Solutions for Physical Science 11th Us Edition by Tillery by 6alsm

2 Motion Contents

2.1DescribingMotion

2.2MeasuringMotion

Speed

Velocity

Acceleration

ScienceandSociety:TransportationandtheEnvironment Forces

2.3HorizontalMotiononLand

2.4FallingObjects

ACloserLook:ABicycleRacer’sEdge

2.5CompoundMotion

VerticalProjectiles

HorizontalProjectiles

ACloserLook:FreeFall

2.6ThreeLawsofMotion

Newton’sFirstLawofMotion

Newton’sSecondLawofMotion

WeightandMass

Newton’sThirdLawofMotion

2.7Momentum

ConservationofMomentum Impulse

2.8ForcesandCircularMotion

2.9Newton’sLawofGravitation

EarthSatellites

ACloserLook:GravityProblems

Weightlessness

PeopleBehindtheScience:IsaacNewton

Overview

ThischapterprimarilycontainsthepatternsofmotiondevelopedbyIsaacNewton(A.D 1642–1727).Newtonmademanycontributionstoscience,buthisthreelawsofmotionand hislawofgravitationarethemostfamous.Thethreelawsofmotionareconcernedwith(1) whathappenstothemotionofasingleobjectwhennounbalancedforcesareinvolved,(2) therelationshipbetweentheforce,themassofanobject,andtheresultingchangeofmotion whenanunbalancedforceisinvolved,and(3)therelationshipbetweentheforceexperienced bytwoobjectswhentheyinteract.Thelawsofmotionareuniversal,thatis,theyapply

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throughouttheknownuniverseanddescribeallmotion.Throughouttheuniversemassisa measureofinertia,andinertiaexistseverywhere.Achangeofmotion,acceleration,always resultsfromanunbalancedforceeverywhereintheknownuniverse.Finally,forcesofthe universealwayscomeinpairs.Ofthetwoforcesoneforceisalwaysequalinmagnitudebut oppositeindirectiontotheother.Thelawofgravitationisalsoapplicablethroughoutthe knownuniverse.AllobjectsintheSolarSystemthesunandtheplanets,theearthandits moon,andallorbitingsatellitesobeythelawofgravitation.Relativisticconsiderations shouldnotbementionedatthistime.ConcentrateonNewton'slawsofmotion,notEinstein's modificationsofthem.

Thekeytounderstandingpatternsofmotionistounderstandsimultaneouslytheideas representedinthethreelawsofmotion.Theseareasfollow:

1.Inertiaistheresistancetoachangeinthestateofmotionofanobjectinthe absenceofanunbalancedforce.Anobjectatrestremainsatrestandanobjectmovingina straightlineretainsitsstraight-linemotionintheabsenceofanunbalancedforce.The analysisofwhyaballmovingacrossasmoothfloorcomestoastop,aspresentedinthe previouschapter,isanimportantpartofthedevelopmentofthisconcept.Newton'sfirstlaw ofmotionisalsoknownasthelawofinertia.

2.Massisdefinedasameasureofinertia,thatis,aresistancetoachangeinthestate ofmotionofanobject.Thusthegreaterthemassthegreatertheresistancetoachangeinthe stateofmotionofanobject.Accelerationisachangeinthestateofmotionofanobject. Accordingtothedefinitiondevelopedinthepreviouschapter,anobjectthatspeedsup,slows down,orchangesitsdirectionoftravelisundergoinganacceleration.Studentswhohave difficultyacceptingthemeaningsofmassandaccelerationoftenhavelessdifficultyifthey aretoldthesearedefinitionsofthequantities.Aforceisapushorapullthatiscapableof causingachangeinthestateofmotionofanobject,thatis,capableofproducingan acceleration.Theresultingaccelerationisalwaysinthesamedirectionasthedirectionofthe appliedforce.Newton'ssecondlawofmotionisarelationshipbetweenmass,acceleration, andanunbalancedforcethatbecomesclearwhentheconceptualmeaningofthesetermsis understood.Therelationshipisthatthegreaterthemass(inertia),thegreatertheforce requiredtobringaboutachangeinthestateofmotion(acceleration).Insymbolform,the relationshipisa F/m,orthemorefamiliarF ma.Sinceanewtonofforceisdefinedin termsofacertainmass(1kg)andacertainacceleration(1m/s2),theunitsarethesameon bothsidesandtherelationshipbecomesanequation,orF=ma.Thisisanexampleofan equationthatdefinesaconcept(seechapter1).

3.Asingleforceneveroccursalone;aforceisalwaysproducedbytheinteractionof twoormoreobjects.Thereisalwaysamatchedandoppositeforcethatoccursatthesame time,andNewton'ssecondlawofmotionisastatementofthisrelationship.

