KEEP IT SIMPLE SCIENCE OnScreen Format. Physics Module 1 Kinematics. Usage & copying is permitted only according to the following

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1 keep i simple science KEEP IT IMPLE CIENCE Oncreen Forma Physics Year 11 Module 1 Kinemaics Usage & copying is permied only according o he following ie Licence Condiions for chools A school (or oher recognised educaional insiuion) may sore he disk conens in muliple compuers (or oher daa rerieval sysems) o faciliae he following usages of he disk conens: 1. chool saff may prin and/or phoocopy unlimied copies a one school and campus only, for use by sudens enrolled a ha school and campus only, for non-profi, educaional use only. 2. chool saff may display he disk conens via compuer neworks, or using projecors or oher display devices, a one school and campus only, for viewing by sudens enrolled a ha school and campus only, for non-profi, educaional use only. 3. chool saff may allow sudens enrolled a ha school and campus only o obain copies of he disk files and sore hem in each suden s personal compuer for non-profi, educaional use only. IN THI CAE, THE CHOOL HOULD MAKE PARTICIPATING TUDENT AWARE OF THEE ITE LICENCE CONDITION AND ADVIE THEM THAT FURTHER COPYING OR DITRIBUTION OF KI FILE BY TUDENT MAY CONTITUTE AN ILLEGAL ACT. 4. The KI logo and copyrigh declaraion mus be included in every usage of KI Resources. 5. NO ime limi applies o he use of KI Resources when used in compliance wih hese condiions. Please Respec Our Righs Under Copyrigh Law lide 1

2 keep i simple science Topic Ouline Wha is his opic abou? To keep i as simple as possible, (K.I... Principle) his opic covers: 1. PEED & VELOCITY Review of disance, ime, speed relaionship & ravel graphs. calars & Vecors. Displacemen & velociy vecors. Displacemen-ime graphs. Velociy-ime graphs. 2. ACCELERATION Concep of acceleraion. Moion graphs of acceleraion. Measuring moion in he laboraory. 3. EQUATION of MOTION Derivaion of he Equaions of Moion by graphical analysis. Using he equaions o analyse & predic moion. 4. VECTOR ANALYI Represening moion wih vecor diagrams in 1-D and in 2-dimensions. Combining vecors o find a resulan. Relaive displacemen & velociy in 1-D & 2-D. Resolving a vecor ino componens. 4. Vecor Analysis Vecors & calars Average & insananeous speed & velociy chool Inspecion only. Copying NOT permied. Kinemaics 1. peed & Velociy Analysing moion using vecor diagrams Vecor analysis in 1-D & 2-D Relaive displacemen & velociy Resolving vecors ino componens lide 2 peed & disance revision Graphing Moion Displacemen-ime graphs Velociy-ime graphs 2. Acceleraion Acceleraion concep Acceleraion on moion graphs Measuring moion... prac. work 3. Equaions of Moion Deriving he equaions of moion Analysing moion using equaions Inspecion Copy for school evaluaion only. Copying NOT permied.

3 keep i simple science Kinemaics Kinemaics is he branch of Physics concerned wih measuring & analysing moion. In his opic you will sudy hings like velociy & acceleraion, bu wihou considering he causes of he moion... ha will be covered in a laer opic. A knowledge of Kinemaics allows you o analyse & predic he moion of an objec. 1. peed & Velociy Average peed for a Journey If you ravelled by car a disance of 300 km in exacly 4 hours, hen your average speed was: average speed = disance ravelled = 300 = 75 km/hr (km.hr -1 ) ime aken 4 chool Inspecion only. Copying NOT permied. However, his does no mean ha you acually ravelled a a speed of 75 km/hr he whole way. You probably wen faser a imes, slower a oher imes, and may have sopped for a res a some poin. Disance-Time Graphs Perhaps your journey was similar o his graph. ar a he boom-lef of he graph and consider each secion A, B, C and D. o alhough he average speed for he enire journey was 75km/hr, in fac you never acually moved a ha speed. This raises he idea of INTANTANEOU PEED: he speed a a paricular insan of ime. The speedomeer in your car gives you a momen-by-momen reading of your curren speed... his is your insananeous speed. On he graph, he GRADIENT a any given poin is equal o INTANTANEOU PEED. DITANCE TRAVELLED (km) A Disance-Time Graph gradien = disance ime = speed B C gradien = zero i.e. sopped D TIME (hours) Graph secion D Travelled 150km in 1.5 hr: Av.peed = 100 km/hr Graph secion C Travelled 50 km in 1.0 hr: Av.peed = 50 km/hr Graph secion B Zero disance moved in 0.5hr: Av.peed = zero. Graph secion A Travelled 100 km in 1.0 hour: Av.peed = 100 km/hr DITANCE-TIME GRAPH show he DITANCE (from he saring poin) a each TIME. The GRADIENT a any poin equals INTANTANEOU PEED. A horizonal secion means ha he objec was no moving. lide 3 Inspecion Copy for school evaluaion only. Copying NOT permied.

