Edexcel GCE Mechanics M1 Advanced/Advanced Subsidiary

Transcription

1 Centre No. Candidate No. Paper Reference(s) 6677/01 Edexcel GCE Mechanics M1 Advanced/Advanced Subsidiary Wednesday 16 May 2012 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Paper Reference Surname Signature Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. Initial(s) Examiner s use only Team Leader s use only Question Leave Number Blank 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 of g is required, take g = 9.8 m s 2. 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. This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy Pearson Education Ltd. Printer s Log. No. P40689A W850/R6677/ /5/3 *p40689a0128* Total Turn over

2 1. Two particles A and B, of mass 5m kg and 2m kg respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of A and B are 3 m s 1 and 4 m s 1 respectively. The direction of motion of A is unchanged by the collision. Immediately after the collision, the speed of A is 0.8 m s 1. Leave blank (a) Find the speed of B immediately after the collision. In the collision, the magnitude of the impulse exerted on A by B is 3.3 N s. (b) Find the value of m. 2 *P40689A0228*

3 2. A P G Q B 0.8 m 0.6 m 3 m Figure 1 A non-uniform rod AB has length 3 m and mass 4.5 kg. The rod rests in equilibrium, in a horizontal position, on two smooth supports at P and at Q, where AP = 0.8 m and QB = 0.6 m, as shown in Figure 1. The centre of mass of the rod is at G. Given that the magnitude of the reaction of the support at P on the rod is twice the magnitude of the reaction of the support at Q on the rod, find Leave blank (a) the magnitude of the reaction of the support at Q on the rod, (b) the distance AG. (4) 4 *P40689A0428*

4 3. Leave blank 20 N 30 Figure 2 A box of mass 5 kg lies on a rough plane inclined at 30 to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N, as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find (a) the magnitude of the normal reaction of the plane on the box, (4) (b) the coefficient of friction between the box and the plane. (5) 6 *P40689A0628*

5 4. A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed 20 m s 1 and is at the point A. The car maintains the speed of 20 m s 1 for 25 s. The car then moves with constant deceleration 0.4 m s 2, reducing its speed from 20 m s 1 to 8 m s 1. The car then moves with constant speed 8 m s 1 for 60 s. The car then moves with constant acceleration until it is moving with speed 20 m s 1 at the point B. Leave blank (a) Sketch a speed-time graph to represent the motion of the car from A to B. (b) Find the time for which the car is decelerating. (2) Given that the distance from A to B is 1960 m, (c) find the time taken for the car to move from A to B. (8) 10 *P40689A01028*

6 5. A particle P is projected vertically upwards from a point A with speed u m s 1. The point A is 17.5 m above horizontal ground. The particle P moves freely under gravity until it reaches the ground with speed 28 m s 1. Leave blank (a) Show that u = 21 At time t seconds after projection, P is 19 m above A. (b) Find the possible values of t. (5) The ground is soft and, after P reaches the ground, P sinks vertically downwards into the ground before coming to rest. The mass of P is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on P. (c) Find the vertical distance that P sinks into the ground before coming to rest. (4) 14 *P40689A01428*

7 6. [In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] Leave blank A ship S is moving with constant velocity ( 12i + 7.5j) km h 1. (a) Find the direction in which S is moving, giving your answer as a bearing. At time t hours after noon, the position vector of S is s km. When t = 0, s = 40i 6j. (b) Write down s in terms of t. (2) A fixed beacon B is at the point with position vector (7i j) km. (c) Find the distance of S from B when t = 3 (4) (d) Find the distance of S from B when S is due north of B. (4) 18 *P40689A01828*