Suggestions

1.Theneedforprecisionandexactunderstandingshouldbeemphasizedasthevarious termssuchasspeed,velocity,rate,distance,acceleration,andothersarepresented. Stressthereasoningbehindeachequation,forexample,thatvelocityisaratiothat describesapropertyofobjectsinmotion.Likewise,accelerationisatimerateofchange ofvelocity,sovf-vi/tnotonlymakessensebutcanbereasonedoutratherthan memorized.Alsostresstheneedtoshowhowunitsarehandledinsolvingproblems. Thecompletemanipulationofunitsmathematicallyisstressedthroughoutthisbook. Typicallystudentsmustbeshownhowunitworkservesasacheckonproblem-solving steps.Studentsaresometimesconfusedbytheuseofthesymbol“v”forbothspeedand velocity.Explainthatspeedisthesamequantityasvelocitybutwithoutdirection,sothe samesymbolisusedtosimplifythings.Onthepointofsimplifyingthings,avoidthe temptationtousecalculusinanyexplanationordiscussion.

2.Studentsaregenerallyinterestedin“relativetowhat”questionsconcerningmotion.For example,whatisthespeedofaninsectflyingat5mphfromthefronttothebackofa busmovingat50mph?Whatdoyouobservehappeningtoanobjectdroppedinsidean airplanemovingat600mph?Whatwouldanobserveroutsidetheairplaneobserve happeningtotheobject?

3.Thediscussionofwhathappenstoaballrollingacrossthefloorisanimportantone,and manystudentswhothinkfroman“Aristotelianframework”aresurprisedbytheanalysis. Whendiscussingtheroleoffrictionandobjectsmovingontheearth’ssurface,itisoften interestingtoaskwhyplanetsdonotstopmovingaroundthesun.Spuronthe discussionbyansweringwithanotherquestion,whyshouldtheystop?Itmightbe helpfultoreviewthemeaningofvectorarrowsthatrepresentforces.

4.Anotherwaytoconsideraccelerationistoask,Howfastdoes“howfast”change?If studentshavelearnedtheconceptofaratiotheywillunderstandtheconceptofuniform straight-linemotion.Theaccelerationconcepts,however,requiretheuseofaratio withinanotherratio,thatis,achangeofvelocity(aratiowithin)perunitoftime(the accelerationratio).Thisunderstandingisnecessary(alongwithsomebasicmathskills) tounderstandthemeaningofsuchunitsasm/s2 .

5.Demonstrationsthatillustratethecharacteristicsofprojectilemotionareillustratedin severaldevicesfoundinscientificcatalogs.Amongthemostimpressiveisthe“monkey andhunter”demonstration.Studentsenjoythisdemonstrationalongwiththehumorthat theinstructorcaninducewhileperformingit.

6.Therearemanydemonstrationsanddevicesavailablefromscientificsuppliersthat readilyillustratethelawsofmotion.However,noneseemsbetterthanthepersonal experiencesofstudentswhohavestoodintheaisleofabusasitstartsmoving,turnsa corner,orcomestoastop.Usethethreelawsofmotiontoanalyzetheinertia,forces, andresultingchangesofmotionofastudentstandinginsuchanaisleofabus.

7.Stressthatweightandmassaretwoentirelydifferentconcepts.Youwillprobablyhave toemphasizemorethanoncethatweightisanothernameforthegravitationalforce actingonanobject,andthatweightvariesfromplacetoplacewhilemassdoesnot.Use thesecondlawofmotiontoshowhowweightcanbeusedtocalculatemass.Alarge demonstrationspringscalecalibratedinnewtonscanbeusedtoshowthata1-kgmass weighs9.8N.Othermassescanbeweighedtoshowthatweightandmassare proportionalinagivenlocation.

8.Insolvingproblemsinvolvingthethirdlawofmotion,itishelpfulforstudentstorealize thatachangeinthestateofmotionalwaysoccursinthesamedirectionasthedirection ofanappliedforce.IfyouapplyanunbalancedforceonaballtowardtheNorth,you wouldexpecttheballtomovetowardtheNorth.Thusifonestartswalkingtowardthe Northaforcemusthavebeenappliedinthesamedirection.Thefootpushedonthe groundintheoppositedirection,soitmustbethattheequalandoppositeforceofthe groundpushingonthefootiswhatcausedthemotiontowardtheNorth.Itseemsalmost anthropomorphictostatethatthegroundpushedonafoot,butnootheransweris possiblewiththisanalysis.Thenextstep,sotospeak,istorealizethatsincetheforceof thefootonthegroundequalstheforceofthegroundonthefoot(thirdlaw).Thenthe massoftheearthtimestheaccelerationoftheearth(secondlaw)mustequalthemassof thepersontimestheaccelerationoftheperson(ma=ma).Thismeansatleasttwo things:(1)thattheearthmustmovewhenyouwalkacrossthesurface(earth's accelerationmustbegreaterthanzero)and(2)thattheearthwouldmovewiththesame accelerationasthepersonifbothhadthesamemass.Studentsaremakingprogress whentheycanunderstandormakethiskindofanalysis.

