Physics for the IB Diploma- 2nd Edition

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JohnAllumandChristopherTalbot2014

Firsteditionpublishedin2012byHodderEducationAnHachetteUKCompany338EustonRoadLondonNW13BH

Thissecondeditionpublished2014

Impressionnumber 54321Year 20182017201620152014

Allrightsreserved.ApartfromanyusepermittedunderUKcopyrightlaw,nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopyingandrecording,orheldwithinanyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisherorunderlicencefromtheCopyrightLicensingAgencyLimited.Furtherdetailsofsuchlicences(forreprographicreproduction)maybeobtainedfromtheCopyrightLicensingAgencyLimited,SaffronHouse,610KirbyStreet,LondonEC1N8TS.

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AcataloguerecordforthistitleisavailablefromtheBritishLibrary

ISBN:9781471829048eISBN:9781471829277

ContentsIntroduction

Core

Chapter1Measurementsanduncertainties1.1Measurementsinphysics1.2Uncertaintiesanderrors1.3Vectorsandscalars

Chapter2Mechanics2.1Motion2.2Forces2.3Work,energyandpower2.4Momentumandimpulse

Chapter3Thermalphysics3.1Thermalconcepts3.2Modellingagas

Chapter4Waves4.1Oscillations4.2Travellingwaves4.3Wavecharacteristics4.4Wavebehaviour4.5Standingwaves

Chapter5Electricityandmagnetism5.1Electricfields5.2Heatingeffectofelectriccurrents5.3Electriccells5.4Magneticeffectsofelectriccurrents

Chapter6Circularmotionandgravitation6.1Circularmotion6.2Newtonslawofgravitation

Chapter7Atomic,nuclearandparticlephysics7.1Discreteenergyandradioactivity7.2Nuclearreactions

7.3Thestructureofmatter

Chapter8Energyproduction8.1Energysources8.2Thermalenergytransfer

Additionalhigherlevel(AHL)

Chapter9Wavephenomena9.1Simpleharmonicmotion9.2Single-slitdiffraction9.3Interference9.4Resolution9.5Dopplereffect

Chapter10Fields10.1Describingfields10.2Fieldsatwork

Chapter11Electromagneticinduction11.1Electromagneticinduction11.2Powergenerationandtransmission11.3Capacitance

Chapter12Quantumandnuclearphysics12.1Theinteractionofmatterwithradiation12.2Nuclearphysics

Options

Availableonthewebsiteaccompanyingthisbook:www.hodderplus.com/ibphysics

OptionAChapter13Relativity13.1Thebeginningsofrelativity13.2Lorentztransformations13.3Spacetimediagrams13.4Relativisticmechanics(AHL)13.5Generalrelativity(AHL)

OptionBChapter14Engineeringphysics

14.1Rigidbodiesandrotationaldynamics14.2Thermodynamics14.3Fluidsandfluiddynamics(AHL)14.4Forcedvibrationsandresonance(AHL)

OptionCChapter15Imaging15.1Introductiontoimaging15.2Imaginginstrumentation15.3Fibreoptics15.4Medicalimaging(AHL)

OptionDChapter16Astrophysics16.1Stellarquantities16.2Stellarcharacteristicsandstellarevolution16.3Cosmology16.4Stellarprocesses(AHL)16.5Furthercosmology(AHL)

Appendix

Graphsanddataanalysis

Answeringexaminationquestions

Answers,glossaryandindexAnswerstotheself-assessmentquestionsandexaminationquestionsinChapters112appearinthebook;answersfortheOptions,Chapters1316,areavailableonthewebsiteaccompanyingthisbook:www.hodderplus.com/ibphysics.

Answerstotheself-assessmentquestionsinChapters1to12

AnswerstotheexaminationquestionsinChapters1to12

Glossary

Acknowledgements

Index

IntroductionWelcometothesecondeditionofPhysicsfortheIBDiploma.Thecontentandstructureofthissecondeditionhasbeencompletelyrevisedtomeetthedemandsofthe2014IBDiplomaProgrammePhysicsGuide.WithintheIBDiplomaProgramme,thephysicscontentisorganizedintocompulsorytopicsplusa

numberofoptions,fromwhichallstudentsselectone.TheorganizationofthisresourceexactlyfollowstheIBPhysicsGuidesequence:Core:Chapters18coverthecommoncoretopicsforStandardandHigherLevelstudents.AdditionalHigherLevel(AHL):Chapters912covertheadditionaltopicsforHigherLevelstudents.

Options:Chapters1316coverOptionsA,B,CandDrespectively.EachoftheseisavailabletobothStandardandHigherLevelstudents.(HigherLevelstudentsstudymoretopicswithinthesameoption.)

EachofthecoreandAHLtopicsisthesubjectofacorrespondingsinglechapterinthePhysicsfortheIBDiplomaprintedbook.TheOptions(Chapters1316)areavailableonthewebsiteaccompanyingthisbook,asareuseful

appendicesandadditionalstudentsupport(includingStartingpointsandSummaryofknowledge):www.hoddereducation.com/IBextrasTherearetwoadditionalshortchaptersofferingphysics-specificadviceontheskillsnecessaryfor

GraphsanddataanalysisandPreparingfortheIBDiplomaPhysicsexamination,includingexplanationsofthecommandterms.Thesechapterscanbefoundontheaccompanyingwebsite.SpecialfeaturesofthechaptersofPhysicsfortheIBDiplomaaredescribedbelow.

Thetextiswritteninstraightforwardlanguage,withoutphrasesoridiomsthatmightconfusestudentsforwhomEnglishisasecondlanguage.

ThedepthoftreatmentoftopicshasbeencarefullyplannedtoaccuratelyreflecttheobjectivesoftheIBsyllabusandtherequirementsoftheexaminations.

TheNatureofScienceisanimportantnewaspectoftheIBPhysicscourse,whichaimstobroadenstudentsinterestsandknowledgebeyondtheconfinesofitsspecificphysicscontent.Throughoutthisbookwehopethatstudentswilldevelopanappreciationoftheprocessesandapplicationsofphysicsandtechnology.SomeaspectsoftheNatureofSciencemaybeexaminedinIBPhysicsexaminationsandimportantdiscussionpointsarehighlightedinthemargins.

TheUtilizationsandAdditionalPerspectivessectionsalsoreflecttheNatureofScience,buttheyaredesignedtotakestudentsbeyondthelimitsoftheIBsyllabusinavarietyofways.Theymight,forexample,provideahistoricalcontext,extendtheoryorofferaninterestingapplication.Theyaresometimesaccompaniedbymorechallenging,orresearch-style,questions.TheydonotcontainanyknowledgethatisessentialfortheIBexaminations.

Scienceandtechnologyhavedevelopedoverthecenturieswithcontributionsfromscientistsfromall

aroundtheworld.Inthemodernworldscienceknowsfewboundariesandtheflowofinformationisusuallyquickandeasy.SomeinternationalapplicationsofsciencehavebeenindicatedwiththeInternationalMindednessicon.

Workedexamplesareprovidedineachchapterwhenevernewequationsareintroduced.Alargenumberofself-assessmentquestionsareplacedthroughoutthechaptersclosetotherelevanttheory.Answerstomostquestionsareprovidedattheendofthebook.

Itisnotanaimofthisbooktoprovidedetailedinformationaboutexperimentalworkortheuseofcomputers.However,ourSkillsiconhasbeenplacedinthemargintoindicatewhereversuchworkmayusefullyaidunderstanding.AnumberofkeyexperimentsareincludedintheIBPhysicsGuideandthesearelistedinChapter18:PreparingfortheIBDiplomaPhysicsexamination,tobefoundonthewebsitethataccompaniesthisbook.

AselectionofIBexamination-stylequestionsisprovidedattheendofeachchapter,aswellassomepastIBPhysicsexaminationquestions.

LinkstotheinterdisciplinaryTheoryofKnowledge(ToK)elementoftheIBDiplomacoursearemadeinallchapters.

ComprehensiveglossariesofwordsandtermsforCoreandAHLtopicsareincludedintheprintedbook.GlossariesfortheOptionsareavailableonthewebsite.

UsingthisbookThesequenceofchaptersinPhysicsfortheIBDiplomadeliberatelyfollowsthesequenceofthesyllabuscontent.However,theIBDiplomaPhysicsGuideisnotdesignedasateachingsyllabus,sotheorderinwhichthesyllabuscontentispresentedisnotnecessarilytheorderinwhichitwillbetaught.Differentschoolsandcollegesshoulddesignacoursebasedontheirindividualcircumstances.Inadditiontothestudyofthephysicsprinciplescontainedinthisbook,IBsciencestudentscarryout

experimentsandinvestigations,aswellascollaboratinginaGroup4Project.Theseareassessedwithintheschool(InternalAssessment),basedonwell-establishedcriteria.ThecontentsofChapter1(Physicsandphysicalmeasurement)haveapplicationsthatrecur

throughouttherestofthebookandalsoduringpracticalwork.Forthisreason,itisintendedmoreasasourceofreference,ratherthanasmaterialthatshouldbefullyunderstoodbeforeprogressingtotherestofthecourse.

AuthorprofilesJohnAllumJohnhastaughtpre-universityphysicscoursesasaHeadofDepartmentinavarietyofinternationalschoolsformorethan30years.HehastaughtIBPhysicsinMalaysiaandinAbuDhabi,andhasbeenanexaminerforIBPhysicsformanyyears.

ChristopherTalbotChristeachesTOKandIBChemistryataleadingIBWorldSchoolinSingapore.HehasalsotaughtIBBiologyandavarietyofIGCSEcourses,includingIGCSEPhysics,atOverseasFamilySchool,RepublicofSingapore.

AuthorsacknowledgementsWeareindebtedtothefollowingteachersandlecturerswhoreviewedearlydraftsofthechapters:DrRobertSmith,UniversityofSussex(Astrophysics);DrTimBrown,UniversityofSurrey(CommunicationsandDigitalTechnology);DrDavidCooper(QuantumPhysics);MrBernardTaylor(TheoryofKnowledge,InternalAssessmentandFieldsandForces);ProfessorChristopherHammond,UniversityofLeeds(ElectromagneticWaves);ProfessorPhilWalker,UniversityofSurrey(NuclearPhysics);DrDavidJenkins,UniversityofYork(NuclearPhysics)andTrevorWilson,BavariaInternationalSchoole.V.,Germany.WealsoliketothankDavidTalbot,whosuppliedsomeofthephotographsforthebook,andTerri

HarwoodandJonHomewood,whodrewanumberofphysicists.Forthissecondedition,wewouldliketothankthefollowingacademicsfortheiradvice,comment

andfeedbackondraftsofthechapters:DrRobertSmith,EmeritusReaderinAstronomy,UniversityofSussex,DrTimBrown,LecturerinRadioFrequencyAntennasandPropagation,UniversityofSurrey,DrAlexanderMerle,DepartmentofPhysicsandAstronomy,UniversityofSouthampton,DrDavidBerman,SchoolofPhysicsandAstronomy,QueenMaryCollege,ProfessorCarlDettmann,SchoolofMathematics,UniversityofBristolandDrJohnRoche,LinacreCollege,UniversityofOxford.WewouldliketothankRichardBurt,WindermereSchool,UKforauthoringOptionARelativity(Chapter13).FinallywewouldalsoliketoexpressourgratitudeforthetirelesseffortsoftheHodderEducation

teamthatproducedthebookyouhaveinfrontofyou,ledbySo-ShanAuandPatrickFox.

1Measurementsanduncertainties

ESSENTIALIDEASSince1948,theSystmeInternationaldUnits(SI)hasbeenusedasthepreferredlanguageofscienceandtechnologyacrosstheglobeandreflectscurrentbestmeasurementpractice.

Scientistsaimtowardsdesigningexperimentsthatcangiveatruevaluefromtheirmeasurementsbut,duetothelimitedprecisioninmeasuringdevices,theyoftenquotetheirresultswithsomeformofuncertainty.

Somequantitieshavedirectionandmagnitude,othershavemagnitudeonly,andthisunderstandingisthekeytocorrectmanipulationofquantities.

1.1MeasurementsinphysicsSince1948,theSystmeInternationaldUnits(SI)hasbeenusedasthepreferredlanguageofscienceandtechnologyacrosstheglobeandreflectscurrentbestmeasurementpractice

FundamentalandderivedSIunitsTocommunicatewitheachotherweneedtoshareacommonlanguage,andtosharenumerical

informationweneedtousecommonunitsofmeasurement.Aninternationallyagreedsystemofunitsisnowusedbyscientistsaroundtheworld.ItiscalledtheSIsystem(fromtheFrenchSystmeInternational).SIunitswillbeusedthroughoutthiscourse.

NatureofScienceCommonterminologyFormuchofthelast200yearsmanyprominentscientistshavetriedtoreachagreementonametric(decimal)systemofunitsthateveryonewoulduseformeasurementsinscienceandcommerce.Acommonsystemofmeasurementisinvaluableforthetransferofscientificinformationandforinternationaltrade.Inprinciplethismayseemmorethansensible,buttherearesignificanthistoricalandculturalreasonswhysomecountries,andsomesocietiesandindividuals,havebeenresistanttochangingtheirsystemofunits.TheSIsystemwasformalizedin1960andtheseventhunit(themole)wasaddedin1971.