4 keep i simple science peed-time Graphs The same journey (from previous slide) could also be represened by a differen graph, showing he PEED a differen imes: udy his graph carefully and compare i wih he oher. You mus no confuse he 2 ypes of graph and how o inerpre hem. This graph is very unrealisic in one way. I shows he speed changing INTANTLY from (say) 100 km/hr o zero (sopped), wihou any ime o slow down. I also shows he car ravelling a exacly 100 km/hr for an hour a a ime... very unlikely wih hills, raffic ec. Changes of speed (ACCELERATION) will be deal wih in he nex secion. For now we re Keeping I imple! chool Inspecion only. Copying NOT permied. erious peed Airforce je a he speed of sound (abou 1,200km/hr). The sonic boom shock wave is causing condensaion of waer vapour o form a visible mis. PEED (km/hr) This graph represens he same journey shown by he previous Disance-Time graph. Graph secions A, B, C & D correspond exacly. A C D TIME (hr) PEED-TIME GRAPH show he PEED of a moving objec a each TIME. The speed a any ime can be read from he verical scale. A horizonal secion means ha he objec was moving a consan speed. The area under he graph is equal o he disance moved in ha ime. B These Fla pars DO NOT mean sopped, bu mean consan speed. opped. peed scale reads zero. Area Under he Graph If you calculae he area of he recangles under each secion: ecion A area = 100 x 1 = 100 ecion B area 0 ecion C area = 50 x 1 = 50 ecion D area = 100 x 1.5 = 150 Toal = 300 = disance moved, in km. This is always rue for a PEED-TIME graph. lide 4 Inspecion Copy for school evaluaion only. Copying NOT permied.

5 A calar quaniy is somehing ha has a size (magniude) bu no paricular direcion. A Vecor quaniy has boh size (magniude) AND DIRECTION. o far we have deal wih only disances & speeds... hese are calar quaniies, since hey do no have any special direcion associaed. Now you mus learn he vecor equivalens: Displacemen = disance in a given direcion, and Velociy = speed in a given direcion. Consider his journey: TART drove 60 km EAT in 1 hour hen keep i simple science drove 30 km WET in 0.5 hour. As a CALAR journey: ravelled a oal 90 km disance in 1.5 hours, average speed = 90/1.5 = 60 km/hr calars & Vecors BUT, consider he NET journey : a he end of he journey you end up 30 km EAT of he saring poin. o, your final displacemen is 30 km eas. The VECTOR journey was: ravelled 30 km eas displacemen in 1.5 hours. average velociy = 30/1.5 = 20 km/hr eas. Noice ha boh displacemen and velociy have a direcion ( eas ) specified... hey are VECTOR! To make beer sense (mahemaically) of he journey, he direcions eas & wes could have (+) or ( - ) signs aached. Le eas be (+) and wes be ( - ). Then he oal journey displacemen was (+60) + (-30) = +30 km. Average = Displacemen Velociy ime V av = Δ Δ Δ means he change in displacemen value during ha ime. chool Inspecion only. Copying NOT permied. Noe: The symbol is used for Displacemen The Greek leer dela (Δ) is ofen used o indicae a change in some quaniy. Δ means he change in ime value, or he amoun of ime ha has elapsed. The small arrow above a symbol indicaes ha his is a vecor quaniy. In his case, velociy & displacemen are vecors & mus have a direcion specified. Time is a scalar quaniy. lide 5 Inspecion Copy for school evaluaion only. Copying NOT permied.