9.Astrongcordattachedtoalargecoffeecanhalffilledwithwatermakesaninteresting demonstrationofcentripetalforceandinertiawhenwhirledoverhead.Practicethis, however,beforetryingbeforeaclass.

10.Additionaldemonstrations:

(a)Showthestroboscopiceffectasameansofmeasuringmotion.Useastrobelightor handstroboscopes,forexample,to“stop”themotionofaspinningwheelofanupsidedownbicycle.

(b)Rollasteelballdownalongrampandmarkthedistanceattheendofeachsecond. Plotdistancevs.timeanddistancevs.timesquaredtoverifytheaccelerationequation.

(c)Crumpleasheetofpapertightlyintoasmallball.Dropthecrumpledballandasheet ofuncrumpledpaperfromthesameheight.Discusswhichisacceleratedat9.8m/s2and therollofairresistance.

(d)Usethecommercialapparatusthatshootsormovesoneballhorizontallyanddrops anotherballverticallyatthesametime.Asingle“click”meansthatbothballshitthe flooratthesametime.Thisillustratestheindependenceofvelocities.

(e)Dropasmallsteelballfromthehighestplacepracticalintoatubofwater.Makesure thisisdoneonadaywithoutwindandwithnopersonnearthetub.Timethefallwitha stopwatch.Measuretheverticaldistanceaccurately,thenfindgfromd=1/2gt2 .

(f)Useaspringscaletoshowthata1.0-kgmassweighs9.8N.Useothermassesto showthattheweightofanobjectisalwaysproportionaltothemassinagivenlocation.

(g)UseanairtracktoillustrateNewton’sfirstandsecondlawofmotion.Ifanairtrack isnotavailable,consideraslaboficeordryiceonasmoothdemonstrationtabletop Woodblockscanbesetontheicetoaddmass.

(h)Willajetplanebackeduptoabrickwalltakeofffasterthanoneoutintheopen? Comparethejetplanetoaballoonfilledwithair,thatis,ajetofescapingairpropelsthe balloon.Thus,themovementisaconsequenceofNewton’sthirdlawandthebrickwall willmakenodifference–ajetplanebackeduptoabrickwilltakeoffthesameasan identicaljetplaneoutintheopen.

(i)SeatyourselfonasmallcartwithaCO2fireextinguisherorabottleofcompressed airfromtheshop.Holdthedevicebetweenyourfeetandlegswiththeescapevalve pointedawayfromyourbody.Withthewayclearbehindyou,carefullydischargea shortburstofgasasyouaccelerate.Thisattention-grabberaffordsanopportunityto reviewallthreeofNewton’sLawsofmotion.

(j)Demonstratethattheaccelerationofafreelyfallingobjectisindependentofweight. Useacommercial“free-falltube”ifoneisavailable.Ifnot,tryalarge-diameter1-meter glassorplastictubewithasolidstopperinoneendandaone-holestopperintheother. Placeacoilandafeatherinthetube,thenconnecttheone-holestoppertoavacuum pump.Invertthetubetoshowhowthecoinandfeatherfallinair.Pumpairfromthe tubethenagaininverttoshowthecoinandfeatherinfreefall.

ForClassDiscussions

1.Neglectingairresistance,aballinfreefallwillhave a.constantspeedandconstantacceleration. b.increasingspeedandincreasingacceleration. c.increasingspeedanddecreasingacceleration. d.increasingspeedandconstantacceleration. e.decreasingspeedandincreasingacceleration.

2.Neglectingairresistance,aballrollingdowntheslopeofasteephillwillhave a.constantspeedandconstantacceleration.

b.increasingspeedandincreasingacceleration.

c.increasingspeedanddecreasingacceleration.

d.increasingspeedandconstantacceleration.

e.decreasingspeedandincreasingacceleration.

3.Againneglectingairresistance,aballthrownstraightupwillcometoamomentarystopat thetopofthepath.Whatistheaccelerationoftheballduringthisstop?

a.9.8m/s2 .

b.zero.

c.lessthan9.8m/s2 .

d.morethan9.8m/s2 .

