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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