Mathematics Stage 5 PAS5.2.5 Graphs of physical phenomena. Distance/time graphs

Transcription

1 Mahemaics Sage 5 PAS5.2.5 Graphs of physical phenomena Par 2 Disance/ime graphs

2 Number: Tile: PAS5.2.5 Graphs of physical phenomena This publicaion is copyrigh New Souh Wales Deparmen of Eucaion an Training (DET), however i may conain maerial from oher sources which is no owne by DET. We woul like o acknowlege he following people an organisaions whose maerial has been use: Oucomes an inicaors from Mahemaics Years 7-10 Syllabus Boar of Suies NSW, Overview, pp. iiiiv Par 1, pp. 3-4 Par 2, pp. 3-4 COMMONWEALTH OF AUSTRALIA Copyrigh Regulaions 1969 WARNING This maerial has been reprouce an communicae o you on behalf of he New Souh Wales Deparmen of Eucaion an Training (Cenre for Learning Innovaion) pursuan o Par VB of he Copyrigh Ac 1968 (he Ac). The maerial in his communicaion may be subjec o copyrigh uner he Ac. Any furher reproucion or communicaion of his maerial by you may be he subjec of copyrigh proecion uner he Ac. CLI Projec Team acknowlegemen: Wrier: Janine Angove Eior: Dr. Ric Morane Illusraors: Thomas Brown, Tim Huchinson Deskop Publisher: Gayle Rey Version Dae: June 17, 2005 All reasonable effors have been mae o obain copyrigh permissions. All claims will be sele in goo faih. Publishe by Cenre for Learning Innovaion (CLI) 51 Wenworh R Srahfiel NSW 2135 Copyrigh of his maerial is reserve o he Crown in he righ of he Sae of New Souh Wales. Reproucion or ransmial in whole, or in par, oher han in accorance wih provisions of he Copyrigh Ac, is prohibie wihou he wrien auhoriy of he Cenre for Learning Innovaion (CLI). Sae of New Souh Wales, Deparmen of Eucaion an Training 2005.

3 Conens Par 2 Inroucion Par Inicaors...3 Preliminary quiz...5 Spee, isance an ime...7 Spee...10 Drawing isance ime graphs...17 Suggese answers Par Exercises Par Par 2 Disance/ime graphs 1

4 2 PAS5.2.5 Graphs of physical phenomena

5 Inroucion Par 2 Graphs provie informaion abou aa an rens. They also provie valuable informaion abou how one hing is changing in relaion o anoher. In his par you will explore graphs ha relae isance o ime an eermine he meaning conveye by he graphs. Inicaors By he en of Par 2, you will have been given he opporuniy o work owars aspecs of knowlege an skills incluing: unersaning erms such as ravel graph, epenen variable an inepenen variable inerpreing isance/ime graphs mae up of sraigh line segmens eermining which variable shoul be place on he horizonal axis rawing isance/ime graphs elling a sory shown by a isance/ime graph escribing how isance changes over ime using he relaive posiions of wo poins on a graph o inerpre informaion. By he en of Par 2, you will have been given he opporuniy o work mahemaically by: escribing he meaning of ifferen graiens isinguishing beween posiive an negaive graiens from a graph maching a graph o a escripion of a paricular even an explaining he reasons for your choice using spreashees o generae graphs Par 2 Disance/ime graphs 3

6 moelling, recoring an skeching graphs from invesigaions comparing an conrasing graphs. Source: Aape from oucomes of he Mahemaics Years 7 10 syllabus < 710_syllabus.pf > (accesse 04 November 2003). Boar of Suies NSW, PAS5.2.5 Graphs of physical phenomena

7 Preliminary quiz Before you sar his par, use his preliminary quiz o revise some skills you will nee. Aciviy Preliminary quiz Try hese. 1 On he axes below, raw a line ha has a negaive graien. 2 The graph below shows he spee (s) of a cyclis over ime (). Each par of he line graph is labelle. Use hese labels o answer he following quesions. s A B C D E o F Par 2 Disance/ime graphs 5

8 a Which par or pars of he graph show an increasing spee? b Which par or pars of he graph show he cyclis slowing own? c Which par or pars show ha he cyclis soppe? 3 Use he informaion below o calculae he average spee of he objec given he formula spee = isance. ime a A car ha ravels 300 km in 4 hours (answer in km/h). b A snail ha ravels 8 cm in 5.5 secons (answer in mm/s). Check your response by going o he suggese answers secion. 6 PAS5.2.5 Graphs of physical phenomena