4.Againneglectingairresistance,theballthrownstraightupcomestoamomentarystopat thetopofthepath,thenfallsfor1.0s.Whatisspeedoftheballafterfalling1.0s?

a.1m/s

b.4.9m/s

c.9.8m/s

d.19.6m/s

5.Yetagainneglectingairresistance,theballthrownstraightupcomestoamomentarystop atthetopofthepath,thenfallsfor2.0s.Whatdistancedidtheballfallduringthe2.0s?

a.1m

b.4.9m

c.9.8m

d.19.6m

6.Aballisthrownstraightupatthesametimeaballisthrownstraightdownfromabridge, withthesameinitialspeed.Neglectingairresistance,whichballwouldhaveagreaterspeed whenithitstheground?

a.Theonethrownstraightup.

b.Theonethrownstraightdown.

c.Bothballswouldhavethesamespeed.

7.Afterbeingreleased,aballthrownstraightdownfromabridgewouldhavean accelerationof

a.9.8m/s2

b.zero.

c.lessthan9.8m/s2 .

d.morethan9.8m/s2 .

8.Agunisaimedatanapplehangingfromatree.Theinstantthegunisfiredtheapplefalls totheground,andthebullet

a.hitstheapple.

b.arriveslate,missingtheapple.

c.mayormaynothittheapple,dependingonhowfastitismoving.

9.Youareatrestwithagrocerycartatthesupermarket,whenyouseean“opening”ina checkoutline.Youapplyacertainforcetothecartforashorttimeandacquireacertain speed.Neglectingfriction,howlongwouldyouhavetopushwithhalftheforcetoacquire thesamefinalspeed?

a.one-fourthaslong.

b.one-halfaslong.

c.fortwiceaslong.

d.forfourtimesaslong.

10.Onceagainyouareatrestwithagrocerycartatthesupermarket,whenyouapplya certainforcetothecartforashorttimeandacquireacertainspeed.Supposeyouhadbought moregroceries,enoughtodoublethemassofthegroceriesandcart.Neglectingfriction, doublingthemasswouldhavewhateffectontheresultingfinalspeedifyouusedthesame forceforthesamelengthoftime?Thenewfinalspeedwouldbe

a.one-fourth.

b.one-half.

c.doubled.

d.quadrupled.

11.Youaremovingagrocerycartataconstantspeedinastraightlinedowntheaisleofa store.Theforcesonthecartare

a.unbalanced,inthedirectionofthemovement.

b.balanced,withanetforceofzero.

c.equaltotheforceofgravityactingonthecart.

d.greaterthanthefrictionalforcesopposingthemotionofthecart.

12.ConsideringthegravitationalattractionbetweentheEarthandMoon,the a.moremassiveEarthpullsharderonthelessmassiveMoon. b.lessmassiveMoonpullsharderonthemoremassiveEarth.

c.attractionbetweentheEarthandMoonandtheMoonandEarthareequal. d.attractionvarieswiththeMoonphase,beinggreatestatafullmoon.

13.Youareoutsideastore,movingaloadedgrocerycartdownthestreetonaverysteep hill.Itisdifficult,butyouareabletopullbackonthehandleandkeepthecartmovingdown thestreetinastraightlineandataconstantspeed.Theforcesonthecartare

a.unbalanced,inthedirectionofthemovement.

b.balanced,withanetforceofzero.

c.equaltotheforceofgravityactingonthecart.

d.greaterthanthefrictionalforcesopposingthemotionofthecart.

14.Whichofthefollowingmustbetrueaboutahorsepullingabuggy?

a.Accordingtothethirdlawofmotion,thehorsepullsonthebuggyandthebuggy pullsonthehorsewithanequalandoppositeforce.Thereforethenetforceiszero andthebuggycannotmove.

b.Sincetheymoveforward,thismeansthehorseispullingharderonthebuggy thanthebuggyispullingonthehorse.

c.Theactionforcefromthehorseisquickerthanthereactionforcefromthebuggy, sothebuggymovesforward.

d.Theaction-reactionforcebetweenthehorseandbuggyareequal,butthe resistingfrictionalforceonthebuggyissmallersinceitisonwheels.

15.Supposeyouhaveachoiceofdrivingyourspeedingcarheadonintoamassiveconcrete wallorhittinganidenticalcarheadon.Whichwouldproducethegreatestchangeinthe momentumofyourcar?

a.Theidenticalcar.

b.Theconcretewall.

c.Bothwouldbeequal.

16.Asmall,compactcarandalargesportsutilityvehiclecollideheadonandsticktogether. Whichvehiclehadthelargermomentumchange?

a.Thesmall,compactcar.

b.Thelargesportsutilityvehicle.

c.Bothwouldbeequal.