8 7. P(0.3 kg) Q(0.5 kg) F(4 N) Leave blank Figure 3 Two particles P and Q, of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time t = 0, a constant force F of magnitude 4 N is applied to Q in the direction PQ, as shown in Figure 3. The system moves under the action of this force until t = 6 s. During the motion, the resistance to the motion of P has constant magnitude 1 N and the resistance to the motion of Q has constant magnitude 2 N. Find (a) the acceleration of the particles as the system moves under the action of F, (b) the speed of the particles at t = 6 s, (c) the tension in the rod as the system moves under the action of F. (2) At t = 6 s, F is removed and the system decelerates to rest. The resistances to motion are unchanged. Find (d) the distance moved by P as the system decelerates, (4) (e) the thrust in the rod as the system decelerates. 22 *P40689A02228*

9 Centre No. Candidate No. Paper Reference Paper Reference(s) 6677/01 Edexcel GCE Mechanics M1 Advanced/Advanced Subsidiary Friday 20 January 2012 Afternoon Time: 1 hour 30 minutes Surname Signature Materials required for examination Items included with question papers Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. Initial(s) Examiner s use only Team Leader s use only Question Leave Number Blank 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 of g is required, take g = 9.8 m s 2. 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 8 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. This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy Pearson Education Ltd. Printer s Log. No. P40096A W850/R6677/ /4/5/4/3 *P40096A0128* Total Turn over

10 1. A railway truck P, of mass m kg, is moving along a straight horizontal track with speed 15 m s 1. Truck P collides with a truck Q of mass 3000 kg, which is at rest on the same track. Immediately after the collision the speed of P is 3 m s 1 and the speed of Q is 9 m s 1. The direction of motion of P is reversed by the collision. Leave blank Modelling the trucks as particles, find (a) the magnitude of the impulse exerted by P on Q, (2) (b) the value of m. 2 *P40096A0228*

11 2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N. The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and R newtons respectively. Leave blank Given that the acceleration of the car and the caravan is 0.88 m s 2, (a) show that R = 860, (b) find the tension in the tow-bar. 4 *P40096A0428*

12 3. Three forces F 1, F 2 and F 3 acting on a particle P are given by Leave blank F 1 = (7i 9j) N F 2 = (5i + 6j) N F 3 = (pi + qj) N where p and q are constants. Given that P is in equilibrium, (a) find the value of p and the value of q. The force F 3 is now removed. The resultant of F 1 and F 2 is R. Find (b) the magnitude of R, (2) (c) the angle, to the nearest degree, that the direction of R makes with j. 6 *P40096A0628*

13 4. A C G D B Leave blank d 5d Figure 1 d A non-uniform rod AB, of mass m and length 5d, rests horizontally in equilibrium on two supports at C and D, where AC = DB = d, as shown in Figure 1. The centre of mass of the rod is at the point G. A particle of mass 5 m is placed on the rod at B and the rod is on the 2 point of tipping about D. (a) Show that GD d = 5 2. (4) The particle is moved from B to the mid-point of the rod and the rod remains in equilibrium. (b) Find the magnitude of the normal reaction between the support at D and the rod. (5) 8 *P40096A0828*

14 5. A stone is projected vertically upwards from a point A with speed u m s 1. After projection the stone moves freely under gravity until it returns to A. The time between the instant that the stone is projected and the instant that it returns to A is seconds. Leave blank Modelling the stone as a particle, (a) show that u = , (b) find the greatest height above A reached by the stone, (2) (c) find the length of time for which the stone is at least m above A. (6) 12 *P40096A01228*

15 6. A car moves along a straight horizontal road from a point A to a point B, where AB = 885 m. The car accelerates from rest at A to a speed of 15 m s 1 at a constant rate a m s 2. The time for which the car accelerates is 1 T seconds. The car maintains the speed of 3 15 m s 1 for T seconds. The car then decelerates at a constant rate of 2.5 m s 2 stopping at B. Leave blank (a) Find the time for which the car decelerates. (b) Sketch a speed-time graph for the motion of the car. (c) Find the value of T. (d) Find the value of a. (2) (2) (4) (2) (e) Sketch an acceleration-time graph for the motion of the car. 16 *P40096A01628*