9 Spee, isance an ime Disance an ime are wo very commonly linke measuremens. They are neee when preparing imeables for rains, planes an buses or for preparing recors for sporing evens. From measures of isance an ime you can also calculae velociy an acceleraion. In his secion, you will re-examine isance/ime graphs o eermine how o gain rich informaion from hem. These graphs are ofen referre o a ravel graphs. As wih oher graphs, isance/ime graphs ell a sory. They give much more informaion han jus a lis of aa poins. In mos insances aa is collece by measuring isance a regular ime inervals. Therefore isance epens on ime. This means ha isance is he epenen variable an is shown on he verical axis. Time is he inepenen variable an is shown on he horizonal axis. Disance Time Par 2 Disance/ime graphs 7

10 There are wo common means for measuring isance: isance ravelle since ime zero (how far you have gone in oal) isance from a fixe poin (such as home). I is very imporan o eermine which of hese has been recore on he graph. For example, he wo graphs below boh show John s journey as he ravelle along a sraigh pah. He walke 400 m away from his car in five minues, hen urne an jogge back o his car in wo minues. In his firs graph, John recors how far he is away from his car a cerain imes. The isance increases o 400 m hen ecreases more quickly back o zero as he reurns. The secon par of he graph is seeper showing ha John was moving a a faser rae. In his secon graph, John recors how far he has ravelle alogeher so he isance is always increasing. The wo ifferen graiens show ha he is ravelling more quickly in he secon par of he journey Disance from car Disance ravelle The firs graph can be use o fin he oal isance by aing he isance ravelle in each secion (400 m ou an 400 m back). However, he secon graph no longer conains informaion abou he irecion John was ravelling, an so is no as useful as he firs. 8 PAS5.2.5 Graphs of physical phenomena

11 Aciviy Spee, isance an ime Try hese. 1 The graph below escribes how far Lauren is away from home over a perio of five hours. Disance () is measure in kilomeres an ime () is measure in hours. 40 Disance from home a How far is Lauren from her home afer one hour an afer wo hours? Use his informaion o wrie a brief explanaion abou wha she migh have one uring his hour? b If she was ravelling along a sraigh roa, wrie a brief sory o escribe her journey. Par 2 Disance/ime graphs 9

12 c Use he aa in he able o fin he oal isance Lauren ravelle over he five hours. Check your response by going o he suggese answers secion. I is imporan o realise ha when recorings of aa are mae, informaion abou wha has happene beween reaings is los. You can only use he aa given. Tha is why i is vial o ake reaings as ofen as possible when conucing an experimen, paricularly when hings are changing in an unpreicable fashion. Spee The graien of a sraigh line is calculae using he formula graien = rise run For isance/ime graphs, he graien of each line segmen gives you he spee because spee = isance ravelle ime aken Spee in always consiere o be posiive, whereas graiens can be posiive or negaive. So o inerpre he spee from a graph, you can ignore he sign of he slope. However, he irecion of he slope oes ell you he irecion of he journey an is herefore very informaive. Using he graph of Lauren s journey again, you can calculae various spees. 10 PAS5.2.5 Graphs of physical phenomena

13 40 30 Disance from home The spee for he firs hour = 30 1, so Lauren ravelle a an average spee of 30 km/h for he firs hour Use he graph o complee he following aciviy. Aciviy Spee, isance an ime Try hese. 2 Calculae Laurens average spee for hese pars of her journey. a The secon hour (beween one an wo). b The final wo hours. c The whole five hours (use oal isance ravelle). Check your response by going o he suggese answers secion. Par 2 Disance/ime graphs 11

14 As wih mos line graphs, he graien or slope of he line provies very useful informaion. For isance/ime graphs, he graien conveys he spee an ofen he irecion of ravel as well. A a glance, you can see he fases or he slowes par of he journey an even ime when he objec is a res (no moving because spee = 0 so he line is horizonal). The following aciviy asks you o inerpre siuaions an mach hem o possible graphs. Aciviy Spee, isance an ime Try hese. 3 The graphs below are suppose o represen one person s journey. One graph shows an impossible siuaion. Sae which graph i is, an escribe why i is impossible. Disance from home Disance from home Disance from home Time Time Time Graph A Graph B Graph C 12 PAS5.2.5 Graphs of physical phenomena

15 4 In each graph below, represens he isance from home an represens ime. Graph A Graph B Graph C Graph D Graph E Graph F Rea he escripions of he journeys below an fin he graph ha bes maches each journey. a b c e f Ivan heae off o school from home, bu half-way here he realise he ha forgoen his lunch. He ran home, picke up his lunch an hen walke quickly o school. Toby walke up he roa from home hen ran ill he reache he highway. He urne aroun an ran owars home, bu walke he las secion. John lef his uncle s place an roe his bike sraigh home. Ali walke o he local café an ha morning ea wih a frien. She hen caugh a axi o he nex suburb an wen shopping. Finally she caugh a axi ha roppe her off ousie her gae. Amana ran from her house o her granmoher s place 2 km way. Georgie jogge away from home for 2 km, walke for he nex km, urne an heae home. Going home she walke for 1 km hen jogge he res of he way. Par 2 Disance/ime graphs 13

