# * * MATHEMATICS (MEI) 4761 Mechanics 1 ADVANCED SUBSIDIARY GCE. Wednesday 21 January 2009 Afternoon. Duration: 1 hour 30 minutes.

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1 ADVANCED SUBSIDIARY GCE MATHEMATICS (MEI) 4761 Mechanics 1 Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination Formulae and Tables (MF2) Other Materials Required: None Wednesday 21 January 2009 Afternoon Duration: 1 hour 30 minutes * * * * INSTRUCTIONS TO CANDIDATES Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet. Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context. The acceleration due to gravity is denoted by g m s 2. Unless otherwise instructed, when a numerical value is needed, use g = 9.8. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used. The total number of marks for this paper is 72. This document consists of 8 pages. Any blank pages are indicated. OCR 2009 [M/102/2649] OCR is an exempt Charity RP 8F10 Turn over

2 2 Section A (36 marks) 1 A particle is travelling in a straight line. Its velocity v m s 1 at time t seconds is given by v = 6 + 4t for 0 t 5. (i) Write down the initial velocity of the particle and find the acceleration for 0 t 5. [2] (ii) Write down the velocity of the particle when t = 5. 5 seconds. Find the distance travelled in the first [3] For 5 t 15, the acceleration of the particle is 3 m s 2. (iii) Find the total distance travelled by the particle during the 15 seconds. [3] 2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. acceleration in m s 2 4 a time in seconds t 8 Fig. 2 At t = 0 the particle has a velocity of 6 m s 1 in the positive direction. (i) Find the velocity of the particle when t = 2. [2] (ii) At what time is the particle travelling in the negative direction with a speed of 6 m s 1? [2] 3 The resultant of the force ( 4 ) N and the force F gives an object of mass 6 kg an acceleration of 8 ( 2 3 ) m s 2. (i) Calculate F. [4] (ii) Calculate the angle between F and the vector ( 0 ). [2] 1

3 3 4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of 35 m s 1 at an angle of α to the horizontal, where cos α = You are given that, t seconds after being thrown, the stone is (9.8t 4.9t 2 ) m higher than O. When descending, the stone hits a plum which is m higher than O. Air resistance should be neglected. Calculate the horizontal distance of the plum from O. [6] 5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N. lift string parcel Fig. 5 (i) Find the upward acceleration. [3] (ii) Find the reaction on the man of the lift floor. [2] 6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height H m directly above A. B A H m Fig. 6 At the instant when B is released from rest, A is projected vertically upwards with a speed of 29.4 m s 1. Air resistance may be neglected. The stones collide T seconds after they begin to move. At this instant they have the same speed, V m s 1, and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that T = 1.5 and find the values of V and H. [7] Turn over

4 4 Section B (36 marks) 7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. 121 N 34 Fig. 7 Initially she pulls with a force of 121 N on the rope inclined at 34 to the horizontal, as shown in Fig. 7, but the sledge does not move. (i) Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N, correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. [7] The sledge is given a small push to set it moving at 0.5 m s 1. The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged. (ii) Describe the subsequent motion of the sledge. [2] The explorer now pulls the rope, still at an angle of 34 to the horizontal, so that the tension in it is 155 N. The frictional force is now 95 N. (iii) Calculate the acceleration of the sledge. [3] In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at 26 to the horizontal. The sledge starts from rest and reaches a speed of 5 m s 1 in 2 seconds. (iv) Calculate the frictional force between the slope and the sledge. [5]

5 5 8 A toy boat moves in a horizontal plane with position vector r = xi + yj, where i and j are the standard unit vectors east and north respectively. The origin of the position vectors is at O. The displacements x and y are in metres. First consider only the motion of the boat parallel to the x-axis. For this motion x = 8t 2t 2. The velocity of the boat in the x-direction is v x m s 1. (i) Find an expression in terms of t for v x and determine when the boat instantaneously has zero speed in the x-direction. [3] Now consider only the motion of the boat parallel to the y-axis. For this motion v y = (t 2)(3t 2), where v y m s 1 is the velocity of the boat in the y-direction at time t seconds. (ii) Given that y = 3 when t = 1, use integration to show that y = t 3 4t 2 + 4t + 2. [4] The position vector of the boat is given in terms of t by r = (8t 2t 2 )i + (t 3 4t 2 + 4t + 2)j. (iii) Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times. [4] (iv) Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times. [5] (v) Plot a graph of the path of the boat for 0 t 2. [3]

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