Beforethat,apartfromSIunits,asystembasedoncentimetres,gramsandseconds(CGS)waswidelyused,whiletheimperial(non-decimal)systemoffeet,poundsandsecondswasalsopopularinsomecountries.Fornon-scientific,everydayuse,peopleinmanycountriessometimesstillprefertousedifferentsystemsthathavebeenpopularforcenturies.ConfusionbetweendifferentsystemsofunitshasbeenfamouslyblamedforthefailureoftheMarsorbiterin1999andhasbeenimplicatedinseveralaviationincidents.

ThefundamentalunitsofmeasurementTherearesevenfundamental(basic)unitsintheSIsystem:kilogram,metre,second,ampere,mole,kelvin(andcandela,whichisnotpartofthiscourse).Thequantities,namesandsymbolsforthesefundamentalSIunitsaregiveninTable1.1.

Theyarecalledfundamentalbecausetheirdefinitionsarenotcombinationsofotherunits(unlikemetrespersecond,forexample).Youdonotneedtolearnthedefinitionsoftheseunits.

NatureofScienceImprovementininstrumentationAccurateandprecisemeasurementsofexperimentaldataareacornerstoneofscience,andsuchmeasurementsrelyontheprecisionofoursystemofunits.Thedefinitionsofthefundamentalunitsdependonscientistsabilitytomakeveryprecisemeasurementsandthishasimprovedsincetheunitswerefirstdefinedandused.Scientificadvancescancomefromoriginalresearchinnewareas,buttheyarealsodriven

byimprovedtechnologiesandtheabilitytomakemoreaccuratemeasurements.Astronomyisagoodexample:controlledexperimentsaregenerallynotpossible,soourrapidlyexpandingunderstandingoftheuniverseisbeingachievedlargelyasaresultoftheimproveddatawecanreceivewiththehelpofthelatesttechnologies(higher-resolutiontelescopes,forexample).

DerivedunitsofmeasurementAllotherunitsinsciencearecombinationsofthefundamentalunits.Forexample,theunitforvolumeism3andtheunitforspeedisms1.Combinationsoffundamentalunitsareknownasderivedunits.Sometimesderivedunitsarealsogiventheirownname(Table1.2).Forexample,theunitofforceis

kgms2,butitisusuallycalledthenewton,N.Allderivedunitswillbeintroducedanddefinedwhentheyareneededduringthecourse.

Notethatstudentsareexpectedtowriteandrecognizeunitsusingsuperscriptformat,suchasms1ratherthanm/s.Acceleration,forexample,hastheunitms2.OccasionallyphysicistsuseunitsthatarenotpartoftheSIsystem.Forexample,theelectronvolt,eV,

isaconvenientlysmallunitofenergythatisoftenusedinatomicphysics.Unitssuchasthiswillbeintroducedwhennecessaryduringthecourse.Studentswillbeexpectedtobeabletoconvertfromoneunittoanother.Amorecommonconversionwouldbechangingtimeinyearstotimeinseconds.

ToKLinkFundamentalconceptsAswellassomeunitsofmeasurement,manyoftheideasandprinciplesusedinphysicscanbedescribedasbeingfundamental.Indeed,physicsitselfisoftendescribedasthefundamentalscience.Butwhatexactlydowemeanwhenwedescribesomethingasfundamental?Wecouldreplacethewordwithelementaryorbasic,butthatdoesnotreallyhelpustounderstanditstruemeaning.

Oneofthecentralthemesofphysicsisthesearchforfundamentalparticlesparticlesthatarethebasicbuildingblocksoftheuniverseandarenot,themselves,madeupofsmallerandsimplerparticles.Itisthesamewithfundamentallawsandprinciples:aphysicsprinciplecannotbedescribedasfundamentalifitcanbeexplainedbysimplerideas.Mostscientistsalsobelievethataprinciplecannotbereallyfundamentalunlessitisrelativelysimpletoexpress(probablyusingmathematics).Ifitiscomplicated,maybetheunderlyingsimplicityhasnotyetbeendiscovered.

Fundamentalprinciplesmustalsobetrueeverywhereandforalltime.Thefundamentalprinciplesofphysicsthatweusetodayhavebeentested,re-testedandtestedagaintocheckiftheyaretrulyfundamental.Ofcourse,thereisalwaysapossibilitythatinthefutureaprinciplethatisbelievedtobefundamentalnowisdiscoveredtobeexplainablebysimplerideas.

Considertwowell-knownlawsinphysics.Hookeslawdescribeshowsomematerialsstretchwhenforcesactonthem.Itisasimplelaw,butitisnotafundamentallawbecauseitiscertainlynotalwaystrue.Thelawofconservationofenergyisalsosimple,butitisdescribedasfundamentalbecausetherearenoknownexceptions.

ScientificnotationandmetricmultipliersScientificnotationWhenwritingandcomparingverylargeorverysmallnumbersitisconvenienttousescientificnotation(sometimescalledstandardform).Inscientificnotationeverynumberisexpressedintheforma10b,whereaisadecimalnumber

largerthan1andlessthan10,andbisawholenumber(integer)calledtheexponent.Forexample,inscientificnotationthenumber434iswrittenas4.34102;similarly,0.000316iswrittenas3.16104.Scientificnotationisusefulformakingthenumberofsignificantfiguresclear(seethenextsection).

Itisalsousedforenteringanddisplayinglargeandsmallnumbersoncalculators.10xortheletterEisoftenusedoncalculatorstorepresenttimestentothepowerof.Forexample,4.62E3represents4.62103,or4620.

Theworldwideuseofthisstandardformforrepresentingnumericaldataisofgreatimportanceforthecommunicationofscientificinformationbetweendifferentcountries.

StandardmetricmultipliersIneverydaylanguageweusethewordsthousandandmilliontohelprepresentlargenumbers.Thescientificequivalentsaretheprefixeskilo-andmega-.Forexample,akilowattisonethousandwatts,andamegajouleisonemillionjoules.Similarly,athousandthandamilliontharerepresentedscientificallybytheprefixesmilli-andmicro-.AlistofstandardprefixesisshowninTable1.3.ItisprovidedinthePhysicsdatabooklet.

ToKLinkEffectivecommunicationneedsacommonlanguageandterminologyWhathasinfluencedthecommonlanguageusedinscience?Towhatextentdoeshavingacommonstandardapproachtomeasurementfacilitatethesharingofknowledgeinphysics?

Therecanbelittledoubtthatcommunicationbetweenscientistsismucheasieriftheyshareacommonscientificlanguage(symbols,units,standardscientificnotationetc.asoutlinedinthischapter).Butareourmodernmethodsofscientificcommunicationandterminologythebest,orcouldtheybeimproved?Towhatextentaretheyjustahistoricalaccident,basedonthespecificlanguagesandculturesthatweredominantatthetimeoftheirdevelopment?

SignificantfiguresThemorepreciseameasurementis,thegreaterthenumberofsignificantfigures(digits)thatcanbeusedtorepresentit.Forexample,anelectriccurrentstatedtobe4.20A(asdistinctfrom4.19Aor4.21A)suggestsamuchgreaterprecisionthanacurrentstatedtobe4.2A.Significantfiguresareallthedigitsusedindatatocarrymeaning,whethertheyarebeforeoraftera

decimalpoint,andthisincludeszeros.Butsometimeszerosareusedwithoutthoughtormeaning,andthiscanleadtoconfusion.Forexample,ifyouaretoldthatitis100kmtothenearestairport,youmightbeunsurewhetheritisapproximately100km,orexactly100km.Thisisagoodexampleofwhyscientificnotationisuseful.Using1.00103kmmakesitclearthattherearethreesignificantfigures.1103kmrepresentsmuchlessprecision.Whenmakingcalculations,theresultcannotbemoreprecisethanthedatausedtoproduceit.Asa

general(andsimplified)rule,whenansweringquestionsorprocessingexperimentaldata,theresult

shouldhavethesamenumberofsignificantfiguresasthedataused.Ifthenumberofsignificantfiguresisnotthesameforallpiecesofdata,thenthenumberofsignificantfiguresintheanswershouldbethesameastheleastpreciseofthedata(whichhasthefewestsignificantfigures).ThisisillustratedinWorkedexample1.

Workedexample1Usetheequation:

todeterminethepower,P,ofanelectricmotorthatraisesamass,m,of1.5kg,adistance,h,of1.128minatime,t,of4.79s.(g=9.81ms2)

Acalculatorwilldisplayananswerof3.4652,butthisanswersuggestsaveryhighprecision,whichisnotjustifiedbythedata.Thedatausedwiththeleastnumberofsignificantfiguresis1.5kg,sotheanswershouldalsohavethesamenumber:P=3.5W

RoundingofftoanappropriatenumberofsignificantfiguresRoundingoff,asinWorkedexample1,shouldbedoneattheendofamulti-stepcalculation,whentheanswerhastobegiven.Iffurthercalculationsusingthisanswerarethenneeded,allthedigitsshownpreviouslyonthecalculatorshouldbeused.Theanswertothiscalculationshouldthenberoundedofftothecorrectnumberofsignificantfigures.Thisprocesscansometimesresultinsmallbutapparentinconsistenciesbetweenanswers.

OrdersofmagnitudePhysicsisthefundamentalsciencethattriestoexplainhowandwhyeverythingintheuniversebehavesinthewaythatitdoes.Physicistsstudyeverythingfromthesmallestpartsofatomstodistantobjectsinourgalaxyandbeyond(Figure1.1).

Physicsisaquantitativesubjectthatmakesmuchuseofmathematics.Measurementsandcalculationscommonlyrelatetotheworldthatwecanseearoundus(themacroscopicworld),butourobservationsmayrequiremicroscopicexplanations,oftenincludinganunderstandingofmolecules,atoms,ionsandsub-atomicparticles.Astronomyisabranchofphysicsthatdealswiththeotherextremequantitiesthatareverymuchbiggerthananythingweexperienceineverydaylife.Thestudyofphysicsthereforeinvolvesdealingwithbothverylargeandverysmallnumbers.When

numbersaresodifferentfromoureverydayexperiences,itcanbedifficulttoappreciatetheirtruesize.Forexample,theageoftheuniverseisbelievedtobeabout1018s,butjusthowbigisthatnumber?Theonlysensiblewaytoanswerthatquestionistocomparethequantitywithsomethingelsewithwhichwearemorefamiliar.Forexample,theageoftheuniverseisabout100millionhumanlifetimes.Whencomparingquantitiesofverydifferentsizes(magnitudes),forsimplicityweoftenmake

approximationstothenearestpowerof10.Whennumbersareapproximatedandquotedtothenearestpowerof10,itiscalledgivingthemanorderofmagnitude.Forexample,whencomparingthelifetimeofahuman(theworldwideaverageisabout70years)withtheageoftheuniverse(1.41010y),wecanusetheapproximateratio1010/102.Thatis,theageoftheuniverseisabout108humanlifetimes,orwecouldsaythatthereareeightordersofmagnitudebetweenthem.Herearethreefurtherexamples:

Themassofahydrogenatomis1.671027kg.Toanorderofmagnitudethisis1027kg.Thedistancetotheneareststar(ProximaCentauri)is4.011016m.Toanorderofmagnitudethisis1017m.(Note:logof4.011016=16.60,whichisnearerto17thanto16.)

Thereare86400secondsinaday.Toanorderofmagnitudethisis105s.

Tables1.4to1.6listtherangesofmass,distanceandtimethatoccurintheuniverse.Youarerecommendedtolookatcomputersimulationsrepresentingtheseranges.

EstimationSometimeswedonothavethedataneededforaccuratecalculations,ormaybecalculationsneedtobemadequickly.Sometimesaquestionissovaguethatanaccurateanswerissimplynotpossible.Theabilitytomakesensibleestimatesisaveryusefulskillthatneedsplentyofpractice.Theworkedexampleandquestions25belowaretypicalofcalculationsthatdonothaveexactanswers.Whenmakingestimates,differentpeoplewillproducedifferentanswersanditisusuallysensibleto

useonlyone(maybetwo)significantfigures.Sometimesonlyanorderofmagnitudeisneeded.

Workedexample2Estimatethemassofairinaclassroom.(densityofair=1.3kgm3)

Atypicalclassroommighthavedimensionsof7m8m3m,soitsvolumeisabout170m3.mass=densityvolume=1701.3=220kg

Sincethisisanestimate,ananswerof200kgmaybemoreappropriate.Toanorderofmagnitudeitwouldbe102kg.

1Estimatethemassof:aapageofabookbairinabottlecadogdwaterintheoceansoftheworld.