16 5 Simon rives up he hill a 40 km/h an own he oher sie a 60 km/h. Which of hese graphs bes escribes his journey? Disance from home Disance from home Disance from home Time Time Time Graph A Graph B Graph C 6 (Harer) The graph below shows Sue s journey where represens her isance from home. Which saemen below bes escribes he journey? Her isance is increasing an her spee is increasing. Her isance is ecreasing an her spee is increasing. Her isance is increasing an her spee is ecreasing. Her isance is ecreasing an her spee is ecreasing. Her isance is increasing an her spee is consan. Her isance is ecreasing an her spee is consan. 14 PAS5.2.5 Graphs of physical phenomena

17 Check your response by going o he suggese answers secion. The spee is given by he graien of he line: he seeper he line, he greaer he spee. Also, a posiive graien means he objec is heaing away from home an a negaive graien shows he objec heaing owars home. Go o he exercises secion an complee Exercise 2.1 Spee, isance an ime. Par 2 Disance/ime graphs 15

18 16 PAS5.2.5 Graphs of physical phenomena

19 Drawing isance ime graphs In his secion you will be aske o creae your own isance/ime graphs from escripions of journeys. When creaing a graph, i is imporan o remember ha he graien inicaes he spee an irecion of he journey. Aciviy Drawing isance/ime graphs Try hese. 1 Draw a rough skech of a isance/ime graph o represen each of hese cases. a Riing a bike away from home. b Riing a bike away from home, sopping because of a fla yre an hen walking home. Par 2 Disance/ime graphs 17

20 2 Draw a graph showing a journey where he isance from home is increasing bu he spee is firs fas an hen slower. Check your response by going o he suggese answers secion. Graphs can be use o compare wo or more ses of aa. For he following aciviy, you will nee o raw more han one line on each graph so ha you can show he ifference beween wo journeys. Aciviy Drawing isance/ime graphs Try hese. Draw wo journeys on each rough skech represening wo cycliss boh ravelling on he same pah. Use he gae a he en of he pah as he zero marker. Label he graph o show which cyclis is represene by each line. 3 a Boh cycliss ravelling a he same spee away from he gae, wih cyclis A saring before cyclis B. 18 PAS5.2.5 Graphs of physical phenomena

21 b Cycliss A an B ravelling in opposie irecion a he same spee hen passing one anoher. Cyclis A sars from he gae. c A race from he gae where cyclis A wins. Cyclis B sars from he gae an a sars from he oher en of he pah a he same ime. When hey mee, hey sop for a brief cha, hen each urns aroun an goes back o where hey sare. Check your response by going o he suggese answers secion. Par 2 Disance/ime graphs 19

22 Creaing a picure from a sory requires you o unersan he meaning of graphs an, in paricular, wha he slope an irecion of he graph is elling you. Graphs are a powerful way o convey informaion an inicae rens, bu are of no use o you unless you know how o rea hem. Go o he exercises secion an complee Exercise 2.2 Drawing isance/ime graphs. 20 PAS5.2.5 Graphs of physical phenomena

23 Suggese answers Par 2 Check your responses o he preliminary quiz an aciviies agains hese suggese answers. Your answers shoul be similar. If your answers are very ifferen or if you o no unersan an answer, conac your eacher. Aciviy Preliminary quiz 1 Your line shoul go own he graph from lef o righ. 2 a A an D (Increasing implies ha he graien is posiive.) b c C an E (Slowing own means ha he spee is ecreasing.) F only (Soppe means ha your spee is zero.) 3 a Spee = isance ime = = 75 km/h b You mus change 8 cm o 80 mm. Spee = isance ime = = mm/s Par 2 Disance/ime graphs 21