6 Displacemen - Time Refer o he previous Disance-Time graph. Wha if he 300km journey had been 150 km norh (secions A, B, C) hen 150 km souh (secion D)? The Displacemen - Time Graph would be: chool Inspecion only. Copying NOT permied. Displacemen NORTH (km) keep i simple science Gradien posiive A B C Gradien negaive TIME (hours) In vecor erms displacemen norh is posiive (+) displacemen souh is negaive ( - ) D MORE GRAPH... Down-sloping line means ravelling OUTH Back a saring poin. (Displacemen = 0 )...and he corresponding Velociy - Time Graph Velociy (km/hr) souh norh A B The velociy values for each par of his graph are equal o he gradiens of he corresponding pars of he Displacemen - Time Graph. C TIME (hrs) Zero velociy means sopped Posiive values mean norh-bound velociy Negaive value: souh-bound velociy Noe: ince he journey ends back a he saring poin, oal displacemen = zero & average velociy = zero for he whole rip. D In secion D, displacemen = -150 km (souh) Now ry velociy = displacemen Workshees ime 1,2,3 = -150 /1.5 = -100 km/hr (i.e. 100km/hr souhward) (However, his simply poins ou how lile informaion an average someimes gives you. The insan-by-insan Physics of he journey is in he graph deails.) If you calculae he area under he graph (beween he graph & he ime axis. ecion D is negaive) you will find ha he oal is zero, equal o he oal displacemen. lide 6 Inspecion Copy for school evaluaion only. Copying NOT permied.

7 keep i simple science Discusssion / Aciviy 1 The following aciviy migh be for class discussion, or here may be paper copies for you o complee. If sudying independenly, please use hese quesions o check your comprehension before moving on. Moion Graphs uden Name In Graph 1, which graph secion(s) represens: a) speed zero? b) he fases speed? c) moving a he slowes speed? Graph 2 Disance Graph 1 A B C D E 2. kech he peed-time graph which would correspond o Graph 1. Label A, B, C, ec. peed chool Inspecion only. Copying NOT permied. Time 3. Graph 1 can be vecorised wih he following informaion: ecions A, C & E all covered he same disance. A & C were norhwards (+ve), par E was souhwards (-ve). kech he shapes for he corresponding Displacemen-Time and Velociy-Time graphs. Label A, B, C, ec. Time Graph 3 Graph 4 Displacemen Velociy Time Time lide 7 Inspecion Copy for school evaluaion only. Copying NOT permied.

8 keep i simple science Change of Velociy = Acceleraion Any change in velociy is an acceleraion. Mahemaically, acceleraion = velociy change = final vel. - iniial velociy ime aken ime aken a = Δv Δ or a = v - u Δ (Greek leer dela ) refers o a change in a quaniy Unis If velociies are in ms -1, and ime in seconds, hen acceleraion is measured in meres/sec/sec (ms -2 ). Explanaion Imagine a car ha acceleraes a 1 ms -2 : 2. Acceleraion ar 1 sec. laer 1 sec.laer 1sec.laer v =0 v = 1 ms -1 v=2 ms -1 v=3ms -1 Every second, is velociy increases by 1 ms -1. Therefore, he rae a which velociy is changing is 1 ms -1 each second, or simply 1 ms -2. Acceleraion is a vecor, so direcion couns. Deceleraion (or negaive acceleraion) simply means ha he direcion of acceleraion is opposie o he curren moion... he objec will slow down raher han speed up. negaive ACCELERATION v = final velociy This is ofen wrien as VECTOR u = iniial velociy = ime involved v = u + a. Example Problem 1 chool Inspecion only. Copying NOT permied. posiive VELOCITY VECTOR THI CAR I LOWING DOWN... DECELERATING A moorcycle ravelling a 10.0 ms -1, acceleraed for 5.00s o a final velociy of 30.0 ms -1. Wha was is acceleraion rae? oluion: a = v - u = ( )/5.00 = 20.0/5.00 = 4.00 ms -2. Example Problem 2 A car moving a 25.0 ms -1 applied is brakes producing an acceleraion of ms -2 (i.e. deceleraion) lasing for 12.0 s. Wha was is final velociy? oluion: a = v - u, so v = u + a = (-1.50) x 12.0 = = 7.00 ms -1. (sill moving forward, bu slower) lide 8 Inspecion Copy for school evaluaion only. Copying NOT permied.

