespeedwithwhichthefeatherdrifts.
Speedwithwhichitdrifts =Distanceitdrifts
time
Example 2
Calculatethedistancetravelledinoneminutebyabirdthatfliesatauniformspeedof5ms-–1. Distanceitflies=speed « time =5ms-–1 « 60s =300 m
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Example 3
Calculatethetimeittakesforacartotravel150kmonahighway,atauniformspeedof60 kmh–1. Timetaken = Distance
Speed
= =
Example 4
Howfardoesamotorcycletravelalongamainroadin5seconds,ifitsspeedometerdisplaysaconstantspeedof36 kmh–1duringthisperiod?
Here,thespeedhasbeengiveninkilometresperhour.Letusconvertittometrespersecond. Sincethespeedis36 kmh–1, distancetravelledduringanhour =36km =36 « 1000m However, 1hour =60 « 60seconds Distancetravelledin 60 « 60seconds =36 « 1000m
Distancetravelledin1second= Distancetravelledbythemotorcycleinonesecond=10m
Distancetravelledin5seconds =10 « 5m =50 m
Example 5
Howlongdoesittakeatrainwhichis75mlongtopassasignpost,ifitistravellingatauniformspeedof60kmh-1?
75 m
Whenitreachesthesignpost Whenitpassesthesignpost
75 m
Thedistancetravelledbythetrainasitpassesthesignpost=75mFirst,letusfindthespeedintermsofmetrespersecond.
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Thespeedofthetrainis60 kmh-1. . .. Distancetravelledinonehour =60 kmDistancetravelledinonehour=
Distancetravelledinonesecond=
=
Speedofthetrain = s
Sincetime=distancespeed ,
timetakenbythetraintopassthesignpost =75 ÷ seconds =75 × seconds
=4.5secondsExample 6
Findthetimeittakesforatrainoflength60mtravellingatauniformspeedof72kmh-1tocrossabridgewhichis100mlong.
100m60m60m
Here,thetimetakenforthetraintotraveladistanceof160mneedstobefound.Forthis,letusfirstfindthespeedinmetrespersecond.
72kmh–1 = ms–1
=20ms–1Thetotaldistancetravelledincrossingthebridge=100m+60m
=160mDistancetravelledbythetrainin1second=20m Thatis,timetakentotravel20m =1second . .. Timetakentotravel160m
8 seconds
seconds
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Average Speed
Avehicletravellingalongamainroadisusuallyunabletomaintainthesamespeedthroughoutthejourney.Theconceptofaveragespeedisimportantinsuchsituations.Thevalueobtainedwhenthetotaldistancetravelledbyanobjectisdividedbythetotaltimetakeniscalledtheaveragespeed.
Example 1
Anintercitybustook anhourtotravelthefirst25kmofajourney.Ifittookthebus1hourtocovertheremaining80kmofthejourney,findtheaveragespeedofthebus.
Total distance travelled by the bus
Total time taken for the journey
The average speed of the bus
h
h
Exercise 22.1
1. Calculatethespeedofanaircraftwhichflies1200kmin4hourswithuniformspeed.
2. If a child runs 200 m in 40 seconds at a uniform speed, find his speed inkilometresperhour.
3. Onacertainday,anelectrictrainmovingatauniformspeed,took6hourstotraveladistanceof300km.Onanotherday,thetraintook8hourstotravelthesamedistance.Find thedifferencebetween thespeedsatwhich the traintravelledduringthetwodays.
4. Howlongwillittakeanaircraftwhichtravelsatauniformspeedof300kmh-1 tofly4500km?
5. Find the distance in metres that a car which travels at a uniform speed of48kmh-1,coversduring30seconds.
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6.Abustravelsfor15minutesataspeedof40kmh-1andthenittravelsafurther30minutesataspeedof70kmh-1.Calculatetheaveragespeedofthebus.
7.Ifthetimetakenbyatraintopassasignpostis10secondswhenitistravellingatauniformspeedof54kmh-1,findthelengthofthetrain.
8.Findthetimeittakesforatrainoflength60mtravellingataspeedof72kmh-1 topassa100mlongplatform.
9.AtrainleavescityAat0800handtravelsatauniformspeedof60kmh-1towardscityB.AnothertrainleavescityBatthesameinstanceandtravelsatauniformspeedof40kmh-1towardscityA.IfthedistancebetweenthetwocitiesAandB is100km,calculatethetimeatwhichthetwotrainspasseachother.
10. Twomotorcyclists, who start their journeys at the same instance from twodifferentcities,travelwithuniformspeedsof40kmh-1and50km-1respectivelytowardseachother.Iftheymeeteachother anhouraftercommencingtheir
journeys,findthedistancebetweenthetwocities.
