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Paper Reference

6 6 8 0 0 1

*P41819A0128*This publication may be reproduced only in accordance with

Pearson Education Ltd copyright policy.2013 Pearson Education Ltd.

Printers Log. No.

P41819AW850/R6680/57570 5/5/5/5/6/

Instructions to Candidates

In the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.Whenever a numerical value ofgis required, takeg= 9.8 m s2.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates

A booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 7 questions in this question paper. The total mark for this paper is 75.There are 28 pages in this question paper. Any blank pages are indicated.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner.Answers without working may not gain full credit.

Paper Reference(s)

6680/01Edexcel GCEMechanics M4

Advanced/Advanced Subsidiary

Tuesday 18 June 2013 Morning

Time: 1 hour 30 minutes

Materials required for examination Items included with question papers

Mathematical Formulae (Pink) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentation/integration, or have retrievablemathematical formulae stored in them.

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*P41819A0228*

1. A particlePof mass 0.5 kg falls vertically from rest. After tseconds it has speed vms1.

A resisting force of magnitude 1.5vnewtons acts onPas it falls.

(a) Show that 3v= 9.8(1 e3t).

(8)

(b) Find the distance thatPfalls in the first two seconds of its motion.(5)

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Q1

(Total 13 marks)

Question 1 continued

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2.

Figure 1

A river is 50 m wide and flows between two straight parallel banks. The river flows with

a uniform speed of2

3ms1parallel to the banks. The pointsAandBare on opposite banks

of the river andABis perpendicular to both banks of the river, as shown in Figure 1.

Keith and Ian decide to swim across the river. The speed relative to the water of both swimmers is

10

9ms1.

Keith sets out fromAand crosses the river in the least possible time, reaching the opposite

bank at the point C. Find

(a) the time taken by Keith to reach C,

(2)

(b) the distanceBC.

(2)

Ian sets out fromAand swims in a straight line so as to land on the opposite bank atB.

(c) Find the time taken by Ian to reachB.

(4)

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50 mm s1

2

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A

B C

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Q2

(Total 8 marks)

Question 2 continued

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3.

Figure 2

Two smooth uniform spheres A and B, of equal radius r, have masses 3m and 2m

respectively. The spheres are moving on a smooth horizontal plane when they collide.

Immediately before the collision they are moving with speeds uand 2u respectively. The

centres of the spheres are moving towards each other along parallel paths at a distance 1.6r

apart, as shown in Figure 2.

The coefficient of restitution between the two spheres is1

6.

Find, in terms of mand u, the magnitude of the impulse received byBin the collision.

(10)

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1.6r

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A(3m)

B(2m)

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Q3

(Total 10 marks)

Question 3 continued

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4.

Figure 3

A small smooth pegPis fixed at a distance dfrom a fixed smooth vertical wire. A particleof mass 3mis attached to one end of a light inextensible string which passes over P. The

particle hangs vertically belowP. The other end of the string is attached to a small ringR

of mass m, which is threaded on the wire, as shown in Figure 3.

(a) Show that whenRis at a distancexbelow the level ofPthe potential energy of the

system is

3mg (x d2 2+ ) mgx+ constant

(4)

(b) Hence findx, in terms of d, when the system is in equilibrium.(3)

(c) Determine the stability of the position of equilibrium.

(3)

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P

(3m)

d

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Q4

(Total 10 marks)

Question 4 continued

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*P41819A01628*

5. A coastguard ship C is due south of a ship S. Ship S is moving at a constant speed of

12 km h1 on a bearing of 140. Ship C moves in a straight line with constant speedVkm h1in order to intercept S.

(a) Find, giving your answer to 3 significant figures, the minimum possible value for V.

(3)

It is now given that V= 14

(b) Find the bearing of the course that Ctakes to intercept S.

(5)

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Q5

(Total 8 marks)

Question 5 continued

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6. A particlePof mass m kg is attached to the end Aof a light elastic string AB, of natural

length ametres and modulus of elasticity 9ma newtons. Initially the particle and the string

lie at rest on a smooth horizontal plane withAB= ametres. At time t = 0 the endBof the

string is set in motion and moves at a constant speed U ms1in the directionAB. The air

resistance acting onPhas magnitude 6mv newtons, where vm s1is the speed of P. At

time tseconds, the extension of the string isxmetres and the displacement ofPfrom its

initial position isy metres.

Show that, while the string is taut,

(a) x+y= Ut

(2)

(b)d

d

d

d

2

2 6 9 6

x

t

x

tx U+ + =

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You are given that the general solution of the differential equation in (b) is

x= (A +Bt)Ue3t+2

3

U

whereAandBare arbitrary constants.

(c) Find the value ofAand the value ofB.

(5)

(d) Find the speed ofPat time tseconds.

(2)

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Q6

(Total 14 marks)

Question 6 continued

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*P41819A02428*

7. [In this question iandjare perpendicular unit vectors in a horizontal plane]

A small smooth ball of mass mkg is moving on a smooth horizontal plane and strikes a

fixed smooth vertical wall. The plane and the wall intersect in a straight line which is

parallel to the vector 2i+ j. The velocity of the ball immediately before the impact is

bims1, where b is positive. The velocity of the ball immediately after the impact is

a(i+j)ms1, where ais positive.

(a) Show that the impulse received by the ball when it strikes the wall is parallel to

( i+ 2j).

(1)

Find

(b) the coefficient of restitution between the ball and the wall,

(8)

(c) the fraction of the kinetic energy of the ball that is lost due to the impact.

(3)

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Question 7 continued

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Question 7 continued

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Question 7 continued

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Q7

(Total 12 marks)

Question 7 continued

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