9 keep i simple science Graphs of Acceleraing Vehicles You may have done laboraory work o sudy he moion of an acceleraing rolley. If you used a Ticker-imer, he paper ape records would look somehing like hese: Tape of rolley moving a consan velociy (for comparison) Tape of rolley acceleraing... dos ge furher apar The graphs ha resul from acceleraion are as follows: THEE 2 GRAPH CORREPOND TO THE AME MOTION chool Inspecion only. Copying NOT permied. Trolley deceleraing (negaive acceleraion)... dos ge closer Remember, Gradien equals Velociy Displacemen DIPLACEMENT-TIME GRAPH Gradiens decreasing (curve flaens ou) Acceleraing Consan Velociy Deceleraing Gradien consan (sraigh line) Gradiens increasing (curve ges seeper) Now ry Workshee 4 +ve Velociy 0 VELOCITY-TIME GRAPH Acceleraing Consan Velociy Velociy increasing Time Velociy decreasing A common error is o hink ha his means he objec is moving backwards. Wrong! I is moving forward, bu slowing down. Deceleraing (If i were o move backwards is velociy would become negaive.) Velociy = 0 opped! Time -ve Gradien posiive Gradien negaive On a Velociy-Time Graph, Gradien = Acceleraion lide 9 Inspecion Copy for school evaluaion only. Copying NOT permied.

10 keep i simple science Discusssion / Aciviy 2 The following aciviy migh be for class discussion, or here may be paper copies for you o complee. If sudying independenly, please use hese quesions o check your comprehension before moving on. Velociy & Acceleraion uden Name Ouline he difference beween: a) a scalar measuremen and a vecor measuremen. b) disance and displacemen. chool Inspecion only. Copying NOT permied. c) speed and velociy. d) average velociy and insananeous velociy. 2. The acceleraion of a car was described as being 5 km/hr per sec. Explain wha his means in erms of changing velociy. 3. For each graph par A, B, C, ec, idenify he moion as eiher sopped, consan velociy, +ve acceleraion or -ve accleraion (deceleraion). (These graphs do NOT correspond o each oher... differen moions) A = Displacemen-Time Graph Velociy-Time Graph B = F C = B E D = A G C E = K Time F = D H J G = H = Time J = K = Displacemen Velociy lide 10 Inspecion Copy for school evaluaion only. Copying NOT permied.

11 keep i simple science chool Inspecion only. Copying NOT permied. Prac Work: Measuring Moion You will probably experience one or more of hese common ways o measure moion. You migh do some measuremens as suggesed by his diagram Time o ravel from A o B measured by sopwach Tape Measure & opwach The simples mehod of all: measure he disance or displacemen involved, and he ime aken. Then use speed (velociy) = disance (displacemen) Typical Resuls ime Disance Time Velociy (m) (s) (ms -1 ) Car Bicycle Landmark A Disance beween landmarks measured wih spors ape Landmark B However, his can only give you he AVERAGE speed or velociy. In Physics we ofen need o consider INTANTANEOU velociy. The Ticker-Timer Every ime he hammer his he moving srip of paper i leaves a do. The sring of dos can be analysed o sudy he moion of he rolley. Moving lab. rolley drags a srip of paper behind i Ticker-imer device has a small hammer which vibraes up and down every 0.02 sec. Alhough his mehod is very ou-daed, i is sill commonly used as a way for sudens o learn how o measure insananeous velociy. A moving objec drags a paper srip on which dos ge prined (usually every 0.02 second) as i goes. The gap beween dos is a record of displacemen and ime. This allows you o calculae he velociy over every 0.02 s. I s sill an average, bu over such small ime inervals i approximaes he insananeous velociy. Elecronic or Compuer Timing Moving rolley equipped wih a sonar reflecor. (An aluminium pie dish will do) onar ransponder gives ou pulses of ulra-sound and picks up any reurning echoes To compuer for analysis You may use devices ha use eiher Ligh Gaes or ONAR o record displacemens and imes for you. Once again, any velociies calculaed are averages, bu he ime inervals are so shor (e.g. as small as s) ha he velociy calculaed is essenially insananeous. lide 11 Inspecion Copy for school evaluaion only. Copying NOT permied.