2Giveanestimateforeachofthefollowing:atheheightofahousewiththreefloorsbhowmanytimesawheelonacarrotatesduringthelifetimeofthecarchowmanygrainsofsandwouldfillacupdthethicknessofapageinabook.3Estimatethefollowingperiodsoftime:ahowmanysecondsthereareinanaveragehumanlifetimebhowlongitwouldtakeapersontowalkaroundtheEarth(ignorethetimenotspent

walking)chowlongittakesforlighttotravelacrossaroom.4Researchtherelevantdatasothatyoucancomparethefollowingmeasurements.(Give

youranswerasanorderofmagnitude.)athedistancetotheMoonwiththecircumferenceoftheEarthbthemassoftheEarthwiththemassofanapplecthetimeittakeslighttotravel1mwiththetimebetweenyourheartbeats.

1.2UncertaintiesanderrorsScientistsaimtowardsdesigningexperimentsthatcangiveatruevaluefromtheirmeasurements,butbecauseofthelimitedprecisioninmeasuringdevices,theyoftenquotetheirresultswithsomeformofuncertainty

NatureofScienceCertaintyAlthoughscientistsareperceivedasworkingtowardsfindingexactanswers,anunavoidableuncertaintyexistsinanymeasurement.Theresultsofallscientificinvestigationshaveuncertaintiesanderrors,althoughgoodexperimentationwilltrytokeeptheseassmallaspossible.Whenwereceivenumericaldataofanykind(scientificorotherwise)weneedtoknowhow

muchbeliefweshouldplaceintheinformationthatwearereadingorhearing.Thepresentationoftheresultsofseriousscientificresearchshouldalwayshaveanassessmentoftheuncertaintiesinthefindings,becausethisisanintegralpartofthescientificprocess.Unfortunatelythesameisnottrueofmuchoftheinformationwereceivethroughthemedia,wheredataaretoooftenpresenteduncriticallyandunscientifically,withoutanyreferencetotheirsourceorreliability.Nomatterhowhardwetry,evenwiththeverybestofmeasuringinstruments,itissimply

notpossibletomeasureanythingexactly.Foronereason,thethingsthatwecanmeasuredonotexistasperfectlyexactquantities;thereisnoreasonwhytheyshould.Thismeansthateverymeasurementisanapproximation.Ameasurementcouldbethe

mostaccurateevermade,forexamplethewidthofarulermightbestatedas2.28389103cm,butthatisstillnotperfect,andevenifitwaswewouldnotknowbecausewewouldalwaysneedamoreaccurateinstrumenttocheckit.Inthisexamplewealsohavetheaddedcomplicationofthefactthatwhenmeasurementsoflengthbecomeverysmallwehavetodealwiththeatomicnatureoftheobjectsthatwearemeasuring.(Whereistheedgeofanatom?)

Theuncertaintyinameasurementistherange,aboveandbelowastatedvalue,overwhichwewouldexpectanyrepeatedmeasurementstofall.Forexample,iftheaverageheighttowhichaballbouncedwhendropped(fromthesameheight)was48cm,butactualmeasurementsvariedbetween45cmand51cm,theresultshouldberecordedas483cm.Theuncertaintyis3cm,butthisissometimesbetterquotedasapercentage,inthisexample6%.Obviously,itisdesirablethatexperimentsshouldproduceresultswithlowuncertaintiessuchmeasurementsaredescribedasbeingprecise.Butitshouldbenotedthatsometimesresultscanbeprecise,butwrong!Themoreprecisethatameasurementis,thegreaterthenumberofsignificantfigures(digits)thatcan

beusedtorepresentit.Ifthecorrect(true)valueofaquantityisknown,butanactualmeasurementismadethatisnotthe

same,werefertothisasanexperimentalerror.Thatis,anerroroccursinameasurementwhenitisnotexactlythesameasthecorrectvalue.Forexample,ifastudentrecordedtheheightofaballsbounceas49cm,butcarefulobservationofavideorecordingshowedthatitwasactually48cm,thentherewasan

errorinthemeasurementof+1cm.Allmeasurementsinvolveerrors,whethertheyarelargeorsmall,forwhichtherearemanypossible

reasons,buttheyshouldnotbeconfusedwithmistakes.Errorscanbedescribedaseitherrandomorsystematic(seebelow),althoughallmeasurementsinvolvebothkindsoferrortosomeextent.Thewordserroranduncertaintyaresometimesusedtomeanthesamething,althoughthiscanonly

betruewhenreferringtoexperimentsthathaveaknowncorrectresult.

ToKLinkScientificknowledgeisprovisionalOneaimofthephysicalscienceshasbeentogiveanexactpictureofthematerialworld.Oneachievementofphysicsinthetwentiethcenturyhasbeentoprovethatthisaimisunattainable.

JacobBronowski

Canscientistseverbetrulycertainoftheirdiscoveries?

Thepopularbeliefisthatsciencedealswithfactsand,toalargeextent,thatisafaircomment,butitalsogivesanincompleteimpressionofthenatureofscience.Thestatementismisleadingifitsuggeststhatscientiststypicallybelievethattheyhaveuncoveredcertainuniversaltruthsforalltime.Scientificknowledgeisprovisionalandfullyopentochangeifandwhenwemakenewdiscoveries.Morethanthat,itistheessentialnatureofscienceandgoodscientiststoencouragethere-examinationofexistingknowledgeandtolookforimprovementsandprogress.

DifferentkindsofuncertaintyTheuncertaintyinexperimentalmeasurementsdiscussedinthischapterisaconsequenceofthelimitationsofscientistsandtheirequipmenttoobtain100%accurateresults.However,weshouldalsoconsiderthattheactofmeasurement,initself,canchangewhatweareattemptingtomeasure.Forexample,connectinganammeterinanelectriccircuitmustaffectthecurrentitistryingtomeasure,althougheveryeffortshouldbemadetoensurethiseffectisnotsignificant.Similarly,puttingacoldthermometerinawarmliquidwillalteritstemperature.Uncertaintyalsoappearsasanimportantconceptinmodernphysics:theHeisenberguncertainty

principledealswiththebehaviourofsub-atomicparticlesandisdiscussedinChapter12(HigherLevelstudents).Oneofitscoreideasisthatthemorepreciselythepositionofaparticleisknown,thelesspreciselyitsmomentumcanbeknown,andviceversa.ButitshouldbestressedthattheHeisenberguncertaintyprincipleisafundamentalfeatureofquantumphysicsandhasnothingtodowiththeexperimentallimitsofcurrentlaboratorytechnology.

RandomandsystematicerrorsRandomerrorsRandomerrorscannotbeavoidedbecauseexactmeasurementsarenotpossible.Measurementscanbe

biggerorsmallerthanthecorrectvalueandarescatteredrandomlyaroundthatvalue.Randomerrorsaregenerallyunknownandunpredictable.Therearemanypossiblereasonsforthem,including:limitationsofthescaleordisplaybeingusedreadingscalesfromwrongpositionsirregularhumanreactiontimeswhenusingastopwatchdifficultyinmakingobservationsthatchangequicklywithtime.

Thereadingobtainedfromameasuringinstrumentislimitedbythesmallestdivisionofitsscale.Thisissometimescalledareadability(orreading)error.Forexample,aliquid-in-glassthermometerwithascalemarkedonlyindegrees(23C,24C,25C,etc.)cannotreliablybeusedtomeasuretoevery0.1C.Itisusuallyassumedthattheerrorforanalogue(continuous)scales,likealiquid-in-glassthermometer,ishalfofthesmallestdivisioninthisexample0.5C.Fordigitalinstrumentstheerrorisassumedtobethesmallestdivisionthatthemetercandisplay.Figure1.2showsanalogueanddigitalammetersthatcanbeusedformeasuringelectriccurrent.

Acommonreasonforrandomerrorsisreadingananaloguescalefromanincorrectposition.ThisiscalledaparallaxerroranexampleisshowninFigure1.3.

SystematicerrorsAsystematicerroroccursbecausethereissomethingconsistentlywrongwiththemeasuringinstrumentorthemethodused.Areadingwithasystematicerrorisalwayseitherbiggerorsmallerthanthecorrectvaluebythesameamount.Commoncausesareinstrumentsthathaveanincorrectscale(wronglycalibrated),orinstrumentsthathaveanincorrectvaluetobeginwith,suchasameterthatdisplaysareadingwhenitshouldreadzero.ThisiscalledazerooffseterroranexampleisshowninFigure1.4.Athermometerthatincorrectlyrecordsroomtemperaturewillproducesystematicerrorswhenusedtomeasureothertemperatures.

AccuracyAmeasurementthatisclosetothecorrectvalue(ifitisknown)isdescribedasaccurate,butinsciencethewordaccuratealsomeansthatasetofmeasurementsmadeduringanexperimenthaveasmall,

systematicerror.Thismeansthatanaccuratesetofmeasurementsareapproximatelyevenlydistributedaroundthecorrectvalues(whethertheyareclosetoitornot),sothatanaverageofthosemeasurementswillbeclosetothetruevalue.Inmanyexperimentsthecorrectresultmightnotbeknown,whichmeansthattheaccuracyof

measurementscannotbeknownwithanycertainty.Insuchcases,thequalityofthemeasurementscanbestbejudgedbytheirprecision:canthesameresultsberepeated?Thedifferencebetweenpreciseandaccuratecanbeillustratedbyconsideringarrowsfiredatatarget,

asinFigure1.5.Theaimispreciseifthearrowsaregroupedclosetogetherandaccurateifthearrowsareapproximatelyevenlydistributedaroundthecentreofthetarget.Thelastdiagramshowsbothaccuracyandprecision,althoughineverydayconversationwewouldprobablyjustdescribeitasaccurate.

Awatchthatisalways5minutesfastcanbedescribedasprecisebutnotaccurate.Thisisanexampleofasystematiczerooffseterror.Usinghand-operatedstopwatchestotimea100mracemightgiveaccurateresults(iftherearenosystematicerrors),buttheyareunlikelytobeprecisebecausehumanreactiontimeswillproducesignificantrandomerrors.

IdentifyingandreducingtheeffectsoferrorsIfasinglemeasurementismadeofaparticularquantity,wemayhavenowayofknowinghowcloseitistothecorrectresult;thatis,weprobablydonotknowthesizeofanyerrorinmeasurement.Butifthesamemeasurementisrepeatedandtheresultsaresimilar(lowuncertainty,highprecision),wewillgainsomeconfidenceintheresultsoftheexperiment,especiallyifwehavecheckedforanypossiblecausesofsystematicerror.Themostcommonwayofreducingtheeffectsofrandomerrorsisbyrepeatingmeasurementsand

calculatingameanvalue,whichshouldbeclosertothecorrectvaluethanmost,orall,oftheindividualmeasurements.Anyunusual(anomalous)valuesshouldbecheckedandprobablyexcludedfromthecalculationofthemean.Manyexperimentsinvolvetakingarangeofmeasurements,eachunderdifferentexperimental

conditions,sothatagraphcanbedrawntoshowthepatternoftheresults.(Forexample,changingthevoltageinanelectriccircuittoseehowitaffectsthecurrent.)Increasingthenumberofpairsofmeasurementsmadealsoreducestheeffectsofrandomerrorsbecausethelineofbest-fitcanbeplacedwithmoreconfidence.Experimentsshouldbedesigned,whereverpossible,toproducelargereadings.Forexample,ametre

rulermightonlybereadabletothenearesthalfamillimetreandthiswillbethesameforall

measurementsthataremadewithit.Whenmeasuringalengthof90cmthiserrorwillprobablybeconsideredasacceptable(itisapercentageerrorof0.56%),butthesamesizederrorwhenmeasuringonly2mmis25%,whichisprobablyunacceptable.Thelargerameasurement(thatismadewithaparticularmeasuringinstrument),thesmallerthepercentageerrorshouldbe.Ifthisisnotpossible,thenthemeasuringinstrumentmightneedtobechangedtoonewithsmallerdivisions.Itispossibletocarryoutanexperimentcarefullywithgoodqualityinstruments,butstillhavelarge

randomerrors.Therecouldbemanydifferentreasonsforthisandtheexperimentmayhavetoberedesignedtogetovertheproblems.Usingastopwatchtotimethefallofanobjectdroppedfromahandtothefloor,ormeasuringtheheightofabouncingball,aretwoexamplesofsimpleexperimentswhichmayhavesignificantrandomerrors.Theeffectsofsystematicerrorscannotbereducedbyrepeatingmeasurements.Instrumentsshouldbe

checkedforerrorsbeforetheyareused,butasystematicerrormightnotevenbenoticeduntilagraphhasbeendrawnoftheresultsandalineofbest-fitfoundnottopassthroughtheexpectedintercept,asshowninFigure1.6.Insuchacaseitmightthenbesensibletoadjustallmeasurementsupordownbythesameamountifthecauseofthesystematicerrorcanbedetermined.

Absolute,fractionalandpercentageuncertaintiesUncertaintiesinexperimentaldataUncertaintiesinexperimentaldatacanbeexpressedinoneofthreeways:Theabsoluteuncertaintyofameasurementistherange,aboveandbelowthestatedvalue,withinwhichwewouldexpectanyrepeatedmeasurementstofall.Forexample,themassofapenmightbestatedas53.2g0.1g,wheretheuncertaintyis0.1g.