16 7. [In this question, the unit vectors i and j are due east and due north respectively. Position vectors are relative to a fixed origin O.] Leave blank A boat P is moving with constant velocity ( 4i + 8j) km h 1. (a) Calculate the speed of P. (2) When t = 0, the boat P has position vector (2i 8j) km. At time t hours, the position vector of P is p km. (b) Write down p in terms of t. (1) A second boat Q is also moving with constant velocity. At time t hours, the position vector of Q is q km, where Find q = 18i + 12j t (6i + 8j) (c) the value of t when P is due west of Q, (d) the distance between P and Q when P is due west of Q. 20 *P40096A02028*

17 8. 36 N Leave blank P Figure 2 A particle P of mass 4 kg is moving up a fixed rough plane at a constant speed of 16 m s 1 under the action of a force of magnitude 36 N. The plane is inclined at 30 to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through P, and acts at 30 to the inclined plane, as shown in Figure 2. The coefficient of friction between P and the plane is μ. Find (a) the magnitude of the normal reaction between P and the plane, (b) the value of μ. (4) (5) The force of magnitude 36 N is removed. (c) Find the distance that P travels between the instant when the force is removed and the instant when it comes to rest. (5) 24 *P40096A02428*

18 Centre No. Candidate No. Paper Reference(s) 6677/01 Edexcel GCE Mechanics M1 Advanced/Advanced Subsidiary Wednesday 18 May 2011 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Paper Reference Surname Signature Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. Initial(s) Examiner s use only Team Leader s use only Question Leave Number Blank 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 of g is required, take g = 9.8 m s 2. 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 24 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. This publication may be reproduced only in accordance with Edexcel Limited copyright policy Edexcel Limited. Printer s Log. No. P38161RRA W850/R6677/ /5/4/3/3/2/e2/5 *p38161rra0124* Total Turn over

19 1. At time t = 0 a ball is projected vertically upwards from a point O and rises to a maximum height of 40 m above O. The ball is modelled as a particle moving freely under gravity. Leave blank (a) Show that the speed of projection is 28 m s 1. (b) Find the times, in seconds, when the ball is 33.6 m above O. (5) 2 *P38161RRA0224*

20 2. Particle P has mass 3 kg and particle Q has mass 2 kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, P has speed 3 m s 1 and Q has speed 2 m s 1. Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 m s 1. Leave blank (a) Find the speed of each particle after the collision. (5) (b) Find the magnitude of the impulse exerted on P by Q. 4 *P38161RRA0424*

21 3. Leave blank 4 N α Figure 1 A particle of weight W newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle, where as shown in Figure 1. The coefficient of friction between the particle and the plane is Given that the particle is on the point of sliding down the plane, (i) show that the magnitude of the normal reaction between the particle and the plane is 20 N, (ii) find the value of W. (9) 6 *P38161RRA0624*

22 4. A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s 1. She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of V m s 1. Leave blank (a) Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race. (2) (b) Find the distance run by the girl in the first 64 s of the race. (c) Find the value of V. (d) Find the deceleration of the girl in the final 20 s of her race. (5) (2) 10 *P38161RRA01024*

23 5. A plank PQR, of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at P and Q, where PQ = 6 m. Leave blank A child of mass 40 kg stands on the plank at a distance of 2 m from P and a block of mass M kg is placed on the plank at the end R. The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at P is equal to the force exerted on the plank by the support at Q. By modelling the plank as a uniform rod, and the child and the block as particles, (a) (i) find the magnitude of the force exerted on the plank by the support at P, (ii) find the value of M. (10) (b) State how, in your calculations, you have used the fact that the child and the block can be modelled as particles. (1) 14 *P38161RRA01424*

24 6. Leave blank Q P Figure 2 Two particles P and Q have masses 0.3 kg and m kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle, where. The coefficient of friction between P and the plane is The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle P is held at rest on the inclined plane and the particle Q hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and Q accelerates vertically downwards at Find (a) the magnitude of the normal reaction of the inclined plane on P, (b) the value of m. (2) (8) When the particles have been moving for 0.5 s, the string breaks. Assuming that P does not reach the pulley, (c) find the further time that elapses until P comes to instantaneous rest. (6) 18 *P38161RRA01824*