12 keep i simple science The syllabus requires you o carry ou a variey of pracical aciviies o measure & analyse he moion of objecs in various forms of moion, such as consan velociy, uniform acceleraion & uniform deceleraion. In all cases, he moion should be in a sraigh line. ome uggesions Consan Velociy Moion A laboraory reacion rolley acceleraes up o is maximum velociy when he spring is fired. This acceleraion occurs wihin he firs 20cm (approx) of ravel. On a smooh, level surface is velociy will hen be very close o consan, a leas over he nex mere (or so) of ravel. Even beer, is he moion of a glider on a horizonal air-rack. There is virually no fricion, so if he glider is given a lile push, is velociy will be consan unil i his he end of he rack. Consan Acceleraion / Deceleraion The diagram shows a very simple se-up: Glass rod or pulley wheel o reduce Lab. rolley acceleraes fricion ring Pracical Work (con.) chool Inspecion only. Copying NOT permied. Measuring he Moion If using a rolley, he old-fashioned icker-imer can be used, bu he analysis of he moion can be quie laborious. The paper ape wih dos on i is a record of boh disance (displacemen) and ime. A sonar device also works well wih a rolley and has he added benefi ha he compuer sofware does mos of he edious number-crunching for you. The icker-imer does no work wih an air-rack because he paper pulls on he glider & causes is moion o be unsable & non-uniform. Air-rack measuremens are bes done using ligh-gae devices and a daa-logger. Analysis of he Measuremens Compuer-assised echnologies can, of course, do ALL he analysis for you, bu ha way you migh learn very lile! Ideally, you wan a se of daa showing he displacemen AND he velociy of he objec a regular ime inervals. The oal elapsed ime migh only be a fracion of a second. Use his daa o consruc a Displacemen-Time graph and a Velociy-Time graph for each moion. Calculae gradiens & areas as suggesed below. Consan Velociy Uniform Acceleraion Uniform Deceleraion Bench op You can produce higher raes of acceleraion by adding more masses on he sring, or lower acceleraions by adding mass o he rolley. (eg add a house brick) Wih an air rack, adjus i so i is NOT level. The glider will accelerae down he slope, or if pushed up-hill i will decelerae uniformly. loed masses are pulled down by graviy. This causes consan acceleraion of he rolley. To decelerae a rolley you could fire i up a ramp. I is bes o il he enire work-bench or able o make a slope. V gradien = velociy gradien = acceleraion lide 12 V gradien a any poin is he gradien of a angen o he curve Area under he graph = displacemen Inspecion Copy for school evaluaion only. Copying NOT permied. V

13 keep i simple science Discusssion / Aciviy 3 The following aciviy migh be for class discussion, or here may be paper copies for you o complee. If sudying independenly, please use hese quesions o check your comprehension before moving on. Measuring Moion uden Name Ouline he advanage(s) and disadvanage(s) of measuring speed/velociy using: a) ape measure & sopwach. chool Inspecion only. Copying NOT permied. b) icker-imer. c) ligh gae or sonar devices wih daa-logger. 2. answer by wriing A,B,C, ec Which graph(s) a he righ: a) show consan velociy? A B C b) have a gradien equal o he acceleraion? c) allow you o deermine he displacemen by calculaion of he area under he graph? d) show an objec deceleraing? e) have gradiens equal o velociy? V V V D E F lide 13 Inspecion Copy for school evaluaion only. Copying NOT permied.

14 keep i simple science 3. Equaions of Moion chool Inspecion only. Copying NOT permied. Analysing moion using graphs is no he only way. There are also a number of mahemaical equaions (formulas) which allow you o calculae velociy, acceleraion, displacemen, ec. so you can solve problems & make predicions regarding he moion of any objec. These equaions work for moion in a sraigh line, wih consan acceleraion only. The Equaions of Moion can be derived by considering he graph of an acceleraing objec, as follows: Graphical Derivaion Make youself familiar wih he following graph. A ime = 0 he objec was moving wih an iniial velociy = u. I is acceleraing a a consan rae, so a ime is velociy has increased o v. Velociy v u gradien = acceleraion Area under he graph = displacemen v-u u v The area under he graph is equal o he displacemen of he objec during ime. This area is a rapezium as in his diagram. Area = (sum of parallel sides) x perp. dis. apar 2 so, displacemen = (v + u) 2 Noe ha he erm (v + u)/2 can be hough of as he average velociy of he moion. Equaion 2 is NOT one you migh use very ofen, bu i will be needed laer o derive an imporan one. Insead of using he rapezium area, we can also find he area under he graph as he sum of a recangle plus a riangle. u = (v + u). 2 Equaion 2 v v-u 0 Time By definiion, he acceleraion is he rae of change of velociy, so i is equal o he gradien of he graph: Equaion 1 gradien = acceleraion = v - u You have seen his equaion before. We shall call his Equaion 1. a = Δv Δ or a = v - u Recangle Area = u. Triangle Area = (v-u). 2 Area under graph = Displacemen so, = u. + (v-u). Tha s a bi messy... 2 From equaion 1, v-u = a. so, = u. + (a.). 2 = u + a 2 2 This one is very useful! = u. + 1 a. 2 2 Equaion 3 u lide 14 Inspecion Copy for school evaluaion only. Copying NOT permied.