Thefractionaluncertaintyistheratiooftheabsoluteuncertaintytothemeasuredvalue.Thepercentageuncertaintyisthefractionaluncertaintyexpressedasapercentage.

Uncertaintiesexpressedinpercentagesareoftenthemostinformative.Experimentsthatproduceresultswithuncertaintiesoflessthan5%maybedesirable,butarenotalwayspossible.

Workedexample3Themassofapieceofmetalisquotedtobe346g2.0%.aWhatistheabsoluteuncertainty?bWhatistherangeofvaluesthatthemasscouldbeexpectedtohave?cWhatisthefractionaluncertainty?

a2.0%of346gis7g(tothenearestgram,asprovidedinthedatainthequestion)b339gto353g(to3significantfigures)c2%isequivalentto

Ideallyuncertaintiesshouldbequotedforallexperimentalmeasurements,butthiscanberepetitiveandtediousinalearningenvironment,sotheyareoftenomittedunlessbeingtaughtspecifically.Itisusuallyeasytodecideonthesizeofanuncertaintyassociatedwithtakingasinglemeasurement

withaparticularinstrument.Itisoftenassumedtobethereadabilityerror,asdescribedearlier.However,theoveralluncertaintyinameasurement,allowingforallexperimentaldifficulties,issometimesmoredifficulttodecide.Forexample,thereadabilityerroronahand-operatedstopwatchmightbe0.01s,buttheuncertaintyinitsmeasurementswillbemuchgreaterbecauseofhumanreactiontimes.Theamountofscatteringofthereadingsaroundameanvalueisausefulguidetorandomuncertainty,

butnotsystematicuncertainty.Afterthemeanvalueofthereadingshasbeencalculated,therandomuncertaintycanbeassumedtobethelargestdifferencebetweenanysinglereadingandthemeanvalue.Thisisshowninthefollowingworkedexample.

Workedexample4Thefollowingmeasurements(incm)wererecordedinanexperimenttomeasurethe

heighttowhichaballbounced:32,29,33,32,37and28.Estimatevaluesfortheabsoluteandpercentagerandomuncertaintiesintheexperiment.

Themeanofthesesixreadingsis31.83cm,butitwouldbesensibletoquotethistotwosignificantfigures(32cm),asintheoriginaldata.Themeasurementthathasthegreatestdifferencefromthisvalueis37cm,soanestimateoftheuncertaintyis5cm,whichmeansapercentageuncertaintyof(5/37)100=14%.

Notethatifthesamedatahadbeenobtainedintheorder28,29,32,32,33,37,itwouldbedifficulttobelievethattheuncertaintieswererandom,andanotherexplanationforthevariationinresultswouldneedtobefound.

UncertaintiesincalculatedresultsWhenmakingfurthercalculationsbasedonexperimentaldata,theuncertaintyinindividualmeasurementsshouldbeknown.Itisthenimportanttoknowhowtousetheseuncertaintiestodeterminetheuncertaintyinanyresultsthatarecalculatedfromthosedata.

Considerasimpleexample:atrolleymovingwithconstantspeedwasmeasuredtotraveladistanceof76cm2cm(2.6%)inatimeof4.3s0.2s(4.7%).Thespeedcanbecalculatedfromdistance/time=76/4.3=17.67,whichis18ms1whenrounded

totwosignificantfigures,consistentwiththeexperimentaldata.Todeterminetheuncertaintyinthisanswerweconsidertheuncertaintiesindistanceandtime.Using

thelargestdistanceandshortesttime,thelargestpossibleanswerforspeedis78/4.1=19.02.Usingthesmallestdistanceandthelongesttime,thesmallestpossibleanswerforspeedis74/4.5=16.44.(Thenumberswillberoundedattheendofthecalculations.)Thespeedisthereforebetween16.44cms1and19.02cms1.Thevalue19.02hasthegreater

difference(1.35)from17.67.Sothefinalresultcanbeexpressedas17.671.35cms1,whichisamaximumuncertaintyof7.6%.Roundingtotwosignificantfigures,theresultbecomes181cms1.Uncertaintycalculationslikethesecanbeverytimeconsumingand,forthiscourse,approximate

methodsareacceptable.Forexample,inthecalculationforspeedshownabove,theuncertaintyinthedatawas2.6%fordistanceand4.7%fortime.Thepercentageuncertaintyinthefinalresultisapproximatedbyaddingthepercentageuncertaintiesinthedata:2.6+4.7=7.3%.Thisgivesapproximatelythesamevalueascalculatedusingthelargestandsmallestpossiblevaluesforspeed.Rulesforfindinguncertaintiesincalculatedresultsaregivenbelow.

RulesforuncertaintiesincalculationsForquantitiesthatareaddedorsubtracted:addtheabsoluteuncertainties.InthePhysicsdatabookletthisisgivenas:

Ify=abtheny=a+b

Forquantitiesthataremultipliedordivided:addtheindividualfractionalorpercentageuncertainties.InthePhysicsdatabookletthisisgivenas:

Forquantitiesthatareraisedtoapower,n,thePhysicsdatabookletgives:

Forotherfunctions(suchastrigonometricfunctions,logarithmsorsquareroots):calculatethehighestandlowestabsolutevaluespossibleandcomparewiththemeanvalue,asshowninthefollowingworkedexample.Butnotethatalthoughsuchcalculationscanoccurinconnectionwithlaboratorywork,theywillnotberequiredinexaminations.

Workedexample5Anangle,,wasmeasuredtobe341.Whatistheuncertaintyintheslopeofthis

angle?

tan34=0.675tan33=0.649tan35=0.700

Largerabsoluteuncertainty=0.6750.649=0.026(0.7000.675=0.025,whichissmallerthan0.026)

So,tan=0.670.03(usingthesamenumberofsignificantfiguresasintheoriginaldata).

5Amassof3462gwasaddedtoamassof1291g.aWhatwastheoverallabsoluteuncertainty?bWhatwastheoverallpercentageuncertainty?6Theequation wasusedtocalculateavalueforswhenawas4.30.2ms2and

twas1.40.1s.aCalculateavaluefors.bCalculatethepercentageuncertaintyinthedataprovided.cCalculatethepercentageuncertaintyintheanswer.dCalculatetheabsoluteuncertaintyintheanswer.7Acertainquantitywasmeasuredtohaveamagnitudeof(1.460.08).Whatisthe

maximumuncertaintyinthesquarerootofthisquantity?

UsingcomputerspreadsheetstocalculateuncertaintiesComputerspreadsheetscanbeveryhelpfulwhenitisnecessarytomakemultiplecalculationsofuncertaintiesinexperimentalresults.Forexample,theresistivity,,ofametalwirecanbecalculatedusingtheequation=Rr2/l,whererandlaretheradiusandlengthofthewire,andRisitsresistance.Figure1.7showstherawdata(shadedgreen)ofanexperimentthatmeasuredtheresistanceofvariouswiresofthesamemetal.Therestofthespreadsheetshowsthecalculationsinvolvedwithprocessingthedatatodetermineresistivityandtheuncertaintyintheresult.Acomputerprogramcanthenbeusedtodrawasuitablegraphoftheresults,andthiscanincludeerrorbars(seepage13).

8aUseacomputerspreadsheettoenterthesamerawdataasshowninFigure1.7.bUsethespreadsheettoconfirmtheresultsofthecalculationsshown.

cWhatdifferencewoulditmaketotheresultsiftheradiusofthewirecouldonlybemeasuredtothenearesthalfamillimetre?

NatureofScienceUncertaintiesAllscientificknowledgeisuncertain

RichardP.Feynman(1998),TheMeaningofItAll:ThoughtsofaCitizen-ScientistItisnotonlymeasurementsthathaveuncertainties.Allscientificknowledgeisuncertainin

thesensethatgoodscientistsunderstandthatanythingwebelievetobetruetoday,mayhavetobechangedinthelightoffuturediscoveriesorinsights.Thisdoubtisfundamentaltothetruenatureofscience.Atanytime,pastorpresent,inthedevelopmentofsciencethereisanacceptedbodyofknowledge,andthegreatestadvancescomefromthosewhoquestionanddoubtthestatusquoofexistingknowledgeandthinking.

RepresentinguncertaintiesongraphsGraphdrawingskillsarediscussedindetailinGraphsanddataanalysisonthefreeaccompanying

website.Therangeofrandomuncertaintyinameasurementoracalculatedresultcanberepresentedona

graphbyusingcrossedlinestomarkthepoint(insteadofadot).

ErrorbarsFigure1.8showsanexampleagraphofdistanceagainsttimeforthemotionofatrain.Verticalandhorizontallinesaredrawnthrougheachdatapointtorepresenttheuncertaintiesinthetwomeasurements.Inthisexample,theuncertaintyintimeis0.5sandtheuncertaintyindistanceis1m.Theselines,whichusuallyhavesmalllinestoindicateclearlywheretheyend,arecallederrorbars(perhapstheywouldbebettercalleduncertaintybars).InFigure1.8thespaceoutlinedbyeacherrorbarhasbeenshadedforemphasisitisexpectedthatalineofbest-fitshouldpasssomewherethrougheachshadedarea.

Insomeexperimentstheerrorbarsaresoshortandinsignificantthattheyarenotincludedonthegraph.Forexample,amasscouldbemeasuredas347.460.01g.Theuncertaintyinthisreadingwouldbetoosmalltoshowasanerrorbaronagraph.(Notethaterrorbarsarenotexpectedfortrigonometricorlogarithmicfunctions.)

UncertaintyofgradientsandinterceptsIftheresultsofanexperimentsuggestastraight-linegraph,itisoftenimportanttodeterminevaluesforthegradientand/ortheintercept(s)withtheaxes.However,itisoftenpossibletodrawarangeofdifferentstraightlines,allofwhichpassthroughtheerrorbarsrepresentingtheexperimentaldata.Weusuallyassumethatthebest-fitlineismidwaybetweenthelinesofmaximumpossiblegradient

andminimumpossiblegradient.Figure1.9showsanexample(forsimplicity,onlythefirstandlasterrorbarsareshown,butinpracticealltheerrorbarsneedtobeconsideredwhendrawingthelines).Figure1.9showshowthelengthofametalspringchangedastheforceappliedwasincreased.We

knowthatthemeasurementswerenotveryprecisebecausetheerrorbarsarelong.Thelineofbest-fithasbeendrawnmidwaybetweentheothertwo.Thisisalineargraph(astraightline)anditisknownthatthegradientofthegraphrepresentstheforceconstant(stiffness)ofthespringandthex-interceptrepresentstheoriginallengthofthespring.Takingmeasurementsfromthebest-fitline,wecanmakethefollowingcalculations:

originallength=x-intercept=1.9cm

Todeterminetheuncertaintyinthecalculationsofgradientandintercept,weneedtoconsidertherange

ofstraightlinesthatcouldbedrawnthroughtheerrorbars.Theuncertaintywillbethemaximumdifferencebetweenvaluesobtainedfromgraphsofmaximumandminimumpossiblegradientsandthevaluecalculatedfromthebest-fitline.Inthisexampleitcanbeshownthat:

forceconstantisbetween14Ncm1and28Ncm1

originallengthisbetween1.1cmand2.6cm.

Thefinalresultcanbequotedas:

forceconstant=199Ncm1

originallength=1.90.8cm.

Clearly,thelargeuncertaintiesintheseresultsconfirmthattheexperimentlackedprecision.

1.3VectorsandscalarsSomequantitieshavedirectionandmagnitude,othershavemagnitudeonly,andthisunderstandingisthekeytocorrectmanipulationofquantities

NatureofScienceModelsinthreedimensionsSpatialawarenessandanappreciationthattheprinciplesofscienceapplytothree-dimensionalspacecaneasilybeoverlookedwhenstudyingthetwo-dimensionalpagesofabookorascreen.Knowingthedirectionsofsomephysicalquantities(intwoorthreedimensions)isimportantforunderstandingtheireffects.Suchquantitiesarecalledvectors.Mathematicaltreatmentofvectorquantitiesinthreedimensions(vectoranalysis)beganintheeighteenthcentury.

VectorandscalarquantitiesThediagramsinFigure1.10showtheforce(s)actingonanobject.InFigure1.10atheobjectisbeingpulledtotherightwithaforceof5N.Thelengthofthearrowrepresentsthesizeoftheforceandtheorientationofthearrowshowsthedirectioninwhichtheforceacts.Thelengthofthearrowisproportionaltotheforce.InFigure1.10bthereisasmallerforce(3N)pushingtheobjecttotheright.Inbothexamplestheobjectwillmove(accelerate)totheright.

InFigure1.10ctherearetwoforcesacting.Wecanaddthemtogethertoshowthattheeffectisthesameasifasingleforceof8N(=3+5)wasactingontheobject.Wesaythattheresultant(net)forceis8N.InFigure1.10dtherearetwoforcesactingontheobject,buttheyactindifferentdirections.The

overalleffectisstillfoundbyaddingthetwoforces,butalsotakingtheirdirectionintoaccount.Thiscanbewrittenas+5+(3)=+2N,whereforcestotherightaregivenapositivesignandforcestotheleftaregivenanegativesign.Theresultantwillbethesameasiftherewasonlyoneforce(2N)actingtotheright.InFigures1.10eand1.10ftherearealsotwoforcesacting,buttheyarenotactingalongthesameline.Fortheseforces,theresultantcanbedeterminedusingascaledrawingortrigonometry(seepage16).Clearly,forceisaquantityforwhichweneedtoknowitsdirectionaswellasitsmagnitude(size).

