Jan Dangerfield Stuart Haring Series Editor: Julian Gilbey Cambridge International AS & A Level Mathematics: Mechanics Coursebook B CAMBRIDGECAMBRIDGE Univesiy Printing House, Cambridge CB2 BBS, United Kingdom (One Liery Plz, kh Floor, New York, NY 1006, USA 477 Willanstowa Road, Poet Melbourne, VIC 307, Astalia 314-22, Flor, Plot 3, Splendor Forum, asla District Centre, New Deli 10025, nd 7 Anson Rod, MOG-0406, Singapore 9906 Cambridge University Press part ofthe Universi of Cambri eGarthers the Univers’ mission by dsennating Knowledge in the pursuit of ‘lucatin, learning and research atthe bighest intertional eel hele, wawambridgeorg Taformaton on his tide: wwwcambrige org SHO8407267 (© Cambridge University Pros 2018 “his publication isin copyright. Sujet to sttstory exception and the provisions of relevant cllect casing aerment, ‘0 reproduction of ny pat may tak place without he wien povmboson of Cambridge Universty Pres Fis published 2018 20 9 817 1S WIS 9RTESAIIL Prot in Maia by Vina Printing “A catalogue record for ths publeation availabe fromthe British Library |SBN97-1-108-407267 Paperback Cambridge University Press has no esponsbiliy forthe petstenee or aceursey OT URLs for external or third party itrne websites refered on this pablo, “and doesnot guarantee that ty content on sack webster oF wl em, {ccuat or appropriate Information garding rcs, taal neta, and other Feta formation sven inthis work icorrat at he tine of st rnin Bt Cambdge University Press doesnot guarame the aecuray of seh information hereafter, 16CSE ia reiterd radeare Past exam paper question throughout ae reproduce by permission ‘of Cambridge Assesment International Eduestion, Cambridge Aseinent International Education bers no respons forthe eample anor to gestions ‘aken om spas question pupers which are contained nth pation, ‘The questions, example ansers, marks avarded andlor comment that appear in this book were written by tear) examination, he way mars woul be trdod wo anower lite thee may bedierent NOTICE To TEACHERS INTHE Uk [Relea to reprodice any part ofthis workin atrial fom ining Photocopying an electronic rage) excep under the following creumstanoes {i) where you ae abiding by scence granted to your choo or insitaion By he Copyright Lensing Agcy: (where no sch iene exists, o where you wiht exes the term of ens, nd you bave gained he writen permsion of Cambridge University Pres, Gi ete youre slloed to reprodace without permision inde the provisions ‘Chapter Sof the Copyright, Designs and Patents Act 1968, which coer, Or ‘Example, the repredution of sort passages wii certain types a educational ‘nthology and reproduction for the parpores of sting examination stnContents Series introduction vi How to use this book Acknowledgements x 1 Velocity and acceleration 1 LL. Displacement and velocity 2 12 Acceleration 9 1.3 Equations of constant acceleration " 2 Force and motion in one dimension 35 3 Forces in two dimensions 53 32 Resolving forces at other angles in equilibrium problems 59 33 The triangle of forces and Lami’ theorem for three-foree «equilibrium problems e 34 Non-equilibrium problems for objects on slopes and known directions of acceleration 66 3 Non-equilibrium problems and finding resultant forces and directions of acceleration n End-of-chapter review exercise 3 n Cross-topie review exercise 1 80rer cen 4 Friction 82 4,1 Friction as part of the contact force B 42 Limit of friction 90 43. Change of direction of friction in different stages of motion 95 44 Angle of friction 100 End-of-chapter review exercise 4 10s 5 Connected particles 108 51 Newton's third law 109, 5.2 Objects connected by rods 110 53. Objects connected by strings ig SA Objects in moving lifts (elevators) ray End-of-chapter review exercise 5 125 6 General motion in a straight line 128 6.1 Velocity as the derivative of displacement with respect to time 130 a 6.2 Acceleration as the derivative of velocity with respect to time 134 6.3 Displacement as the integral of velocity with respect to time 139 64 Velocity as the integral of acceleration with. respect to time 148 End-of-chapter review exercise 6 133 Cross-topic review exercise 2 7 Momentum 157 71 Momentum 159 12. Collisions and conservation of momentum 161 End-of-chapter review exercise 7 167 8 Work and energy 169 8.1 Work done by a force m 82 Kinetic energy 7 83 Gravitational potential energy 179 End-of-chapter review exercise 8 1839 ‘The work-energy principle and power 185 8.1 The work-energy principle 186 92 Conservation of energy in a system of conservative forces 193 9.3. Conservation of energy in a system with non-conservative forces 196 94 Power 201 End-of-chapter review exercise 9 206 Cross-topic review exercise 3 208 Practice exam-style paper 210 Answers 212 Glossary 232 Index 234Series introduction ‘Cambridge Intemational AS & A Level Mathematics can bea life-changing course. On the one hand, iis a fucitating subject: thete are many university courses that either require an A Level or equivalent quaication in mathematics or prefer applicants who have it. Oa the other hand, wil help you to lara t think more precisely and lozically, while also encouraging creativity, Doing mathematics canbe like doing at: just as an artist needs to ‘master her tools (ase of the paintbrush, for example) and understand theoretical ideas (perspective, colour wheels and so on), 50 does a mathematician (using tools such as algebra and callus, which you will lear abut inthis course), But thisis only the technical side: the joy in art comes through creativity, when the artist uses her tools, ‘oexpress ideas in novel ways. Mathematics i very similat: the tools are needed, but the deep joy inthe subject ‘comes through solving problems. ‘You might wonder what a mathematical ‘problems. This isa very good question, and many people have offered diferent answers. You might like to write down your own thoughis on this question, and reflect on how they change as you progress through this course. One possible idea is that a mathematical problem sa mathematical question that you do not immediatly know how to answer. (Ifyou do know how to answer it immediately, then ‘we might cali an ‘exercise’ instead) Such a problem will take time to answer: you may have to try diferent approaches, using diferent tools or ideas, on your own or with athes, unl you finally discover & way into i, This may take minutes, hours, days or weeks to achieve, and your sense of achievement may well gow with he effort it has taken, Im addition to the mathematical tools that you will learn inthis course, the problem-solving sills that sou ‘will develop will also help you throughout life, whatever you end up doing. Iti very conimon toe faced with problems, be iin scence, engineering, mathematies, accountancy, law or beyond, and having the confidence to systematically work your way through them willbe very useful ‘This series of Cambridge International AS & A Level Mathematics coursebooks, written forthe Cambridge Assessment International Education syllabus for examination from 2020, will support you both to lear the ‘mathematics required for these examinations and to develop your mathematical problem-solving skills. The new examinations may well include more unfamiliar questions than inthe past, and having these skills will allow you {o approach such questions with curiosity and confidence. In addition to problem solving, there are two other key concepts that Cambridge Assessment Internatinal dueation have introduced inthis ylabus: namely communication and mathematical modelling. These appear in various forms throughout the coursebooks Communication in speech, writing and drawing lies atthe heart of what itis tobe human, snd this sno less true in mathematies. While there is a temptation to tink of mathematics as only existing in a dry, written form in textbooks, nothing could be further from the truth: mathematical communication comes in many fom, and discussing mathematical ideas with colleagues isa major part of every mathematician’s working lf. As you study this course, you will work on many problems. Exploring them orstrugaling wit them together with a classmate will help you both to develop Your understanding and thinking, aswell as improving your (mathematical) ‘communication skills. And being ale to convince someone that your reasoning is correct, initially vrtally and thea in writing forms the heart ofthe mathematical skill proof,