15 keep i simple science o far we have: Equaion 1 a = Δv Δ or a = v - u Equaions of Moion (con.) Equaion 2 Equaion 3 = (v + u). 2 = u. + 1 a. 2 2 From Equaion 1, = v-u a If we subsiue ha ino equaion 2, = (v + u). = (v + u)(v - u) 2 2 a chool Inspecion only. Copying NOT permied. Example Problem 3 A bulle in he chamber of a gun is iniially a res. (u=0) When fired, i reaches a velociy of 150 ms -1 in 0.100s. Wha is he acceleraion rae? (use Eq.1) oluion: a = v - u = (150-0)/0.100 = 1,500 ms -2 Example Problem 4 The bulle is fired verically upwards (up = +ve) leaving he barrel a u=150ms -1. I rises verically unil (momenarily) i sops (v=0) before falling again. Acceleraion due o graviy -10ms -2 (down = -ve) a) How high will i go? (use Eq.4) b) How long will i ake o reach he apex? (use Eq.1) This is usually wrien as Equaion 4 v 2 = u a. = v 2 - u 2 2a Equaion 4 can be very useful in cases where you do NOT know he ime involved. Noe ha he variable is no in i. Armed wih hese equaions (especially 1, 3 & 4) you can now calculae & predic he oucomes of all sors of moion. udy he example problems a he righ, hen ge ready for los of pracice workshees from your eacher. oluion: a) v 2 = u 2 + 2a so = (v 2 - u 2 ) / 2a = ( ) / 2 x -10 = - 22,500/ -20 = 1,125 m b) a = v - u so = v - u = (0-150) / -10 Now ry a Workshees = 15 s 5 & 6 Example Problem 5 A car ravelling a 28 ms -1 applies brakes & deceleraes (-ve) a -4.0 ms -2 for 5.0 s. How far does i ravel in his ime? (Eq.3) oluion: = u + 1a 2 = 28 x x -4 x = = 90 m lide 15 Inspecion Copy for school evaluaion only. Copying NOT permied.

16 keep i simple science Vecors in 1-Dimension The idea is ha we can represen vecors by diagrams involving arrows. The arrow poins in he direcion of he vecor quaniy. (Remember ha vecors have a magniude AND a direcion) Les sar wih he simples vecor; displacemen. Imagine ha own Q is 30 km eas of own P. Anoher own, R is 50km eas of Q. We can skech a simple diagram for his: P 30km Q 50km R These arrows represen he displacemen vecors for hese owns. I is suddenly obvious ha (for example) P is 80 km wes of R. Tha is a simple analysis of hese vecors! Easy as! Velociy Vecors Now consider hese 2 cars ravelling EAT on he same sraigh road. 4. Vecor Analysis As well as graphing moion, and using he Equaions of Moion o calculae aspecs of moion, anoher useful echnique is o consider he moion in erms of is vecor quaniies. If we look only a moion in a sraigh line, his is relaively easy. I becomes a lile more challenging when we look a vecor analysis in a 2-dimenional plane. In 3-dimensions... forge i; we re no going here! A Diag. 1 B W N E Relaive Velociy Imagine you are in Car A in diag.1. I will be obvious ha you are caching up o car B, bu wha is he relaive velociy of car B as seen from car A? (Anoher way o hink abou his is: wha if you couldn ell ha you are moving? All you can see is ha car B is geing closer o you. If you measured car B s velociy somehow, wha would you find?) The way o solve his wih vecors is: This doesn really need a fancy formula: all you have o do is sar wih he observed vecor & subrac he observer s vecor. (The only ricky hing is ha o subrac a vecor, you mus ADD is opposie vecor.) o, V B = 90 km/hr EAT (V A ) = 100 km/hr WET) chool Inspecion only. Copying NOT permied. Relaive Velociy of B, seen from A V rel = V B - V A = V B + (- V A ) Noe ha an opposie vecor has he same magniude, bu opposie direcion. v = 100 km/hr v = 90 km/hr I is easy o see wha is happening above, or below, A Diag. 2 B The resulan vecor is a relaive velociy of 10 km/hr WET. Car A will see he rear of car B approaching hem (apparenly heading wes) a 10 km/hr. Viewed from car B, he relaive velociy of car A is 10 km/hr EAT. They see car A coming up behind hem, easward a 10 km/hr. v = 60 km/hr v = 80 km/hr In diag.2, he relaive velociy of B (seen from A) is 20 km/hr EAT. bu wha do you see if you are in one of hose cars? The relaive velociy of A (seen from B) is 20 km/hr WET. lide 16 Inspecion Copy for school evaluaion only. Copying NOT permied. Now ry Workshee 7