17.Againconsiderthesmall,compactcarandlargesportsutilityvehiclethatcollidedhead onandstucktogether.Whichexperiencedthelargerdecelerationduringthecollision?

a.Thesmall,compactcar.

b.Thelargesportsutilityvehicle.

c.Bothwouldbeequal.

18.Certainprofessionalfootballplayerscanthrowafootballsofastthatitmoves horizontallyinaflattrajectory.

a.True

b.False

Answers:1d,2c(a=gstraightdown,butdecreasestozeroonalevelsurface),3a(accelerationisarateofchangeof velocityandgravityisacting,F=ma,soamustbeoccurring),4b(initialspeedwaszero,averagespeedisone-halfoffinal speed),5d,6c,7a(afterreleaseonlygravityactsonball),8a(theappleandbulletacceleratedownwardtogether,nomatter howfastthebulletismoving),9c,10b,11b,12c,13b,14d,15c,16c,17a,18b.

AnswerstoQuestionsforThought

1.Thespeedoftheinsectrelativetothegroundisthe50.0mi/hofthebusplusthe5.0mi/h oftheinsectforatotalof55mi/h.Relativetothebusalonethespeedoftheinsectis5.0 mi/h.

2.Afteritleavestheriflebarrel,theforceofgravityactingstraightdownistheonlyforce actingonthebullet.

3.Gravitydoesnotdependuponsomemediumsoitcanoperateinavacuum.

4.Yes,thesmallcarwouldhavetobemovingwithamuchhighervelocity,butitcanhave thesamemomentumsincemomentumismasstimesvelocity.

5.Anetforceofzeroisrequiredtomaintainaconstantvelocity.Theforcefromtheengine balancestheforceoffrictionasacardriveswithaconstantvelocity.

6.Theactionandreactionforcesarebetweentwoobjectsthatareinteracting.An unbalancedforceoccursonasingleobjectastheresultofoneormoreinteractionswith otherobjects.

7.Bendingyourkneesasyouhitthegroundextendsthestoppingtime.Thisisimportant sincethechangeofmomentumisequaltotheimpulse,whichisforcetimesthetime.A greatertimethereforemeanslessforcewhencomingtoastop.

8.Yourweightcanchangefromplacetoplacebecauseweightisadownwardforcefrom gravitationalattractiononyourmassandtheforceofgravitycanvaryfromplaceto place.

9.Nothing!ThereisnoforceparalleltothemotiontoincreaseordecreaseEarth'sspeed, sothespeedremainsconstant.

10.Ifyouhavesomethingtothrow,suchascarkeysorasnowball,youcaneasilygetoffthe frictionlessice.Sincetheforceyouapplytothethrownobjectresultsinanequaland oppositeforce(thethirdlawofmotion),youwillmoveintheoppositedirectionasthe objectisthrown(thesecondlawofmotion).Thesameresultcanbeachievedby blowingapuffofairinadirectionoppositetothewayyouwishtomove.

11.Consideringeverythingelsetobeequal,thetworocketswillhavethesameacceleration. Inbothcases,theaccelerationresultsasburningrocketfuelescapestherocket,exerting anunbalancedforceontherocket(thirdlaw)andtherocketacceleratesduringthe appliedforce(secondlaw).Theaccelerationhasnothingtodowiththeescapinggases havingsomethingto“pushagainst.”

12.Theastronautistravelingwiththesamespeedasthespaceshipasheorsheleaves.Ifno netforceisappliedparalleltothedirectionofmotionofeithertheastronautorthe spaceship,theywillbothmaintainaconstantvelocityandwillstaytogether.

ForFurtherAnalysis

1.Similar–bothspeedandvelocitydescribeamagnitudeofmotion,thatis,howfast somethingismoving.Differences–velocitymustspecifyadirection;speeddoesnot.

2.Similar–bothvelocityandaccelerationdescribemotion.Differences–velocity specifieshowfastsomethingismovinginaparticulardirection;acceleration specifiedachangeofvelocity(speed,direction,orboth).

3.Thisrequiresacomparisonofbeliefsandananalysisandcomparisonwithnew contexts.Answerswillvary,butshouldshowunderstandingofNewton’sthreelaws ofmotion.

4.Thisquestionrequiresbothclarifyingbeliefsandcomparingperspectives.Answers willvary.

5.Requiresrefiningofunderstanding.Massisameasureofinertia,meaningaresistance toachangeofmotion.Weightisgravitationalaccelerationactingonamass.Since gravitycanvaryfromplacetoplace,theweightasaresultofgravitywillalsovary fromplacetoplace.

6.Requiresclarifyingandanalyzingseveralconceptualunderstandings.Newton’sfirst lawofmotiontellsusthatmotionisunchangedinastraightlinewithoutan unbalancedforce.Anobjectmovingontheendofastringinacircularpathispulled outofastraightlinebyacentripetalforceonthestring.Theobjectwillmoveoffina straightlineifthestringbreaks.Itwouldmoveoffinsomeotherdirectionifother forceswereinvolved.