22.2 Distance - Time Graphs
Agraphcanbeusedtoillustratethechangeinthedistancetravelledbyanobjectinmotion,with respect to time. In such a graph, the x axis represents the timeand theyaxis represents thedistance travelled.Agraphof this form iscalledadistance-time graph.
A table prepared with the information collected by observing the motion of asatellitetravellingwithuniformspeedisgivenbelow.
Timethathaspassedfromthecommencementofthejourney(seconds)
5 10 15 20 25 30 35 40
Distancefromthestartingpoint(metres)
100 200 300 400 500 600 700 800
Thedistance-timegraphdrawnwiththisinformationisgivenbelow.
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²
²
²5
A
B
0
100
200
300
400
500
600
700
10 15 20 25 30
Time(Seconds)
Distance
^m&
800
35 40
²
²
²
²
²
²
Thespeedofthesatellitecanbecalculatedbydividingthetotaldistancetravelledbythetotaltimetaken.
Speedofthesatellite
ObservethatthegradientofthestraightlineAB
Sincethesatelliteistravellingwithuniformspeed,thespeedcanalsobeobtainedbyconsideringthedistancetravelledperunitoftime.
Accordingly,youcanobservethatthegradientofthegraphandthespeedofthesatelliteareequal.Therefore,foranobjectmovingwithuniformspeed,astraightline is obtained as the distance-time graph, and the speed of the object can beobtainedfromthegradientofthisline.
Gradientofthedistance-timegraph=Speedoftheobjectinmotion
Example 1
Adistance-timegraphillustratingthemotionofNimalwhocycledtohisfriend’shouseand then returnedbackhomeafter spending some timewithhis friend isgivenbelow.(i)CalculatethespeedatwhichNimalcycledtohisfriend’shouse.(ii)CalculatethespeedatwhichNimalreturnedhome.
= =200 2010
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² ²
40A
BC
D
0
2
6
4
8
10
12
10 6020 30
²
50 70 80
Time(minutes)
Distance ^km&
Accordingtotheabovegraph, thedistancefromNimal’shousetohisfriend’shouse =6 km
timetakenbyNimaltocycletohisfriend’shouse =30 minutes =
. .. ThespeedatwhichNimalcycledtohisfriend’shouse =
=
ThedistanceisthesameduringtheperiodthatNimalspenttimewithhisfriend
AmountoftimeNimalspentathisfriend’shouse=20minutesTimetakenforNimaltoreturnhome =20minutes
=
SpeedatwhichNimalcycledbackhome =
=
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Exercise 22.21.The following table provides information on the distance travelled by a carmovingatauniformspeedalongahighway,andthetimetakenforthejourney.
Time(hours) 0 1 2 3 4 5 6Distance(km) 0 60 120 180 240 300 360
(i)Drawadistance-timegraphwiththeaboveinformation.(ii)Findthegradientofthegraph.(iii)Hencecalculatethespeedofthecar.
2.Thechangeindistancewithtimeofanobjectinmotionisgiveninthefollowingtable.
Time(s) 0 2 4 6 8 10Distance(m) 0 6 12 18 24 30
(i)Drawadistance-timegraphwiththeaboveinformation.(ii)Findthegradientofthegraph.(iii)Hencecalculatethespeedoftheobject.
3. Acoach,movingwithuniformspeedfromthecommencementof its journey,travelsadistanceof60kmin2hours.Itthentravelsanother40kmin2hours,alsowithuniformspeed,andreachesitsdestination.Representthemotionofthecoachinadistance-timegraph.
4.Adistance-timegraphofthemotionofamanwhotravelsfromhishometothecityonhismotorcycleisgivenbelow.
²
²
²
²5
A
B
C
D
0
1
2
3
4
5
6
7
10 15 20 25 30Time(Minutes)
Distance(km)
35
(i)Howfarisitfromhishometothecity?(ii)Howlongdidittakehimtoreachthecity?(iii)Calculatehisaveragespeed.
(iv)SeparatelycalculatethespeedsatwhichhetravelledfromAtoB,fromBtoC andfromCtoD.
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22.3 Volume and TimeWedefinedspeedasthedistancetravelledperunitoftime.Anotherwayofsayingthisisthatspeedistherateofchangeofdistancewithrespecttotime.Thisideaofratecanalsobeusedtodescribevariousotherprocessesthatwecomeacrossindaytodaylife.Letusconsidertheexampleofwaterflowingoutofatap.Ifwecollectthewaterthatflowsoutfromatapduringperiodsofonesecondeach,andifbymeasuringwediscoverthatthevolumeofwaterthatflowsoutduringeachsecondisaconstant,thenwesaythatthewaterflowsoutatauniformrate.Further,wecallthisconstantvaluetherateatwhichwaterflowsoutfromthetap.