Quantitiesthathavebothmagnitudeanddirectionarecalledvectors.

Everythingthatwemeasurehasamagnitudeandaunit.Forexample,wemightmeasurethemassofabooktobe640g.Here640gisthemagnitudeofthemeasurement,butmasshasnodirection.

Quantitiesthathaveonlymagnitude,andnodirection,arecalledscalars.

Mostquantitiesarescalars.Somecommonexamplesofscalarsusedinphysicsaremass,length,time,energy,temperatureandspeed.However,whenusingthefollowingquantitiesweneedtoknowboththemagnitudeandthedirectioninwhichtheyareacting,sotheyarevectors:displacement(distanceinagivendirection)velocity(speedinagivendirection)force(includingweight)accelerationmomentumandimpulsefieldstrength(gravitational,electricandmagnetic).

Thesymbolsforvectorquantitiesaresometimesshowninbolditalic(forexample,F).Scalarquantitiesareshownwithanormalitalicfont(forexample,m).Indiagrams,allvectorsareshownwithstraightarrows,pointinginthecorrectdirection,whichhave

alengthproportionaltothemagnitudeofthevector(asshownbytheforcesinFigure1.11).Inthiscoursevectorcalculationswillbelimitedtotwodimensions.

Theimportanceofvectorsiseasilyillustratedbythedifferencebetweendistanceanddisplacement.Thepilotofaninternationalflightfrom,say,IstanbultoCaironeedstoknowmorethanthatthetwocitiesareadistanceof1234kmapart.Ofcourse,thepilotalsoneedstoknowtheheading(direction)inwhichtheplanemustflyinordertoreachitsdestination.Similarly,inordertodrawaccuratemapsormakelandsurveys,thedistanceanddirectionofachosenpositionfromareferencepointmustbemeasured.

CombinationandresolutionofvectorsAddingvectorstodeterminearesultantWhentwoormorescalarquantitiesareaddedtogether(forexamplemassesof25gand50g),thereisonlyonepossibleanswer(resultant):75g.Butwhenvectorquantitiesareadded,thereisarangeofdifferentresultantspossible,dependingonthedirectionsinvolved.TodeterminetheresultantofthetwoforcesshowninFigures1.10eor1.10ftherearetwopossible

methods:bydrawing(graphicalmethod)orbytrigonometry.

GraphicalmethodThetwovectorsshowninFigure1.10faredrawncarefullytoscale(forexample,byusing1cmtorepresent1N),withthecorrectangle(140)betweenthem.Aparallelogramisthencompleted.Theresultantisthediagonaloftheparallelogram(seeFigure1.11).Rememberthatthemagnitudeandthedirectionshouldbothbedeterminedfromthediagram.Inthisexampletheresultantforceisrepresentedbythelinedrawninred.Itslengthis3.4cm,whichrepresents3.4N,atanangleof36tothe5.0Nforce.

TrigonometricmethodTheforcesinFigure1.11eareatrightanglestoeachother.Thismeansthataparallelogramdrawntorepresenttheseforceswillbearectangle(Figure1.12)andthemagnitudeoftheresultantoftheforces,F,canbefoundusingPythagorasstheorem:

F2=3.02+5.02=34

F=5.8N

Thedirectionofthisforcecanbedeterminedbyusingtrigonometry:

tan (istheanglethattheresultantmakeswiththedirectionofthe5.0Nforce)

=31

Youwillnotbeexpectedtodeterminetrigonometricalsolutionsiftheparallelogramisnotarectangle.

SubtractingvectorstofindtheirdifferenceWemayneedtoknowthedifferencebetweentwovectorswhenweareconsideringbyhowmuchavectorquantityhaschanged.Thisisdeterminedbysubtractingonevectorfromtheother.Anegativevectorhasthesamemagnitude,butoppositedirection,asapositivevector,sowhenfindingthedifferencebetweenvectorsPandQwecanwrite:

PQ=P+(Q)

Figure1.13showshowvectorsaresubtractedgraphically.Theredlinerepresentsthedifferencewhena

particularvectorchangedinmagnitudeanddirectionfromPtoQ.

MultiplyinganddividingvectorsbyscalarsIfavectorPismultipliedordividedbyascalarnumberk,theresultantvectorsaresimplykPorP/k.Ifkisnegative,thentheresultantvectorbecomesnegative,meaningthatthedirectionisreversed.

ResolvingasinglevectorintotwocomponentsWehaveseenthattwoindividualvectorscanbecombinedmathematicallytofindasingleresultantthathasthesameeffectasthetwoseparatevectors.Thisprocesscanbereversed:asinglevectorcanbeconsideredashavingthesameeffectastwoseparatevectors.Thisiscalledresolvingavectorintotwocomponents.Resolvingcanbeveryusefulbecause,ifthetwocomponentsarechosentobeperpendiculartoeachother(oftenhorizontalandvertical),theywillthenbothbeindependentofeachother,sothattheycanbothbeconsideredtotallyseparately.Figure1.14showsasinglevector,A,actingatanangletothehorizontal.Ifwewanttoknowthe

effectsofthisvectorinthehorizontalandverticaldirections,wecanresolveitintotwocomponents:

and

sothat

AH=Acos

and

AV=Asin

BothoftheseequationsandtheassociateddiagramaregiveninthePhysicsdatabooklet.

Workedexample6Figure1.15showsaboxrestingonaslopingsurface(aninclinedplane).Theboxhasa

weightof585N.Whatarethecomponentsofweight:adowntheslope?bperpendicularlyintotheslope?

acomponentdowntheslope=585sin23=230Nbcomponentintotheslope=585cos23=540N

ToKLinkPhysicsandmathematicsWhatisthenatureofcertaintyandproofinmathematics?

Scienceismostlybasedonknowledgegainedfromexperimentationandmeasurement,althoughithasbeenmadeveryclearinthischapterthatabsoluteaccuracyandcertaintyinthegatheringofdataisnotpossible.Incontrast,theessentialtheoriesandmethodsofpuremathematicsseemtodealwithcertainty.Mathematicsisanindispensibletoolforaphysicistformanyreasons,includingitsconciseness,itslackofambiguityanditsusefulnessinmakingpredictions.Mostimportantprinciplesinphysicscanbesummarizedinmathematicalform.

ExaminationquestionsaselectionPaper1IBquestionsandIBstylequestions

Q1Thediameterofawirewasmeasuredthreetimeswithaninstrumentthathasazerooffseterror.Theresultswere1.24mm,1.26mmand1.25mm.Theaverageoftheseresultsis:

AaccuratebutnotpreciseBprecisebutnotaccurateCaccurateandpreciseDnotaccurateandnotprecise.

Q2Theapproximatethicknessofapageinatextbookis:A0.02mmB0.08mmC0.30mmD1.00mm.

Q3Whichofthefollowinganapproximateconversionofatimeof1monthintoSIunits?A0.08yB30dC3106sDalloftheabove

Q4Themassesandweightsofdifferentobjectsareindependentlymeasured.Thegraphisaplotofweightversusmassthatincludeserrorbars.

Theseexperimentalresultssuggestthat:AthemeasurementsshowasignificantsystematicerrorbutsmallrandomerrorBthemeasurementsshowasignificantrandomerrorbutsmallsystematicerrorCthemeasurementsareprecisebutnotaccurateDtheweightofanobjectisproportionaltoitsmass.

Q5WhichofthefollowingisafundamentalSIunit?Anewton

BcoulombCampereDjoule

Q6Thedistancetravelledbyacarinacertaintimewasmeasuredwithanuncertaintyof6%.Iftheuncertaintyinthetimewas2%,whatwouldtheuncertaintybeinacalculationofthecarsspeed?

A3%B4%C8%D12%

Q7Whichofthefollowingquantitiesisascalar?ApressureBaccelerationCgravitationalfieldstrengthDdisplacement

Q8Thecurrentinaresistorismeasuredas2.00A0.02A.Whichofthefollowingcorrectlyidentifiestheabsoluteuncertaintyandthepercentageuncertaintyinthecurrent?

Absoluteuncertainty PercentageuncertaintyA 0.02A 1%B 0.01A 0.5%C 0.02A 0.01%D 0.01A 0.005%

IBOrganization

Q9Whichofthefollowingisareasonableestimateoftheorderofmagnitudeofthemassofalargeaircraft?

A103kgB105kgC107kgD109kg

Q10WhichofthefollowingisequivalenttotheSIunitofforce(thenewton)?Akgms1Bkgm2s1Ckgms2Dkgm2s2

IBOrganization

2Mechanics

ESSENTIALIDEASMotionmaybedescribedandanalysedbytheuseofgraphsandequations.Classicalphysicsrequiresaforcetochangeastateofmotion,assuggestedbyNewtoninhislawsofmotion.

Thefundamentalconceptofenergylaysthebasisonwhichmuchofscienceisbuilt.Conservationofmomentumisanexampleofalawthatisneverviolated.

2.1MotionMotionmaybedescribedandanalysedbytheuseofgraphsandequations

Kinematicsisthestudyofmovingobjects.Theideasofclassicalphysicspresentedinthischaptercanbeappliedtothemovementofallmasses,fromtheverysmall(freelymovingatomicparticles)totheverylarge(stars).Tocompletelydescribethemotionofanobjectatanyonemomentweneedtosaywhereitis,how

fastitismovingandinwhatdirection.Forexample,wemightobservethatacaris20mtothewestofanobserver,andmovingnortheastataspeedof8ms1(seeFigure2.1).

Ofcourse,anyorallofthesequantitiesmightbechanging.Inreallifethemovementofmanymovingobjectscanbecomplicated;theydonotoftenmoveinstraightlinesandtheymightevenrotateorhavedifferentpartsmovingindifferentdirections.Inthischapterwewilldevelopanunderstandingofthebasicprinciplesofkinematicsbydealingwith

singleobjectsmovinginstraightlines,andcalculationswillbeconfinedtothoseobjectsthathavearegularmotion.Wewillconsidertheeffectsofairresistancelaterinthischapter.

NatureofScienceEverythingismovingThestudyofmotionmustbeacornerstoneofsciencebecauseeverythingmoves.Starsandgalaxiesaremovingapartfromeachotheratenormousspeeds,theEarthorbitstheSunandeverythingonEarthisrotatingaroundtheaxisonceeveryday.Atomsandmoleculesareinconstantmotion,asarethesub-atomicparticleswithinthem.Ofcourseineverydaylifemanyobjectsappeartobestationary,butonlybecauseweareonlycomparingthemwiththeirsurroundings.Ifweweretoimaginethatanobjectwastruly,absolutely,notinmotion,wewouldhavenowaytoproveitbecauseallmotionisrelativetosomethingelse.

Distanceanddisplacement

Displacementisdefinedasthedistanceinagivendirectionfromafixedreferencepoint.

Thedisplacementofanobjectisitspositioncomparedwithaknownreferencepoint.Forexample,the

displacementofthecarinFigure2.1is20mtothewestoftheobserver.Tospecifyadisplacementweneedtostateadistanceandadirectionfromthereferencepoint.Thereferencepointisoftenomittedbecauseitisobviousforexample,wemightjustsaythatanairportis50kmtothenorth.Althoughadisplacementcanbeanywhereinthreedimensions,inthistopicwewillusuallyrestrictourthinkingtooneortwodimensions.Displacementanddistancearebothgiventhesymbols.Thisshouldnotbeconfusedwiththesymbol

forspeed(andvelocity),whichisv.Thesymbolhisalsowidelyusedforverticaldistances(heights).TheSIunitfordistanceisthemetre,m,althoughotherunits,suchasmm,cmandkm,areincommonuse.Becauseadirectionisspecifiedaswellasamagnitude(size),displacementisavectorquantity.

Distanceisascalarquantitybecauseithasmagnitude,butnodirection.Figure2.2showstherouteofsomepeoplewalkingaroundapark.Thetotaldistancewalkedwas4

km,butthedisplacementfromthereferencepointvariedandisshowneveryfewminutesbythevectorarrows(ae).Thefinaldisplacementiszerobecausethewalkersreturnedtotheirstartingplace.

Thetransportofvariousvehicles,goodsandpeoplearoundtheworldisbigbusiness,andismonitoredandcontrolledbymanycountriesandinternationalcompanies.Thisrequiresaccuratemeansoftrackingthelocationandmovementofalargenumberofvehicles(ships,aircraftetc.)andtherapidcommunicationofthisinformationbetweencountries.

Speedandvelocity

Speedisdefinedastherateofchangeofdistancewithtime.