25 7. [In this question i and j are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin O.] Leave blank Two ships P and Q are moving with constant velocities. Ship P moves with velocity and ship Q moves with velocity (a) Find, to the nearest degree, the bearing on which Q is moving. (2) At 2 pm, ship P is at the point with position vector with position vector At time t hours after 2 pm, the position vector of P is Q is and ship Q is at the point and the position vector of (b) Write down expressions, in terms of t, for (i) (ii) (iii) PQ. (5) (c) Find the time when (i) Q is due north of P, (ii) Q is north-west of P. (4) 22 *P38161RRA02224*

26 Mark Scheme (Results) Summer 2012 GCE Mechanics M1 (6677) Paper 1

27 June Mechanics M1 Mark Scheme Question Number Scheme Marks N s A( 5m ) B ( 2m ) 3.3 N s Before 3m s 1 4 m s 1 After v m s m s 1 (a) CLM 5m 3 2m 4 = 5m mv 1 Leading to 1.5 Speed is 1.5 m s A1 v = ( ) (b) Impulse for A 5m( 0.8 3) = 3.3 Leading to m = 0.3 A1 [6] Alternative for (b) Impulse for B 2 m(1.5 4 ) = 3.3 Leading to m = 0.3 A1 Question 1(a) M1 for attempt at CLM equation, with correct no.of terms, correct masses and dimensionally consistent. Allow consistent extra g s, consistent missing m s and sign errors. However, M0 if masses are not paired with the correct speeds. First A1 for a correct equation. Second A1 for v = 1.5. (-1.5 A0) N.B. Allow M1 for an attempt to equate the impulses on the particles but must have 5m (0.8 3) or 5m (3 0.8) on one side of the equation and 2m ( ± v ± 4 ) on the other. Question 1(b) M1 for attempt at impulse = difference in momenta, for either particle, (must be considering one particle) (M0 if g s are included or if mass omitted or if just m used) Allow Initial Momentum Final Momentum. A1 cao (i.e. no ft on their v) for a correct equation in m only. A1 for m = 0.3

28 Question Number Scheme Marks 2. 2X X A P G Q B 0.8 m 4.5g 0.6 m (a) 2X + X = 4.5g 3g Leading to X = or 14.7 or 15 ( N) A1 2 (b) M ( A ) ( ) 4.5g AG = 2X X 2.4 M1 A2 ft (1,0) 4 AG = 3 ( m ), 1.3, 1.33,... A1 (4) [7] Question 2(a) First M1 for a complete method for finding R Q, either by resolving vertically, or taking moments twice, with usual criteria (allow M1 even if R P =2R Q not substituted) First A1 for a correct equation in either R Q or R P ONLY. Second A1 for 1.5g or 14.7 or 15 (A0 for a negative answer) Question 2(b) First M1 for taking moments about any point, with usual criteria. A2 ft for a correct equation (A1A0 one error, A0A0 for two or more errors, ignoring consistent omission of g s) in terms of X and their x (which may not be AG at this stage) Third A1 for AG = 4/3, 1.3, 1.33,.. (any number of decimal places, since g cancels) need AG = or x marked on diagram N.B. if R Q = 2R P throughout, mark as a misread as follows: (a) M1A1A0 (resolution method) (b) M1A0A1A1, assuming all work follows through correctly..