24 Aciviy Spee, isance an ime 1 a Afer 1 hour she is 30 km from home, an afer 2 hours she is also 30 km from home. There are many explanaions as o shy Lauren is in he same place a boh imes. For example, Lauren may have soppe for lunch for an hour. b Lauren sare from home, ravelle 30 km in one hour, soppe for 1 hour, ravelle 10 km furher away from home in he nex hour hen urne aroun an ravelle 30 km back owars home in he nex 2 hours. c Toal isance ravelle = = 70 km 2 a Lauren goes nowhere, so her average spee is zero. The calculaion below shows his using he formula. Average spee = isance ime = 0 1 = 0 km/h This shows ha a horizonal line in a isance/ime graph implies a zero spee. Horizonal lies have a graien of zero. b c She ravelle 30 km in wo hours so her average spee is 15 km/h. oal isance Average spee = ime = 70 5 = 14 km/h I is imporan o noe ha Lauren may no have acually ravelle a his spee uring he journey. This is her average spee over he five hours. 22 PAS5.2.5 Graphs of physical phenomena

25 3 Graph C is impossible since a any ime he person seems o be in wo places. Disance from home The person is a boh hese places a he same ime. Time Graph C 4 a Graph C b Graph D c Graph A Graph F e Graph B f Graph E 5 Graph B is correc because he car is always ravelling away from home, an he slope shoul be seeper for he secon secion because he spee is shown by he graien. 6 Her isance is ecreasing an her spee is consan. (The las saemen.) Her spee is consan because he graien is he same for whole lengh of a single sraigh line. Her isance from home is ecreasing because he graien is negaive (she is heaing owars home). Aciviy Drawing isance/ime graphs 1 Your graphs shoul be similar o hese. No scales are require. a In his graph, i is imporan o check ha your line goes up from lef o righ. The escripion i no sae ha he rier acually sare a home so eiher of hese graphs woul be correc. Par 2 Disance/ime graphs 23

26 b There are hree pars o he journey: fas going away from home, soppe hen a slower secion going home. Two possible answers are shown below. 2 isance increasing means a posiive slopes. firs fas hen slow means wo ifferen graiens: seep firs hen less seep. Two correc examples are shown below. 3 Your graphs shoul look similar o hese. The imporan aspecs o check are iscusse in each example. a a same spee means he slopes of he wo lines shoul be he same. away from gae means posiive slope for boh lines. A saing before B means ha B oes no move immeiaely so he line B s line is iniially horizonal a = 0. A B 24 PAS5.2.5 Graphs of physical phenomena

27 b opposie irecions means one line going up an he oher going own. same spee means he slopes have he same seepness. passing one anoher means he lines mus cross. Two possible answers are: B A B A c from gae means saring a he origin (he corner of he axes). A wins means he line for A will be seeper because A s average spee will be faser. A B opposie en of he pah means ha he line for A sars high up he isance axis. hey mee an sop means he lines mee an are hen horizonal for a shor perio. go back o where hey sare means hey change irecion so heir slopes change irecion. B ens up a isance = 0 an A ens up high up he graph. The graph below shows one possible answer. B A A B A B Par 2 Disance/ime graphs 25

28 26 PAS5.2.5 Graphs of physical phenomena

29 Exercises Par 2 Exercises 2.1 o 2.2 Name Teacher Exercise 2.1 Spee, isance an ime 1 The graph below represens John s isance from home one morning. Disance is measure in meres an ime is measure in minues. 600 A B C D 8 a Calculae he average spee for each secion of he graph. Par 2 Disance/ime graphs 27

30 b Calculae he oal isance ravelle an he average spee for he whole journey. c Secion A of he graph shows John walking owars his home. Wrie a brief saemen ha escribes wha may be happening in he oher secions of he graph. 2 Which of hese graphs shows Toby walking home from school a a consan spee? Disance from home Disance from home Disance from home Time Time Time Graph A Graph B Graph C 28 PAS5.2.5 Graphs of physical phenomena

31 3 Mach each graph below o one of he escripions. In each graph, represens he isance from home an represens ime. Disance from home Disance from home Disance from home Time Time Time Graph A Graph B Graph C a b c Leave home, walk a a seay pace own he riveway an along he roa o he shop. Leave home, run, walk hen sop. Leave he shop an walk home. 4 (Harer) Georgie is riing her bike backwars an forwars along her sree. Her journey is graphe wih ime measure in minues an isance from her home measure in meres. The graien of a secion of he isance/ime graph is foun o be 8. Wha oes his mean in erms of he journey? Par 2 Disance/ime graphs 29

32 Exercise 2.2 Drawing isance/ime graphs 1 Draw a line graph epicing his informaion. A man jogs o he corner sore, buys a newspaper an hen walks home. 2 Draw a isance/ime graph where he isance is ecreasing an he spee is consan. 3 Ali an Toby are cycling owars one anoher along a bike rack. Before hey mee hey boh change irecion. Ali coninues back o her saring poin bu Toby urns again an ries o where Ali is. Draw a graph showing boh journeys. Label he graph o show which journey belongs o each person. 30 PAS5.2.5 Graphs of physical phenomena