Speedisascalarquantityanditisgiventhesymbolv.ItsSIunitismetrespersecond,ms1.Speediscalculatedfrom:

Thedeltasymbol()isusedwhereverwewanttorepresenta(small)changeofsomething,sowecandefinespeedinsymbols,asfollows:

Ifanobjectismovingwithaconstantspeed,determiningitsvalueisastraightforwardcalculation.However,thespeedofanobjectoftenchangesduringthetimeweareobservingit,andthecalculatedvalueisthenanaveragespeedduringthattime.Forexample,ifacarisdrivenadistanceof120kmin1.5h,itsaveragespeedis80kmh1,butitsactualspeedwillcertainlyhavevariedduringthejourney.Atanyonetimewecouldlookatthecarsspeedometertofindouttheinstantaneousspeedthatis,thespeedatthatexactinstant(moment).Inkinematicsweareusuallymoreinterestedininstantaneousvaluesofspeed(andvelocityandacceleration)thanaveragevalues.Averagespeedsarecalculatedoverlengthsoftimethatarelongenoughfortheactualspeedstohave

changed.Instantaneousvalueshavetobecalculatedfrommeasurementsmadeoververyshorttimeintervals(duringwhichtimewecanassumethatthespeedwasconstant).Speediscalculatedusingthedistancetravelledinthetimebeingconsidered,regardlessofthe

directionofmotion.IfthewalkersinFigure2.2took2hourstowalkaroundthepark,theiraveragespeedwouldbes/t(=4/2)=2kmh1.

Utilizations

TraveltimetablesFigure2.4showsatimetablefortheGhan,atrainthattravelsacrossAustraliabetweenAdelaideandDarwin,adistanceof2979kmalongthetrack.

1aCalculatethejourneytimeandhencetheaveragespeed.bWhyisyouranswertoamisleading?

Weareoftenconcernednotonlywithhowfastanobjectismoving,butalsowiththedirectionofmovement.Ifspeedanddirectionarestatedthenthequantityiscalledvelocity.

Velocityisdefinedastherateofchangeofdisplacementwithtime(speedinagivendirection):

Notethatsinthisequationreferstodisplacementandnottotheoveralldistance.(Toavoidconfusion,itisoftenbettertodefinespeedandvelocityinwords,notsymbols.)Velocityhasthesamesymbolandunitasspeed,butthedirectionshouldusuallybestatedaswell,

sincevelocityisavectorquantity.However,ifthedirectionofmotiondoesnotchange,itisnotuncommontorefertoaspeed,ofsay4ms1,asvelocitybecausethedirectionisunderstoodfromthecontext.Returningtothewalkersintheparkattheendoftheirwalktheiraveragespeedwas2kmh1,but

theiraveragevelocitywaszerobecausethefinaldisplacementwaszero.Thismightnotbeaveryusefulpieceofinformation;wearemorelikelytobeinterestedintheinstantaneousvelocityatvarioustimesduringthewalk.Whenthevelocity(orspeed)ofanobjectchangesduringacertaintime,thesymboluisusedforthe

initialvelocityandvisusedforthefinalvelocityduringthattime.Thesevelocitiesarenotnecessarilyat

thebeginningandendoftheentiremotion,justthevelocitiesatthestartandendoftheperiodoftimethatisbeingconsidered.

Thedistancetravelledintimetcanbedeterminedusingtheequation:

distance=averagespeedtime

Foranobjectwithconstantacceleration:

averagespeed=(initialspeed+finalspeed)

Forexample,ifacaracceleratesuniformlyfrom12ms1to16ms1,thenitsaveragespeedduringthattimewas14ms1.Insymbols,thisisshownas:

ThisequationisgiveninthePhysicsdatabooklet.

DatalogginginmotioninvestigationsTheuseofmotionsensorsanddataloggers(seeFigure2.5),lightgatesandelectronictimers,and

videorecordinghaveallmadetheinvestigationofvariouskindsofmotionmoreinteresting,mucheasierandmoreaccurate.

AccelerationAnyvariationfrommovingataconstantspeedinastraightlineisdescribedasanacceleration.Itisveryimportanttorealisethatgoingfaster,goingslowerand/orchangingdirectionarealldifferentkindsofacceleration(changingvelocities).

Acceleration,a,isdefinedastherateofchangeofvelocitywithtime:

(iftheaccelerationisconstantovertimet)

TheSIunitofaccelerationismetrespersecondsquared,ms2(thesameastheunitsofvelocity/time,ms1/s).Accelerationisavectorquantity.Accelerationcanbe:

anincreaseinvelocity(positiveacceleration)adecreaseinvelocity(negativeaccelerationsometimescalledadeceleration)achangeofdirection.

AdditionalperspectivesReactiontimeswhentimingmotionsThedelaybetweenseeingsomethinghappenandrespondingwithsomekindofactionisknownasreactiontime.Atypicalvalueisabout0.20s,butitcanvaryconsiderablydependingontheconditionsinvolved.Asimplewayofmeasuringapersonsreactiontimeisbymeasuringhowfarametrerulefallsbeforeitcanbecaughtbetweenthumbandfinger.Thetimecanthenbecalculatedusingtheequations=5t2.

Themeasurementcanberepeatedwiththepersontestedbeingblindfoldedtoseeifthereactiontimechangesifthestimulus(tocatchtheruler)iseithersoundortouch,ratherthansight.Whatevertestsarecarriedout,ourreactiontimesarelikelytobeinconsistent.Thismeansthatwheneverweusestopwatchesoperatedbyhand,theresultswillhaveanunavoidableuncertainty(seeChapter1).Itissensibletomaketimemeasurementsaslongaspossibletodecreasethesignificanceofthisproblem.(Thisreducesthepercentageuncertainty.)Repeatingmeasurementsandcalculatinganaveragewillalsoreducetheeffectofrandomerrors.1Usethemethoddescribedabove(oranyother)tomeasureyourreactiontimewhenthe

stimulusissight.Repeatthemeasurement10times.aWhatwasthepercentagevariationbetweenyouraverageresultandyourquickest

reactiontime?bDidyourreactiontimesimprovewithpractice?

GraphsdescribingmotionGraphscanbedrawntorepresentanymotionandtheyprovideextraunderstandingandinsight(ataglance)thatveryfewpeoplecangetfromwrittendescriptionsorequations.Furthermore,thegradientsofgraphsandtheareasundergraphsoftenprovideadditionalvaluableinformation.

DisplacementtimegraphsanddistancetimegraphsDisplacementtimegraphs,similartothoseshowninFigure2.6,showhowthedisplacementsofobjectsfromareferencepointvarywithtime.AlltheexamplesshowninFigure2.6arestraightlinesandcan

bedescribedasrepresentinglinearrelationships.

LineArepresentsanobjectmovingawayfromareferencepointsuchthatequaldisplacementsoccurinequaltimes.Thatis,theobjecthasaconstantvelocity.Anylineardisplacementtimegraphrepresentsaconstantvelocity(itdoesnotneedtostartorendattheorigin).

LineBrepresentsanobjectmovingwithahighervelocitythanA.LineCrepresentsanobjectthatismovingclosertothereferencepoint.LineDrepresentsanobjectthatisstationary(atrest).Ithaszerovelocityandstaysatthesamedistancefromthereferencepoint.

Displacementisavectorquantity,butdisplacementtimegraphsliketheseareusuallyusedinsituationswherethemotionisinaknowndirection,sothatthedirectionmaynotneedtobestatedagain.Displacementinoppositedirectionsisrepresentedbytheuseofpositiveandnegativevalues.ThisisshowninFigure2.7,inwhichthesolidlinerepresentsthemotionofanobjectmovingwithaconstant(positive)velocity.Theobjectmovestowardsareferencepoint(whenthedisplacementiszero),passesit,andthenmovesawayintheoppositedirectionwiththesamevelocity.Thedottedlinerepresentsanidenticalspeedintheoppositedirection(oritcouldalsorepresenttheoriginalmotionifthedirectionschosentobepositiveandnegativewerereversed).

Anycurved(non-linear)lineonadisplacementtimegraphrepresentsachangingvelocity,inotherwords,anacceleration(ordeceleration).ThisisillustratedinFigure2.8.

Figure2.8ashowsmotionawayfromareferencepoint.LineArepresentsanobjectaccelerating.LineBrepresentsanobjectdecelerating(negativeacceleration).Figure2.8bshowsmotiontowardsareferencepoint.LineCrepresentsanobjectaccelerating.LineD

representsanobjectdecelerating(negativeacceleration).Thevaluesoftheaccelerationsrepresentedbythesegraphsmay,ormaynot,beconstant(thiscannot

bedeterminedwithoutamoredetailedanalysis).Inphysics,weareusuallymoreconcernedwithdisplacementtimegraphsthandistancetimegraphs.

Inordertoexplainthedifference,considerFigure2.9.Figure2.9ashowsadisplacementtimegraphforanobjectthrownverticallyupwardswithaninitialspeedof20ms1,withoutairresistance.Ittakes2storeachamaximumheightof20m.Atthatpointithasaninstantaneousvelocityofzero,beforereturningtowhereitbeganafter4sandregainingitsinitialspeed.Figure2.9bshowshowthesamemotionwouldappearonanoveralldistancetimegraph.

Gradientsofdisplacementtimegraphs

ConsiderthemotionatconstantvelocityshowninFigure2.10.

Fromthegraph,thevelocityvisgivenby:

Notethatthevelocityisnumericallyequaltothegradient(slope)oftheline.Thisisalwaystrue,whatevertheshapeoftheline.

Theinstantaneousvelocityofanobjectisequaltothegradientofthedisplacementtimegraphatthatinstant.

Figure2.11showsanobjectmovingwithincreasingvelocity.Thevelocityatanytime(forexamplet1)canbedeterminedbycalculatingthegradientofthetangenttothelineatthatinstant.

Thetriangleusedshouldbelarge,inordertomakethisprocessasaccurateaspossible.Thetangentdrawnattimet2hasasmallergradientbecausethevelocityissmaller.Attimet3thevelocityishigherandthegradientsteeper.So,inthisexample:

1Figure2.12representsthemotionofatrainonastraighttrackbetweentwostations.aDescribethemotion.bHowfarapartarethestations?cCalculatethemaximumspeedofthetrain.dWhatwastheaveragespeedofthetrainbetweenthetwostations?

2aDrawadisplacementtimegraphforaswimmerwhoswims50mataconstantspeedof1.0ms1iftheswimmingpoolis25mlongandtheswimmertakes1stoturnaroundhalfwaythroughtherace.

bFindouttheaveragespeedoftheworldfreestylerecordholderwhenthe100mrecordwaslastbroken.

cTheworldrecordforswimming50minapooloflength25misquickerthanforswimminginapooloflength50m.Suggestwhy.

3Drawadisplacementtimegraphforthefollowingmotion:astationarycaris25maway;2slateritstartstomovefurtherawayinastraightlinefromyouwithaconstantaccelerationof1.5ms2for4seconds;thenitcontinueswithaconstantvelocityforanother8s.

4DescribethemotionoftherunnershownbythegraphinFigure2.13.

5aDescribethemotionrepresentedbythegraphinFigure2.14.

bComparethevelocitiesatpointsAandB.cWhenistheobjectmovingwithitsmaximumandminimumvelocities?dEstimatevaluesforthemaximumandminimumvelocities.eSuggestwhatkindofobjectcouldmoveinthisway.

VelocitytimegraphsAnyvelocitytimegraph,likethoseshowninFigure2.15,showshowthevelocityofanobjectvarieswithtime.Anystraight(linear)lineonanyvelocitytimegraphshowsthatequalchangesofvelocityoccurinequaltimesthatis,aconstantacceleration.

LineAshowsanobjectthathasaconstantpositiveacceleration.LineBrepresentsanobjectmovingwithahigherpositiveaccelerationthanA.LineCrepresentsanobjectthatisdecelerating(negativeacceleration).LineDrepresentsanobjectmovingwithaconstantvelocitythatis,ithaszeroacceleration.

Curvedlinesonvelocitytimegraphsrepresentchangingaccelerations.Velocitiesinoppositedirectionsarerepresentedbypositiveandnegativevalues.ThesolidlineinFigure2.16representsanobjectthatdeceleratesuniformlytozerovelocityandthenmovesintheoppositedirectionwithanaccelerationofthesamemagnitude.Thisgraphcouldrepresentthemotionofastonethrownintheair,reachingitsmaximumheightandthenfallingdownagain.Theaccelerationremainsthesamethroughout(9.81ms

2downwards).Inthisexamplevelocityandaccelerationupwardshavebeenchosentobenegative,andvelocityandaccelerationdownwardsarepositive.Thedashedlinewouldrepresentexactlythesamemotionifthedirectionschosentobepositiveandnegativewerereversed.

GradientsofvelocitytimegraphsConsiderthemotionatconstantaccelerationshowninFigure2.17.

Fromthegraph:

Notethattheaccelerationisnumericallyequaltothegradient(slope)oftheline.Thisisalwaystrue,whatevertheshapeoftheline.

Theinstantaneousaccelerationofanobjectisequaltothegradientofthevelocitytimegraphatthatinstant.

Workedexample1TheredlineinFigure2.18showsanobjectdecelerating(withadecreasingnegative

acceleration).Usethegraphtofindtheinstantaneousaccelerationat10s.