17 keep i simple science Vecors in 2-Dimensions When 2 or more vecors are all in one line, adding hem ogeher is simple arihmeic. Wha if hey ac in differen direcions? Here is he mehod o use, explained wih example problems. Displacemen Vecors An aircraf flies 200km eas, hen 100km souh. Where is i in relaion o is saring poin? φ 200km Resulan Displacemen 100km chool Inspecion only. Copying NOT permied. Velociy Vecors A ship is ravelling due eas a velociy 5.0ms -1. The ide is flowing from he souh a 1.8ms -1. Wha is he ship s acual velociy? Resulan Velociy φ R 2 = = 50,000 R = 50,000 = 224 km Tan φ = 100/200 = φ 27 o Final displacemen = 224 km, direcion 27 o of E (bearing from norh = 117 o ) R 2 = = R = = 5.3ms -1 Tan φ = 1.8/5.0 = 0.36 φ 20 o N of E (his angle is 70 o clockwise from norh, bearing = 70 o ) Acual Velociy = 5.3ms -1, on bearing 70 o Represen each vecor by an arrow, poined appropriaely. To add vecors, join hem head-o-ail. In his opic we will keep i simple by adding only 2 vecors, bu here can be any number. The resulan vecor (he sum of all he vecors) is an arrow which joins he begining of he firs vecor o he head (poin) of he las vecor. Undersand he Technique? Graphical cale Diagrams If he diagram is drawn accuraely o scale, he size of he resulan can be measured using he scale. Direcion angles can be measured by proracor. This echnique is covered in he workshees. Mahemaical (Algebraic) Mehods If he 2 saring vecors are a righ angles, (as above) he resulan forms a righ riangle. The magniude of he resulan can be found using Pyhagorus Rule. imple rigonomery finds he direcion. lide 17 Inspecion Copy for school evaluaion only. Copying NOT permied.

18 keep i simple science Relaive Displacemen Imagine 3 owns P, Q & R, locaed as shown: 15 km P Q Relaive Displacemen & Velociy in 2-Dimensions Remember ha o deermine relaive vecors you mus sar wih he observed vecor hen subrac he observer s vecor. (Also remember ha o subrac a vecor, you mus add he opposie vecor.) o, he following vecor diagram shows he relaive displacemen of P, as seen from R. The resulan vecor can be found by Pyhagorus, ec. P 20 km R 2 = = 625 R = 625 = 25 km P is 15 km due norh of Q. R is 20 km due eas of Q. Wha is he relaive displacemen of P as seen from R? 20 km o, own P is 25 km from R on bearing 307 o. The exac opposie vecor (25 km on bearing 127 o ) would be he relaive posiion of R as seen from P. R Resulan Tan φ = 20 / 15 = φ 53 o W of N (his angle is 307 o clockwise from norh, bearing = 307 o ) W N φ E 15 km R Relaive Velociy Two cars are approaching he same inersecion a righ angles, as shown. Wha are heir relaive velociies? The vecor diagram below shows he relaive velociy of K, as seen from J. V k = 70 φ Resulan -V J = 80 car K Tan φ = 80 / 70 = φ 49 o E of (his angle is 131 o clockwise from norh, bearing = 131 o ) 70 km/hr 80 km/hr chool Inspecion only. Copying NOT permied. o, he relaive velociy of K, seen from J is 106 kmhr -1 on a bearing of 131 o. In oher words, he people in car J see he car K approaching hem from (roughly) NW a over 100 km/hr. car J Meanwhile, passengers in car K would see J approaching hem a he same velociy from roughly E; i would seem o hem ha J was heading roughly NW owards hem. (The acual reciprocal bearing of he relaive velociy vecor is 311 o ) W N E R 2 = = 11,300 R = 11,300 = 106 km/hr Now ry Workshees 7 & 8 lide 18 Inspecion Copy for school evaluaion only. Copying NOT permied.