Whentimeismeasuredinsecondsandthevolumeofwaterismeasuredinlitres,theunitoftherateofflowislitrespersecond(ls-1).
Supposeittakes20minutesforatankofcapacity1000ltobefilledcompletelyusingapipethroughwhichwaterflowsatauniformrate.
Then,thevolumeofwaterthatflowedoutofthepipeduring20minutes
Theamountofwaterthatflowedoutduring1minute
Accordingly,theamountofwaterthatflowsoutofthepipeperunitoftime,thatis,duringoneminute,is50litres.Therefore,wecanexpresstherateatwhichwaterflowsoutofthepipeas50litresperminute.
ChangeofvolumeRateofchangeofvolume = Time
Thiscanalsoberepresentedasfollows.
Changeofvolume=Rateofchangeofvolume × Time
Example 1
Thetimetakenfor30litresofpetroltobepumpedintoacarthroughapumpatacertainpetrolshedwas60seconds.Findtherateatwhichpetrolflowsoutofthepump.
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Rateatwhichpetrolflowsoutofthepump= Volumeofpetrol Time
=
=
Example 2
Thelength,breadthandheightofacuboidshapedindoorwatertankare2m, and1mrespectively.Onanoccasionwhen the tankwascompletely filledwithwater,ittook50minutesforthetanktobeemptiedbyapipe.Findtherateatwhichwaterflowedoutthroughthepipe.(Assumethatthewaterflowedthroughthepipeuniformly) Volumeofthetank
Since1m3 = 1000 l,
thevolumeofwaterthatcanbefilledintothetank= 3 « 1000 l = 3000 l
. .. Rateatwhichwaterflowedoutthroughthepipe = capacityofthetanktime
= 3000 l 50 minutes
= 60litresperminute
Example 3
Asalinesolutionwasadministeredtoapatientatarateof0.2 mls-1. Calculatethetimeittakesfor450mlofsalinesolutiontobeadministered.
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Changeofvolume² Rateofchangeofvolume = Time
Since rate volume
seconds
minutes
minutes
Volume of Saline TimeRate of administration
time
3
ll
Exercise 22.31.Acuboidshapedtankbuilt toprovidewater toahousingschemeisof length3m,breadth2mandheight1.5m.
(i)Calculatethevolumeofthetank.(ii)Howmanylitresisthevolumeequalto?(iii)Howmuchtimewillittaketofillthistankcompletelyusingapipethrough
whichwaterflowsatauniformrateof300litresperminute?
2. Ifittook40minutestocompletelyfillacubeshapedtankofsidelength2musingapipe,whatistherateatwhichwaterflowsthroughthepipeinlitresperminute?(Hint:1m3=1000l)
3.How longwill it take to fill a fish tankof length80 cm,breadth60 cmandheight40cmusingapipethroughwhichwaterflowsatauniformrateof6 lperminute?(Hint:1cm3=1ml)
4. The volume of a tank at a water distribution centre is 1800m3. If water isdistributedfromthistankatarateof500ls-1,howmanyminuteswillittaketoemptyhalfthetank?
5.Ittook40minutestofillanemptytankusingapumpthroughwhichpetrolflowsatauniformrateof120litresperminute.Findthecapacityofthetank.
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Miscellaneous Exercise
1. Acylindricalwatertankofcross-sectionalarea0.5m2isfilledtoaheightof70cmin1minuteand10secondsbyapipe throughwhichwaterflowsatauniformrate.Calculatetherateatwhichwaterflowsoutofthepipe.
2. ThedistancebetweenrailwaystationsXandYis420km.AtrainleavesstationXat7.00p.m.andtravelstowardsstationYwithauniformspeedof100kmh-1.Anhourlater,anothertrainleavesstationYandtravelstowardsstationXwithauniformspeedof60kmh-1.Atwhattimedothetwotrainspasseachother?
3. TherailwaystationsAandBare300kmapart.Acertaintraintakes12hourstotravelfromAtoBandthenbacktoA,afterspending2hoursatB.AnothertrainleavesstationAtenhoursafterthefirsttrainleftA,andtravelstowardsB atthesameuniformspeed.Howfarhasthesecondtraintravelledwhenthetwotrainspasseachother?
Summary
²Speed = Distancetravelledbytheobject
Timetaken
Changeofvolume² Rateofchangeofvolume = Time