29 Question Number Scheme Marks 3. R µ R 20 5g 30 (a) plane R = 20 cos g cos 30 M1 A2(1,0) = 52.4 ( N) or 52 A1 (4) (b) Fr = µ R B1 P plane F + 20 cos 30 o = 5gcos60 o M1 A2(1, 0) Leading to µ = or 0.14 A1 (5) [9] Question 3(a) First M1 for resolving perpendicular to plane with usual criteria First A2 for a correct equation (A1A0 one error, A0A0 for two or more errors) Second A1 for either 52 or 52.4 N.B. In part (a), the M1 is for a complete method, so they must have sufficient equations to be able to solve for R. The A2 marks are then for all the equations. Question 3(b) B1 for use of F=µR (could just be on diagram) First M1 (allow if F is used rather than µr) for resolving parallel to the plane with usual criteria First A2 for a correct equation (A1A0 one error, A0A0 for two or more errors) Second A1 for either 0.14 or N.B. If they resolve vertically AND horizontally, there are max 6 marks available (M1A2, M1A2) for the TWO equations, but if they only have one equation, there are no marks available for that equation. The marks for the horizontal resolution should be entered first on epen.

30 Question Number Scheme Marks 4. (a) v 1 ( m s ) 20 B1 B1 20, 8, 25 B1 8 O 25 t ( s) (b) v = u + at 8 = t M1 t = 30 ( s) A1 (2) (c) 1960 = (25 x 20) +(30 x 8) + (½ x 30 x 12) + (60 x 8) + 8 x t + ½ x t x 12 M1A3 ft (2,1, 0) 1960 = t D T = = 155 DM1 (8) N.B. SEE ALTERNATIVES [13] A1 Question 4(a) First B1 for 1 st section of graph Second B1 for 2 nd section Third B1 for the figures 20, 8 and 25 Question 4(b) M1 for a complete method to produce an equation in t only; allow (20 8)/0.4 A1 for 30 N.B. Give A0 for t = - 30, even if changed to 30, but then allow use of 30 in part (c), where full marks could then be scored.

31 Question 4(c) First M1 (generous) for clear attempt to find whole area under their graph (must include at least one 1/2 ), in terms of a single unknown time (t say),and equate it to First A3, ft on their (b), for a correct equation. Deduct 1 mark for each numerical error, or omission, in each of the 4 sections of the area corresponding to each stage of the motion. (they may slice it, horizontally into 3 sections, or a combination of the two) Second DM1, dependent on first M1, for simplifying to produce an equation with all their t terms collected. Fourth A1 for a correct equation for t or T Third DM1, dependent on second M1. for solving for T Fifth A1 155 Please note that any incorrect answer to (b) will lead to an answer of 155 in (c) and can score max 6/8; Solutions with the correct answer of 155 will need to be checked carefully. Solutions to 4 (c) N.B. t = T A = (25 x 20) +(30 x 8) + (½ x 30 x 12) + (60 x 8) + 8 x t + ½ x t x 12 M1 A3 ft 1960 = t T = M1 = 155 A1 B = (25 x 20) + ½ x 30 x (20 + 8) + (60 x 8) + ½ x t x (20 + 8) M1 A3 ft 1960 = t T = M1 = 155 A1 C = 8T + ½ x 12 x ( ) + ½ x 12 x (T 115) M1 A3 ft 1960 = 8T T = 14T = T D = 20T - ½ x 12 x (60 + T 25) M1 A3 ft 1960 = 20T - 6T = 14T = T E = (55 x 20) - ½ x 30 x 12 + (60 x 8) + ½ x t x (20 + 8) M1 A3 ft 1960 = t T = M1 = 155 A1 F = (8 x 115) + ½ x 12 x ( ) + ½ x 28 x (T 115) M1 A3 ft 1960 = T = 14T = T

32 Question Number Scheme Marks 5. (a) (b) v = u + 2as 28 = u Leading to u = 21 cso A1 1 s ut at t t 2 4.9t 2 21t + 19 = = + 19 = t = 21 ± 212 4x4.9.x t = 2.99 or 3.0 t = 1.30 or 1.3 D A1 (5) (c) N2L 4g 5000 = 4a OR Work-Energy: ( a = ) v = u + 2as 0 = s Leading to s = ( m) or 0.32 (4) [12] 1 x 4 x gs = 5000s s = or 0.32