19 keep i simple science Resolving a Vecor All he previous examples have involved adding ogeher wo perpendicular vecors o find one resulan vecor. I can also be useful o do he opposie... urn one vecor ino 2 perpendicular componens. Firs, imagine a displacemen or velociy vecor which is represened by his diagram: vecor M θ The angle θ migh be above he horizonal, or i migh represen a map direcion beween norh & eas. To resolve his vecor, we hink of i as being he resulan of 2 vecors a righ angles o each oher. We simply build a righ-riangle around our vecor: vecor M θ componen M x chool Inspecion only. Copying NOT permied. These imaginary componens of M add ogeher o be exacly equivalen o M. In fac, here are an infinie se of possible componens which can add up o M; we chose hese 2 because hey are a righ angles o each oher. M x is horizonal (or eas) while M y is verical (or norh). The magniudes of each componen can be found from he righriangle as follows: Warning! If he angle is measured sinθ = M y / M so M y = M sinθ beween vecor and and y-axis, he link o cosθ = M x / M so M x = M cosθ he sine or cosine raios is reversed. Le s see how o use his... Always consider he diagram carefully! componen M y Example Problems 1. Raffa s house, as seen from Fred s house, is 350m away on bearing 300 o, as shown in he skech diagram. How much furher norh, and wes, is Raffa s house? Easy! Jus rea he arrow RF as a displacemen vecor and resolve i ino norherly & weserly componens: RF y Raffa s house is 175m furher norh, and 303m furher wes, han Fred s house. 2. A rocke was launched from a ramp so ha i rises a an angle of 70 o above he horizonal. While is moor is burning, is velociy hrough he air is 420ms -1. A wha rae is i gaining aliude? ie wha is is verical velociy? v =420ms o V x V y RF=350m RF x 30 o The rocke is gaining aliude a 395ms -1. R RF=350m Norherly componen = RF y RF y = 350.sin30 = 175m Weserly componen = RF x RF x = 350.cos30 = 303m Rae of gaining verical heigh = V y V y = V.sin70 = 395ms -1 (V x is is horizonal velociy... no required in his quesion.) 30 o W F N E Now ry Workshee 9 lide 19 Inspecion Copy for school evaluaion only. Copying NOT permied.

20 keep i simple science Discusssion / Aciviy 4 The following aciviy migh be for class discussion, or here may be paper copies for you o complee. If sudying independenly, please use hese quesions o check your comprehension before moving on. Vecor Analysis uden Name Ouline he general algebraic mehod for adding ogeher 2 perpendicular vecors o find a resulan. 2. a) Wha is he basic concep for finding a relaive displacemen or velociy. b) How can you subrac a vecor? chool Inspecion only. Copying NOT permied. 3. Wha is mean by resolving a vecor? 4. a) Ouline he graphical mehod for adding or resolving vecors. (see uorial workshee 10 if no sure) b) Wha is/are he advanage(s) and disadvanage(s) of he graphical mehod compared o an algebraic mehod? lide 20 Inspecion Copy for school evaluaion only. Copying NOT permied.

21 keep i simple science ome Final Vecor Analysis Problems We finish he explanaions for his opic wih 2 final examples. However, you need o pracice on los more! 1. A boa is seering due norh and crossing a river a a velociy of 12ms -1. There is a curren flowing wes a 5.0ms -1. Wha is he boa s acual velociy across he river? (Relaive o he river bank poin where i sared from.) V C = 5 W N curren E 2. A car ravelling norh a 110kmhr -1 is approaching a level crossing. There is a rain on he rack ravelling wes a 75kmhr -1. een from he rain, wha is he car s relaive velociy? (Remember, you mus add he car s vecor o he negaive of he observer s vecor) Car N rain chool Inspecion only. Copying NOT permied. Resulan φ V B = 12 R 2 = = 169 R = 169 = 13 ms -1 Tan φ = 5 / 12 = φ 23 o W of N (his angle is 337 o clockwise from norh, bearing = 337 o ) V C = 110kmhr -1 - (V T ) = 75kmhr -1 φ Resulan Tan φ = 75 / 110 = φ 34 o E of N ( bearing = 34 o ) R 2 = = 5, ,100 = 17,725 R = 17,725 = 133 kmhr -1 People on he rain see he car approaching from (roughly) he W, a a velociy of 133kmhr -1. Now ry Workshee 10 lide 21 Inspecion Copy for school evaluaion only. Copying NOT permied.