GroupBSolutions

1.Thedistanceandtimeareknownandtheproblemaskedfortheaveragevelocity.Listing thesequantitieswithheirsymbols,wehave

=4.5h

Thesearethequantitiesinvolvedintheaveragespeedequation,whichisalreadysolvedfor theunknownaveragespeed: 400.0km = 4.5h km =89 h d v= t

2.Listingthequantitiesgiveninthisproblem,wehave =16.0km =45min =? d t v

Theproblemspecifiesthattheanswershouldbeinkm/h.Weseethat45minutesis45/60,or 3/4,or0.75ofanhour,andtheappropriateunitsare: =16.0km =0.75h =? d t v

Substitutingtheknownquantities,wehave 16.0km = 0.75h km =21.3333 h km =21 h d v= t

3.Weightisthegravitationalforceonanobject.Newton’ssecondlawofmotionisF=ma, andsinceweight(w)isaforce(F),thenF=wandthesecondlawcanbewrittenasw=ma

Theacceleration(a)istheaccelerationduetogravity(g),sotheequationforweightisw= mg. (a)

(b)

4.Listingtheknownandunknownquantities,

ThesearethequantitiesfoundinNewton’ssecondlawofmotion,F=ma,whichisalready solvedforforce(F).Thus,

5.Listingtheknownandunknownquantities,

6.Weseethat30.0minutesis1/2or0.50ofanhour,and 15.0km 0.50h 30.0km/h d v= t

7.Weseethatthedistanceunitsarekilometers,butthevelocityunitsarem/s.Weneedto convertkmtom,then

8.Thedistancethatasoundwiththisvelocitytravelsinthegiventimeis

Sincethesoundtraveledfromyoutothecliffandthenback,thecliffmustbe172m/2= 86.0maway.

9.Notethatthetwospeedsgiven(80.0km/hand90.0km/h)areaveragespeedsfortwo differentlegsofatrip.Theyarenottheinitialandfinalspeedsofanacceleratingobject, soyoucannotaddthemtogetheranddivideby2.Theaveragespeedforthetotal(entire) tripcanbefoundfromdefinitionofaveragespeed,thatis,averagespeedisthetotal distancecovereddividedbythetotaltimeelapsed.Therefore,westartbyfindingthe distancecoveredforeachofthetwolegsofthetrip:

Leg1distance80.01.00h h 80.0km d vdvt

Leg2distance90.02.00h h

180.0km

Totaldistance(leg1plusleg2)=260.0km

Totaltime=3.00h

260.0km86.7km/h 3.00h d v t

12.Therelationshipbetweenaveragevelocity(v),distance(d),andtime(t)canbesolved fortime: 380,000,000m 11,000m s

13.Therelationshipbetweenaveragevelocity(v),distance(d),andtime(t)canbesolved fordistance:

14.“Howmanyhours...”isaquestionabouttimeandthedistanceisgiven.Sincethe distanceisgiveninkmandthespeedinm/s,aunitconversionisneeded.Theeasiestthing todoistoconvertkmtom.Thereare1,000minakm,and

Therelationshipbetweenaveragevelocity(v),distance(d),andtime(t)canbesolved fortime:

15.Theinitialvelocity(vi)isgivenas724m/s,thefinalvelocity(vf)isgivenas675m/s,and thetimeisgivenas5.00s.Acceleration,includingadecelerationornegative acceleration,isfoundfromachangeofvelocityduringagiventime.Thus,

(Thenegativesignmeansanegativeacceleration,ordeceleration.)

16.Arockthrownstraightupdeceleratestoavelocityofzero,andthenacceleratesbackto thesurfacejustasadroppedballwoulddofromtheheightreached.Thusthetime deceleratingupwardisthesameasthetimeacceleratingdownward.Theballreturnsto thesurfacewiththesamevelocitywithwhichitwasthrown(neglectingfriction).