33 Question 5(a) First M1 for a complete method for finding u e.g = u 2 + 2gx17.5 or 28 2 = u 2 + 2(-g)x(-17.5) or 28 2 = 2gs s=40 then 0 2 = u 2 + 2(-g)x(22.5) condone sign errors First A1 for a correct equation(s) with g = 9.8 Second A1 for u = 21 PRINTED ANSWER N.B. Allow a verification method, but they must state, as a conclusion, that u = 21, to score the final A1. Question 5(b) First M1 for a complete method for finding at least one t value i.e. for producing an equation in t only. (condone sign errors but not missing terms) First A1 for a correct quadratic equation in t only or TWO correct linear equations in t only. Second DM1, dependent on first M1, for attempt to solve the quadratic or one of the linear equations. Second A1 for 3.0 or 3 or 2.99 Third A1 for 1.3 or 1.30 Question 5(c) First M1 for resolving vertically with usual rules. First A1 for a correct equation Second M1 for use of v 2 = u 2 + 2as, with v = 0, u = 28 or u = 0 and v = 28 and their a, (or any other complete method which produces an equation in s,which could be negative) M0 if they haven t calculated a value of a. Second A1 for 0.32 or (must be positive since it s a distance)

34 Question Number Scheme Marks 6. (a) 7.5 arctan = Bearing is 302 (allow more accuracy) A1 (b) = t ( ) s i j i j (2) (c) t = 3, s = 4i j M1 s b = 3i + 4j M1 2 2 SB = (( 3 ) + 4 ) = 5 ( km ) D (4) (d) Equating i components 40 12t = 7 or t = 0 M1 3 t = 2 4 A1 When t =, ( ) 5 s = 7i + 14 j M1 8 OR When 1 SB = 2 8 ( km) 2.125, 2.13 A1 (4) [13] 3 t = 2, t = 2.125, 2.13

35 Question 6(a) ±7.5 arctan( First M1 for ±12 ) either way up First A1 for a correct value from their expression, usually 32 o or 58 o Second A1 for 302 (allow more accurate answers) Question 6(b) M1 for a clear attempt at (40i 6j)+t(-12i + 7.5j) A1 for any correct expression Question 6(c) First M1 is really B1 for 4i j (seen or implied but can be in unsimplified form) Second M1 is for a subtraction, s b or b s. Third DM1, dependent on second M1, for finding magnitude of their s b or b s A1 for 5 Question 6(d) First M1 for equating i-component of their answer in part (b) to 7 or the i-component of their s b or b s to zero First A1 for 2.75 cao Second M1 (independent) for attempt to find j-component of their s at their t = 2.75 Second A or 2.13 cao

36 Question Number Scheme Marks 7. P ( 0.3 kg) Q ( 0.5 kg) T N T N 1 N 2 N 4 N (a) For system N2L 4 3 = 0.8a 2 a = 1.25 m s, 1.3 A1 ( ) 1 (b) v u at v ( m s ) = + = + = (2) (c) For P N2L T 1 = ft their a ft T = ( N) 1.38, 1.4 A1 OR For Q N2L 4-2 T = 0.5 x 1.25 P ( 0.3 kg) Q ( 0.5 kg) T T 1 N 2 N (d) For system N2L 3 = 0.8a a = v = u + 2as 0 = s M1 s = 7.5 ( m) A1 (4) (e) For P N2L T + 1 = T = ( N), 0.13 A1 [15] Alternative for (e) For Q N2L 2 T = T = N, 0.13 A1 ( )

37 Mark Scheme (Results) January 2012 GCE Mechanics M1 (6677) Paper 1

38 January Mechanics M1 Mark Scheme Question Number 1 (a) 15 m s 1 Scheme Marks P m kg Q 3000 kg (b) 1 3 m s 9 m s 1 For Q I = = ( N s ) Conservation of linear momentum 15m = 3m Leading to m = 1500 A1 (2) 5 Alternative to (b) For P = m( 15 ( 3) ) Leading to m = 1500 A1