Atangentdrawnatthetimeof10scanbeusedtodeterminethevalueoftheaccelerationatthatinstant:

Inthisexamplethelargetriangleusedtodeterminethegradientaccuratelywasdrawnbyextendingthetangenttotheaxesforconvenience.

AreasundervelocitytimegraphsConsideragainthemotionrepresentedinFigure2.17.Thechangeofdisplacement,s,betweenthefourthandninthsecondcanbefoundfrom(averagevelocity)time.

Thisisnumericallyequaltotheareaunderthelinebetweent=4.0sandt=9.0s(asshadedinFigure2.17).Thisisalwaystrue,whatevertheshapeoftheline.

Theareaunderavelocitytimegraphisequaltothechangeofdisplacementinthechosentime.

Workedexample

2Figure2.19ashowshowthevelocityofacarchangedinthefirst5safterstarting.Usethegraphtoestimatethedistancetravelledinthistime.

InFigure2.19bthebluelinehasbeendrawnsothattheareaunderitandtheareaundertheoriginallinearethesame(asjudgedbyeye).

6aDescribethemotionrepresentedbythegraphinFigure2.20.bCalculateaccelerationsforthethreepartsofthejourney.cWhatwasthetotaldistancetravelled?dWhatwastheaveragespeed?

7Thevelocityofacarwasreadfromitsspeedometeratthemomentitstartedandevery2

safterwards.Thesuccessivevalues(ms1)were:0,1.1,2.4,6.9,12.2,18.0,19.9,21.3and21.9.Plot

agraphofthesereadingsanduseittoestimatethemaximumaccelerationandthedistancecoveredin16s.

8aDescribethemotionoftheobjectrepresentedbythegraphinFigure2.21.bCalculatetheaccelerationduringthefirst8s.cWhatwasthetotaldistancetravelledin12s?dWhatwasthetotaldisplacementafter12s?eWhatwastheaveragespeedduringthe12sinterval?

9Sketchavelocitytimegraphofthefollowingmotion:acaris100mawayandtravellingalongastraightroadtowardsyouataconstantvelocityof25ms1.Twosecondsafterpassingyou,thedriverdeceleratesuniformlyandthecarstops62.5mawayfromyou.

UtilizationsBiomechanicsand100msprintersWorld-classsprinterscanrun100minabout10s(seeFigure2.22).Theaveragevelocityiseasytocalculate:v=100/10=10ms1.Clearlytheystartfrom0ms1,sotheirhighestinstantaneousvelocitymustbegreaterthan10ms1.

Trainersusethescienceofbiomechanicstoimproveanathletestechniques,andthelatestcomputerizedmethodsareusedtoanalyseeverymomentoftheirraces.Theaccelerationofftheblocksatthestartoftheraceisallimportant,sothatthehighestvelocityisreachedassoonaspossible.Fortherestoftheracetheathleteshouldbeabletomaintainthesamespeed,althoughtheremaybeaslightdecreasetowardstheendoftherace.Figure2.23showsatypicalvelocitytimegraphfora100mracecompletedin10s.

1aEstimatethehighestaccelerationachievedduringtheraceillustratedinFigure2.23.bWhendoestheathletereachtheirgreatestvelocity?cExplainwhythetwoshadedareasonthegraphareequal.dUsingtheinternettocollectdata,drawagraphshowinghowtheworld(orOlympic)

recordforthe100mhaschangedoverthelast100years.ePredictthe100mrecordfortheyear2040.

AccelerationtimegraphsAnaccelerationtime(at)graph,likethoseshowninFigure2.24,showshowtheaccelerationofanobjectchangeswithtime.Inthischapter,wearemostlyconcernedwithconstantaccelerations(itislesscommontoseemotiongraphsshowingchangingacceleration).ThegraphsinFigure2.24showfivelinesrepresentingconstantaccelerations.

LineAshowszeroacceleration,constantvelocity.LineBshowsaconstantpositiveacceleration(uniformlyincreasingvelocity).LineCshowstheconstantnegativeacceleration(deceleration)ofanobjectthatisslowingdownata

constantrate.LineDshowsa(linearly)increasingpositiveacceleration.LineEshowsanobjectthatisacceleratingpositively,butata(linearly)decreasingrate.

AreasunderaccelerationtimegraphsFigure2.25showstheconstantaccelerationofamovingcar.Usinga=v/t,betweenthefifthandthirteenthseconds,thevelocityofthecarincreasesby:

Thechangeinvelocityisnumericallyequaltotheareaunderthelinebetweent=5sandt=13s(shadedinFigure2.25).Thisisalwaystrue,whatevertheshapeoftheline.

Theareaunderanaccelerationtimegraphisequaltothechangeofvelocityinthechosentime.

10Drawanaccelerationtimegraphforacarthatstartsfromrest,acceleratesat2ms2for5s,thentravelsatconstantvelocityfor8s,beforedeceleratinguniformlytorestagaininafurther2s.

11Figure2.26showshowtheaccelerationofacarchangedduringa6sinterval.Ifthecarwastravellingat2ms1after1s,estimateasuitableareaunderthegraphanduseittodeterminetheapproximatespeedofthecarafter5s.

12Figure2.27showsatennisballbeingstruckbyaracquet.Sketchapossiblevelocitytimegraphandanaccelerationtimegraphfrom1sbeforeimpactto1saftertheimpact.

13Sketchpossibledisplacementtimeandvelocitytimegraphsforabouncingballdroppedfromrest.Continuethesketchesuntilthethirdtimethattheballcontactstheground.

Graphsofmotion:summaryIfanyonegraphofmotionisplotted(st,vtorat),thenthemotionisfullydefinedandtheothertwographscanbedrawnwithinformationaboutgradientsand/orareastakenonlyfromthefirstgraph.ThisissummarizedinFigure2.28.

Toreproduceonegraphfromanotherbyhandisalongandrepetitiveprocess,becauseinordertoproduceaccurategraphsalargenumberofsimilarmeasurementsandcalculationsneedtobemadeovershortintervalsoftime.Ofcourse,computersareidealforthispurpose.Inmoremathematicallyadvancedwork,whichisnotpartofthiscourse,calculuscanbeusedto

performtheseprocessesusingdifferentiationandintegration.

UtilizationsKinematicequations:vehiclebrakingdistancesFigure2.29representshowthevelocitiesoftwoidenticalcarschangedfromthemomentthattheirdriverssawdangerinfrontofthemandtriedtostoptheircarsasquicklyaspossible.Ithasbeenassumedthatbothdrivershavethesamereactiontime(0.7s)andbothcarsdecelerateatthesamerate(5.0ms2).

Thedistancetravelledatconstantvelocitybeforethedriverreactsanddepressesthebrakepedalisknownasthethinkingdistance.Thedistancetravelledwhiledeceleratingiscalled

thebrakingdistance.Thetotalstoppingdistanceisthesumofthesetwodistances.

CarB,travellingattwicethevelocityofcarA,hastwicethethinkingdistance.Thatis,thethinkingdistanceisproportionaltothevelocityofthecar.Thedistancetravelledwhenbraking,however,isproportionaltothevelocitysquared.Thiscanbeconfirmedfromtheareasunderthevtgraphs.TheareaundergraphBisfourtimestheareaundergraphA(duringthedeceleration).Thishasimportantimplicationsforroadsafetyandmostcountriesmakesurethatpeoplelearningtodrivemustunderstandhowstoppingdistanceschangewiththevehiclesvelocity.Somecountriesmeasurethereactiontimesofpeoplebeforetheyaregivenadrivinglicence.

Setupaspreadsheetthatwillcalculatethetotalstoppingdistanceforcarstravellingatinitialspeeds,u,between0and40ms1withadecelerationof6.5ms2.(Makecalculationsevery2ms1.)Thethinkingdistancecanbecalculatedfromst=0.7u(reactiontime0.7s).Inthisexamplethebrakingtimecanbecalculatedfromtb=u/6.5andthebrakingdistancecanbecalculatedfromsb=(u/2)tb.Usethedataproducedtoplotacomputer-generatedgraphofstoppingdistance(y-axis)againstinitialspeed(x-axis).

EquationsofmotionforuniformaccelerationThefivequantitiesu,v,a,sandtareallthatisneededtofullydescribethemotionofanobjectmovingwithuniform(constant)acceleration.u=velocity(speed)atthestartoftimetv=velocity(speed)attheendoftimeta=acceleration(constant)s=distancetravelledintimett=timetakenforvelocity(speed)tochangefromutovandtotraveladistances

Ifanythreeofthequantitiesareknown,theothertwocanbecalculatedusingthetwoequationsbelow.Ifweknowtheinitialvelocityuandaccelerationaofanobject,andtheaccelerationisuniform,thenwecandetermineitsfinalvelocityvafteratimetbyrearrangingtheequationusedtodefineacceleration.Thisgives:

v=u+at

ThisequationisgiveninthePhysicsdatabooklet.Wehavealsoseenthatthedistancetravelledwhileacceleratinguniformlyfromavelocityutoa

velocityvinatimetcanbecalculatedfrom:

ThisequationisgiveninthePhysicsdatabooklet.Thesetwoequationscanbecombinedmathematicallytogivetwofurtherequations,shownbelow,

whicharealsofoundinthePhysicsdatabooklet.Theseveryusefulequationsdonotinvolveanyfurtherphysicstheory;theyjustexpressthesamephysicsprinciplesinadifferentway.

Rememberthat,thefourequationsofmotioncanonlybeusediftheaccelerationisuniformduringthetimebeingconsidered.TheequationsofmotionarecoveredintheIBMathematicscourse(andalsotreatedincalculus

form).

Workedexample3AFormulaOneracingcar(seeFigure2.30)acceleratesfromrest(i.e.itwasstationaryto

beginwith)at18ms2.aWhatisitsspeedafter3.0s?bHowfardoesittravelinthistime?cIfitcontinuestoaccelerateatthesamerate,whatwillitsvelocitybeafterithastravelled

200mfromrest?dConvertthefinalvelocitytokmh1.

Butnotethatthedistancecanbecalculateddirectly,withoutfirstcalculatingthefinalvelocity,asfollows:

4Atraintravellingat50ms1(180kmh1)needstodecelerateuniformlysothatitstopsatastation2kilometresaway.aWhatisthenecessarydeceleration?bHowlongdoesittaketostopthetrain?

Assumethatallaccelerationsareconstant.14Aballrollsdownaslopewithaconstantacceleration.WhenitpassesapointPitsvelocity

is1.2ms1andashorttimelateritpassespointQwithavelocityof2.6ms1.aWhatwasitsaveragevelocitybetweenPandQ?bIfittook1.4stogofromPtoQ,whatisthedistancePQ?cWhatistheaccelerationoftheball?15Aplaneacceleratesfromrestalongarunwayandtakesoffwithavelocityof86.0ms1.

Itsaccelerationduringthistimeis2.40ms2.aWhatdistancealongtherunwaydoestheplanetravelbeforetake-off?bHowlongafterstartingitsaccelerationdoestheplanetakeoff?16Anocean-goingoiltankercandeceleratenoquickerthan0.0032ms2.aWhatistheminimumdistanceneededtostopiftheshipistravellingat10knots?(1knot

=0.514ms1)bHowmuchtimedoesthisdecelerationrequire?17Anadvertisementforanewcarstatesthatitcantravel100mfromrestin8.2s.aWhatistheaverageacceleration?bWhatisthespeedofthecarafterthistime?18Acartravellingataconstantvelocityof21ms1(fasterthanthespeedlimitof50kmh

1)passesastationarypolicecar.Thepolicecaracceleratesaftertheothercarat4.0ms2for8.0sandthencontinueswiththesamevelocityuntilitovertakestheothercar.

aWhendidthetwocarshavethesamevelocity?bHasthepolicecarovertakentheothercarafter10s?cByequatingtwoequationsforthesamedistanceatthesametime,determineexactly

whenthepolicecarovertakestheothercar.19Acarbrakessuddenlyandstops2.4slater,aftertravellingadistanceof38m.aWhatwasitsdeceleration?bWhatwasthevelocityofthecarbeforebraking?20Aspacecrafttravellingat8.00kms1acceleratesat2.00103ms2for100hours.aWhatisitsfinalspeed?bHowfardoesittravelduringthisacceleration?21Combinethefirsttwoequationsofmotion(givenonpage33)toderivethesecondtwo(v2

=u2+2asand

NatureofScienceObservationsScientificknowledgeonlyreallydevelopedaftertheimportanceofexperimentalevidencewasunderstood.Theequationsofmotion(andNewtonslawsofmotion)areaveryimportantpartof

classicalphysicsthatallstudentsshouldunderstandwell.Theywerefirstproposedatanearlystageinthehistoricaldevelopmentofphysics,whenexperimentaltechniqueswerenotasdevelopedastheyaretoday.However,thesebasicideasaboutmotionstillremainjustasimportantinthemodernworld.Earlyscientists,likeGalileoandNewton,wereabletomakecarefulobservationsand

gatherenoughevidencetosupporttheirtheoriesaboutidealizedmotiondespitethefactthatfrictionandairresistancealwayscomplicatethestudyofmovingobjects.Thisisespeciallyimpressivebecausesomeoftheirtheoriescontradictedideasthathadbeenacceptedfor2000years.