Therefore:

17.Thesethreequestionsareeasilyansweredbyusingthethreesetsofrelationships,or equationsthatwerepresentedinthischapter:

18.Notethatthisproblemcanbesolvedwithaseriesofthreestepsasintheprevious problem.Itcanalsobesolvedbytheequationthatcombinesalltherelationshipsintoone step.Eithermethodisacceptable,butthefollowingexampleofaonestepsolutionreduces thepossibilitiesoferrorsincefewercalculationsareinvolved:

21.Massofball:

22.Listingtheknownandunknownquantities:

Shellm=30.0kg

Shell v=500m/s

Cannonm=2,000kg

Cannonv=?m/s

Thisisaconservationofmomentumquestion,wheretheshellandcannoncanbe consideredasasystemofinteractingobjects:

(a)

20.0m/s10.0m/s 1,000.0kg5.00s 1,000.010.0kgm s 5.00s 2,000kgm1 ss 2.0010N

Name Section Date

Experiment 2: Ratios

Introduction

The purpose of this introductory laboratory exercise is to investigate how measurement data are simplified in order to generalize and identify trends in the data Data concerning two quantities will be compared as a ratio, which is generally defined as a relationship between numbers or quantities A ratio is usually simplified by dividing one number by another

Procedure

Part A: Circles and Proportionality Constants

1 Obtain three different sizes of cups, containers, or beakers with circular bases Trace around the bottoms to make three large but different-sized circles on a blank sheet of paper.

2. Mark the diameter on each circle by drawing a straight line across the center. Measure each diameter in mm and record the measurements in Data Table 2.1. Repeat this procedure for each circle for a total of three trials.

3. Measure the circumference of each object by carefully positioning a length of string around the object’s base, then grasping the place where the string ends meet. Measure the length in mm and record the measurements for each circle in Data Table 2 1 Repeat the procedure for each circle for a total of three trials. Find the ratio of the circumference of each circle to its diameter. Record the ratio for each trial in Data Table 2 1 on page 23

4 The ratio of the circumference of a circle to its diameter is known as pi (symbol π), which has a value of 3.14… (the periods mean many decimal places). Average all the values of π in Data Table 2 1 and calculate the experimental error

Figure 2.1

Part B: Area and Volume Ratios

1. Obtain one cube from the supply of same-sized cubes in the laboratory. Note that a cube has six sides, or six units of surface area. The side of a cube is also called a face, so each cube has six identical faces with the same area. The overall surface area of a cube can be found by measuring the length and width of one face (which should have the same value) and then multiplying (length)(width)(number of faces). Use a metric ruler to measure the cube, then calculate the overall surface area and record your finding for this small cube in Data Table 2 2 on page 23

2. The volume of a cube can be found by multiplying the (length)(width)(height). Measure and calculate the volume of the cube and record your finding for this small cube in Data Table 2.2.

3 Calculate the ratio of surface area to volume and record it in Data Table 2 2

4 Build a medium-sized cube from eight of the small cubes stacked into one solid cube Find and record (a) the overall surface area, (b) the volume, and (c) the overall surface area to volume ratio, and record them in Data Table 2 2

5. Build a large cube from 27 of the small cubes stacked into one solid cube. Again, find and record the overall surface area, volume, and overall surface area to volume ratio and record your findings in Data Table 2.2.

6. Describe a pattern, or generalization, concerning the volume of a cube and its surface area to volume ratio For example, as the volume of a cube increases, what happens to the surface area to volume ratio? How do these two quantities change together for larger and larger cubes?

Part C: Mass and Volume

1. Obtain at least three straight-sided, rectangular containers. Measure the length, width, and height inside the container (you do not want the container material included in the volume). Record these measurements in Data Table 2.3 (page 23) in rows 1, 2, and 3. Calculate and record the volume of each container in row 4 of the data table.

2. Measure and record the mass of each container in row 5 of the data table. Measure and record the mass of each container when “level full” of tap water Record each mass in row 6 of the data table Calculate and record the mass of the water in each container (mass of container plus water minus mass of empty container, or row 6 minus row 5 for each container) Record the mass of the water in row 7 of the data table.

Meas ure the volume here

3. Use a graduated cylinder to measure the volume of water in each of the three containers. Be sure to get all the water into the graduated cylinder. Record the water volume of each container in milliliters (mL) in row 8 of the data table.

4 Calculate the ratio of cubic centimeters (cm3) to mL for each container by dividing the volume in cubic centimeters (row 4 data) by the volume in milliliters (row 8 data). Record your findings in the data table

5 Calculate the ratio of mass per unit volume for each container by dividing the mass in grams (row 7 data) by the volume in milliliters (row 8 data). Record your results in the data table.

Figure 2.2

Figure 2.3

6. Make a graph of the mass in grams (row 7 data) and the volume in milliliters (row 8 data) to picture the mass per unit volume ratio found in step 5. Put the volume on the x-axis (horizontal axis) and the mass on the y-axis (the vertical axis) The mass and volume data from each container will be a data point, so there will be a total of three data points.

7. Draw a straight line on your graph that is as close as possible to the three data points and the origin (0, 0) as a fourth point. If you wonder why (0, 0) is also a data point, ask yourself about the mass of a zero volume of water!