AccelerationduetogravityWeareallfamiliarwiththemotionofobjectsfallingtowardsEarthbecauseoftheforceofgravity.

Figure2.31showsanexperimenttogatherdataondistancesandtimesforafallingmass,sothatavalueforitsaccelerationcanbecalculated.Theelectronictimerstartswhentheelectriccurrenttotheelectromagnetisswitchedoffandthesteelballstartstofall.Whentheballhitsthetrapdooratthebottom,asecondelectricalcircuitisswitchedoffandthetimingstops.Alternatively,apositionsensorcouldbeusedtotrackthefalloftheball.

Workedexample5Supposethatwhenthemassfell0.84mthetimewasmeasuredtobe0.42s.Calculateits

gravitationalacceleration.

Ofcourse,obtaininganaccurateandreliableresultwillrequirefurthermeasurements.Themeasurementcouldberepeatedforthesameheight,sothataveragescouldbecalculated.Butitwouldbebettertotakemeasurementsfordifferentheights,sothatanappropriategraphcanbedrawn,whichwillprovideabetterwayofassessingrandomandsystematicerrors.Ifaccuratemeasurementsaremadeinavacuum(tobesurethatthereisnoairresistance),theresults

areverysimilar(butnotidentical)atalllocationsontheEarthssurface.SomeexamplesareshowninTable2.1.TheaccelerationduetogravityinavacuumneartheEarthssurfaceisgiventhesymbolg.Thisis

alsocalledtheaccelerationoffreefall.Theacceptedvalueofgis9.81ms2.ThisvalueshouldbeusedincalculationsandislistedinthePhysicsdatabooklet.AnywhereontheEarthssurface(orinanairplane)canbeconsideredasneartotheEarthssurface.

ItisveryimportanttorememberthatallfreelymovingobjectsclosetotheEarthssurfaceexperiencethissameacceleration,g,downwards.Thisistruewhethertheobjectislargeorverysmall,orwhetheritismovingupwards,downwards,sidewaysorinanyotherdirection.Freelymovingmeansthattheeffectsofairresistancecanbeignoredandthattheobjectisnotpoweredinanyway.Inreality,however,theeffectsofairresistanceusuallycannotbeignored,exceptforlarge,densemassesmovingshortdistancesfromrest.But,asisoftenthecaseinscience,weneedtounderstandsimplifiedexamplesfirstbeforewemoveontomorecomplicatedsituations.

Workedexample

6Acoinfallsfromrestoutofanopenwindow16mabovetheground.Assumingthatthereisnoairresistance:

awhatisitsvelocitywhenithitstheground?bhowlongdidittaketofallthatdistance?

7Aballisthrownverticallyupwardsandreachesamaximumheightof21.4m.aCalculatethespeedwithwhichtheballwasreleased.bWhatassumptiondidyoumake?cWherewilltheballbe3.05safteritwasreleased?dWhatwillitsvelocitybeatthistime?

av2=u2+2asWhentheballhastravelledadistances=21.4m,itsspeed,v,atthehighestpointwill

bezero.02=u2+(29.8121.4)u2=419.9u=20.5ms1Inthisexample,thevectorquantitiesdirectedupwards(u,v,s)areconsideredpositive

andthequantitydirecteddownwards(a)isnegative.Thesameanswerwouldbeobtainedbyreversingallthesigns.Usingpositiveandnegativesignstorepresentvectors(likedisplacement,velocityandacceleration)inoppositedirectionsiscommonpractice.

bItwasassumedthattherewasnoairresistance.c

dv=u+atv=20.5+(9.813.05)v=9.42ms1(movingdownwards)

Inallofthefollowingquestions,ignorethepossibleeffectsofairresistance.Useg=9.81ms2.22Suggestpossiblereasonswhytheaccelerationduetogravityisnotthesameeverywhere

ontheEarthssurface.23aHowlongdoesittakeastonedroppedfromrestfromaheightof2.1mtoreachthe

ground?bIfthestonewasthrowndownwardswithaninitialvelocityof4.4ms1,withwhatspeed

wouldithittheground?cIfthestonewasthrownverticallyupwardswithaninitialvelocityof4.4ms1,withwhat

speedwouldithittheground?24Asmallrockisthrownverticallyupwardswithaninitialvelocityof22ms1.Whenwillits

speedbe10ms1?(Therearetwopossibleanswers.)25Afallingballhasavelocityof12.7ms1asitpassesawindow4.81mabovetheground.

Whenwillithittheground?26Aballisthrownverticallyupwardswithaspeedof18.5ms1fromawindowthatis12.5

mabovetheground.aWhenwillitpassthesamewindowmovingdown?bWithwhatspeedwillithittheground?cHowfarabovethegroundwastheballafterexactly2s?27Twoballsaredroppedfromrestfromthesameheight.Ifthesecondballisreleased0.750

safterthefirst,andassumingtheydonothittheground,howfarapartarethetwoballs:a3.00safterthesecondballwasdropped?b2.00slater?28Astoneisdroppedfromrestfromaheightof34m.Anotherstoneisthrowndownwards

0.5slater.Iftheybothhitthegroundatthesametime,whatwastheinitialvelocityofthesecondstone?

29InWorkedexample3anaccelerationof18ms2wasquotedforaFormulaOneracingcar.Thedriverofthatcarcouldbesaidtoexperienceag-forceofnearly2g,andduringthecourseofatypicalraceadrivermayhavetoundergog-forcesofnearly5g.Explainwhatyouthinkismeantbyag-forceof2g.

30StoneAisdroppedfromrestfromacliff.Afterithasfallen5m,stoneBisdropped.aHowdoesthedistancebetweenthetwostoneschange(ifatall)astheyfall?bExplainyouranswer.31aAfleaacceleratesattheenormousaveragerateof1500ms2duringaverticaltake-off

thatlastsonlyabout0.0012s.Whatheightwillthefleareach?bMeasurehowhighyoucanjumpvertically(standinginthesameplace),andusethe

resulttocalculateyourtake-offvelocity.cInordertojumpupyouhadtobendyourkneesandreduceyourheight.Measurebyhow

muchyourheightwasreducedjustbeforejumping,thenusetheresulttoestimateyouraverageaccelerationduringtake-off.

dWhatwasthedurationofyourtake-off?eCompareyourperformancewiththefleas.32UsetheinternettolearnmoreabouttheGOCEproject,whichendedin2013(Figure

2.32).

33Figure2.33showsthetallestbuildingintheworld:BurjKhalifainDubai.aHowlongwouldittakeanobjecttoreachthegroundifitwasdroppedfrom828m(the

heightofBurjKhalifa)?bWithwhatspeedwouldithittheground?

34Thetimesoffallforaballdroppedfromdifferentheights(Figure2.31)weremeasured.aSketchtheheighttimegraphyouwouldexpecttogetfromtheseresults.bByconsideringtheequations=ut+ whatwouldbethebestgraphtodrawto

produceastraightbest-fitlinefromwhichtheaccelerationduetogravitycouldbedetermined?

FluidresistanceandterminalspeedAsanyobjectmovesthroughair,theairisforcedtomoveoutofthepathoftheobject.Thiscausesa

forceopposingthemotioncalledairresistance,ordrag.Similarforceswillopposethemotionofanobjectmovinginanydirectionthroughanygasorliquid.

(Gasesandliquidsarebothdescribedasfluidsbecausetheycanflow.)Suchforcesopposingmotionaregenerallydescribedasfluidresistance.Figure2.34representsthemotionofanobjectfallingtowardsEarth.LineAshowsthemotion

withoutairresistanceandlineBshowsthemotion,morerealistically,withairresistance.

Whenanyobjectfirststartstofall,thereisnoairresistance.Theinitialacceleration,g,isthesameasifitwasinavacuum.Astheobjectfallsfaster,theairresistanceincreases,sothattherateofincreaseinvelocitybecomesless.ThisisshownintheFigure2.34bythelineBbecominglesssteep.Eventuallytheobjectreachesaconstant,maximumspeedknownastheterminalspeedorterminalvelocity(terminalmeansfinal).Thevalueofanobjectsterminalspeedwilldependonitscross-sectionalarea,shapeandweight,asdiscussedinSection2.2.Theterminalspeedofaskydiverisusuallyquotedatabout200kmh1(56ms1)Figure2.35.TerminalspeedalsodependsonthedensityoftheairinOctober2012FelixBaumgartner(Figure2.36),anAustrianskydiver,reachedaworldrecordspeedof1358kmh1bystartinghisjumpfromaheightofabout39kmabovetheEarthssurfacewherethereisverylittleair.

Thedesignandmotionofsimpleparachutesmakeinterestinginvestigations,especiallyiftheycanbevideoedfallingneartoverticalscales.Themovementofanobjectfallingverticallythroughaliquid(oilforexample)isslowerandcanalsobeinvestigatedinaschoollaboratory.Itmayalsoreachaterminalspeed,andhaveapatternofmotionsimilartothatshowninFigure2.34.Computersimulationsarealsousefulforgainingaquickappreciationofthefactorsthataffectterminalspeeds.Airresistanceisdiscussedingreaterdetaillaterinthischapter(page52).

AdditionalPerspectivesGalileoItisamatterofcommonobservationthatheavierobjectsfalltoEarthquickerthanlighterobjects.Thisiseasilydemonstratedbydropping,forexample,aballandapieceofpapersidebyside.Theunderstandablebeliefthatheavierobjectsfallfasterwasafundamentalprincipleinnaturalphilosophy(thenameforearlystudiesofwhatisnowknownasscience)

formorethan2000yearsofcivilization.InancientGreece,AristotlehadcloselylinkedthemotionoffallingobjectstothebeliefthatallprocesseshaveapurposeandthattheEarthwasthenaturalandrightfulrestingplaceforeverything.

InthesixteenthcenturytheItalianscientistGalileo(Figure2.37)wasamongthefirsttosuggestthatthereasonwhyvariousobjectsfalldifferentlyisonlybecauseofairresistance.Hepredictedthat,iftheexperimentcouldberepeatedinavacuum(withoutair),allobjectswouldhaveexactlythesamepatternofdownwardsmotionundertheeffectsofgravity.

Inoneofthemostfamousstoriesinscience,GalileodroppeddifferentmassesoffabalconyontheTowerofPisainItalytoshowtothosewatchingonthegroundbelowthatfallingobjectsareactedonequallybygravity.Thisstorymayormaynotbetrue,butoneofthereasonsthatGalileoissorespectedasagreatscientististhathewasoneofthefirsttoactuallydoexperiments,ratherthanjustthinkaboutthem.

Itwasmanyyearslater,aftertheinventionofthefirstvacuumpumps,thatIsaacNewtonandotherswereabletoremovetheeffectsofairresistanceanddemonstratethatacoin(aguinea)andafeatherfalltogether.

In1971thatfamousexperimentwasrepeatedontheMoon(Figure2.38)whenastronautDavidScottdroppedahammerandfeathersidebyside.MillionsofpeopleallovertheworldwerewatchingwhileheremindedthemofGalileosachievements.ThestrengthofgravityislessontheMoonthanontheEarthbecausetheMoonissmaller.ObjectsacceleratetowardstheMoonataboutoftheratethattheywouldontheEarth(g=1.6ms2).

1GalileosachievementswerespecificallymentionedwhentheexperimentwasrepeatedontheMoon,butdoyouthinkthattherewereotherscientistswhowereequallydeservingofcreditforadvancingunderstandingofmotionandgravity?Givethenamesoftwosuchpioneersofscienceandlisttheirgreatestachievements.

NatureofScienceWhatisscience?TheItalianscientistGalileoGalilei(15641642)isfamousforhispioneeringworkonkinematicsandfallingobjects,andithasbeenacknowledgedthathewasoneofthefirstpracticalscientists(inthemodernmeaningoftheword).Butwhat,exactly,isscienceandwhatmakessciencedifferentfromotherhumanactivities?Thisisnotaneasyquestiontoanswerinafewwords,althoughtherearecertainly

importantcharacteristicsthatmostscientificactivitiesshare:Scienceattemptstoseesomeunderlyingsimplicityinthevastcomplexityaroundus.Sciencelooksforthelogicalpatternsandrulesthatcontrolevents.Scienceseekstoaccumulateknowledgeand,whereverpossible,tobuildonexistingknowledgetomakeanever-expandingframeworkofunderstanding.

Mostimportantly,scienceisbasedonexperimentationandevidencethatis,sciencereliesonfactsthatare,atthecurrenttime,acceptedtobetrue.Nogoodscientistwouldeverclaimthatsomethingmustbeabsolutelytrueforalltimeoneoftheleadingcharacteristicsofscienceistheconstantindependentandwidespreadtestingofexistingtheoriesbyexperiment.Nofactortheorycaneverbeproventobetrueforalltimesandallplaces,soscienceoftenadvancesthroughexperimentsthattryt