8 Calculate the slope of your graph (See appendix II on page 397 for information on calculating a slope.)

9 Calculate your experimental error Use 1 0 g/mL (grams per milliliter) as the accepted value

10 Density is defined as mass per unit volume, or mass/volume The slope of a straight line is also a ratio, defined as the ratio of the change in the y-value per the change in the x-value. Discuss why the volume data was placed on the x-axis and mass on the y-axis and not vice versa

11. Was the purpose of this lab accomplished? Why or why not? (Your answer to this question should show thoughtful analysis and careful, thorough thinking.)

Results

1. What is a ratio? Give several examples of ratios in everyday use.

A r e l a t i o n s h i p b e t w e e n n u m b e r s o r q u a n t i t i e s . E x a m p l e s : 1 0 0 c e n t s p

2. How is the value of π obtained? Why does π not have units?

B y t a k i n g t h e r a t i o o f t h e c i r c u m f e r e n c e o f a c i r c l e t o t h e d i a m e t e r .

B o t h c i r c u m f e r e n c e a n d

c i r c u m f e r e n c e b y t h e d i a m e t e r t h e u n i t s c a n c e l o u t .

3. Describe what happens to the surface area to volume ratio for larger and larger cubes.

Predict if this pattern would also be observed for other geometric shapes such as a sphere

Explain the reasoning behind your prediction.

Su r f a c e a r e a t o v o l u m e r a t i o a p p r o a c h e s z e r o f o r l a r g e r a n d l a r g e r c u b e s .

T h i s p a t t e r n w o u l d a l s o b e t r u e f o r o t h e r s h a p e s b e c a u s e s u r f a c e a r e a i s p r o p o r t i o n a l t o l e n g t h

s q u a r e d a n d v o l u m e i s p r o p o r t i o n a l t o l e n g t h c u b e

1 / l e n g t h w h i c h g o e s t o w a r d z e r o a s t h e o b j e c t g e t s l a r g e r .

4. Why does crushed ice melt faster than the same amount of ice in a single block?

T h e r e i s m o r e s u r f a c e a r e a f o r t h e s m a l l e r p i e c e s o f i c e t h a n t h e s i n g l e b l o c k , t h e a i r i s i n

c o n t a c t w i t h m o r e o f t h e i c e , s o i t m e l t s f a s t e r .

5. Which contains more potato skins: 10 pounds of small potatoes or 10 pounds of large potatoes? Explain the reasoning behind your answer in terms of this laboratory investigation.

T h e 1 0 l b s o f s m a l l p o t a t o e s h a v e m o r e p o t a t o s k i n s . T h e r e i s m o r e t o t a l s u r f a c e a r e a f o r

t h e s a m e s m a l l e r p o t a t o e s t h a n t h e l a r g e r p o t a t o e s .

6. Using your own words, explain the meaning of the slope of a straight-line graph. What does it tell you about the two graphed quantities? T h e s l o

7. Explain why a slope of mass/volume of a particular substance also identifies the density of that substance

Problems

An aluminum block that is 1 m × 2 m × 3 m has a mass of 1.62 × 104 kilograms (kg). The following problems concern this aluminum block:

Figure 2 4

l. What is the volume of the block in cubic meters (m3)?

2 What are the dimensions of the block in centimeters (cm)?

3. Make a sketch of the aluminum block and show the area of each face in square centimeters (cm2).

4. What is the volume of the block expressed in cubic centimeters (cm3)?

5. What is the mass of the block expressed in grams (g)?

6. What is the ratio of mass (g) to volume (cm3) for aluminum?

7. Under what topic would you look in the index of a reference book to check your answer to question 6? Explain

Invitation to Inquiry

If you have popped a batch of popcorn, you know that a given batch of kernels might pop into big and fluffy popcorn. But another batch might not be big and fluffy and some of the kernels might not pop. Popcorn pops because each kernel contains moisture that vaporizes into steam, expanding rapidly and causing the kernel to explode, or pop Here are some questions you might want to consider investigating to find out more about popcorn: Does the ratio of water to kernel mass influence the final fluffy size of popped corn? (Hint: measure mass of kernel before and after popping). Is there an optimum ratio of water to kernel mass for making bigger popped kernels? Is the size of the popped kernels influenced by how rapidly or how slowly you heat the kernels? Can you influence the size of popped kernels by drying or adding moisture to the unpopped kernels? Is a different ratio of moisture to kernel mass better for use in a microwave than in a convention corn popper? Perhaps you can think of more questions about popcorn.

Data Table 2.1 Circles and Ratios

Small CircleMedium CircleLarge Circle

Data Table 2.2 Area and Volume Ratios

Data Table 2.3 Mass and Volume Ratios

2. Width of container

3. Height of container

4. Calculated volume