39 Question Number Scheme Marks 2 (a) T T 750 kg 1000 kg 3200 N R N 800 N (b) For the whole system R ( ) R = Leading to R = 860 A1 For the caravan R ( ) T 860 = Leading to T = 1520 ( N ) A1 Alternative for (b) For the car ( ) R T = T = ( N ) Leading to 1520 A1 6

40 Question Number Scheme Marks 3 (a) p = 0 or q = 0 M1 p = -12 A1 q = 3 A1 (b) R = 12i 3j (c) 2 2 R = ( 12 + ( 3) ) = 153 or 3 17 or 12.4 or better ( ) tanθ= 3 12 θ = Angle with j is 104, to the nearest degree cao j 12 θ 3 N M1 A1 A1 (2) 8

41 Question Number Scheme Marks 4 (a) A d G Y d B C mg D 5 2 mg (b) M ( D ) A 5 mg GD = mg d 2 5 GD = d 2 D d G Y d B (4) M ( C ) C mg 5 2 mg D d 5 3 mg + mg d = Y 3d M1 A2(1, 0) Leading to Y = mg D 12 (5) 9

42 Question Number Scheme Marks 5 (a) v = u + at( ) => 0 = u g( 25 ) 14 M1 M(A)1 u = 17 ½ A1 (b) v 2 = u 2 + 2as( ) => 0 2 = gs M1 s = 15.6 ( m ) or 16 (m) A1 (c) s = ut at 2 ( ) => 6.6 = 17.5t 1 2 gt 2 M t 17.5t = A1 2 ( ) 17.5 ± ± 13.3 t = = t = (22/7) or (3/7) DM1 A1 (2) T = t2 t1 = 2.71 (2.7) D (6) OR v 2 = u 2 + 2as( ) => v 2 = gx6.6 v = ±13.3 v = u + at( ) => ±13.3 = 17.5 gt 17.5 ± 13.3 t = 9.8 = (22/7) or (3/7) T = = 2.71 or 2.7 M1A1 DM1 A1 D (6) OR v 2 = u 2 + 2as( ) => v 2 = gx6.6 or 0 2 = u 2 2gx( ) v = 13.3 u = 13.3 v = u + at( ) => 0 = 13.3 gt t = 13.3 g T = 2 x 13.3 g = 2.7 or 2.71 D D (6) 11

43 Question Number Scheme Marks 6 (a) v = u + at 0 = t M1 (b) v 1 ( m s ) 15 O t = 6 ( s) A1 1 T T 6 t ( s) 3 Shape B1 15, T B1 (2) (2) (c) a 2 ( m s ) T T + + = T = ft their 6 ft 3 T = 112 = (d) a = 1 T = 16,0.9375, 0.938, 0.94 (e) 3 (4) (2) 3 horizontal lines B1 Correctly placed;no cts vert line B , ft their t ( s ) 15 B

44 Question Number Scheme Marks 7 (a) ( ) ( ) 2 = 80 ( km h 1 ) accept exact equivalents or 8.9 or better (b) = ( 2 8 ) + t ( ) (c) (d) p i j i j B1 Equating j components 8 + 8t = 12 8t 5 t = oe 4 A1 Using their t from (c) to find the i-cpts of p and q and subtract them M ( 3) = 13 ( km) A1 ft A1 2 2 (2) (1) 9

45 Question Number Scheme Marks 8 (a) R 36 F r 30 4g 30 (b) (c) R + 36sin 30 = 4g cos30 R 15.9, 16 Use of Fr = µ R B1 36cos30 = F + 4g sin 30 36cos30 4g sin 30 µ = R 0.73 After force is removed R = 4g cos 30 B1 µ 4g cos 30 4g sin 30 = 4a a = ( ) v u 2as s = + = M1 s = ( ) 11.6 m 12 A1 (4) (5) (5) 14

46 Mark Scheme (Results) June 2011 GCE Mechanics M1 (6677) Paper 1

47 June 2011 Mechanics M Mark Scheme Question Number Scheme Marks 1. (a) 0 2 = u 2 2x9.8x40 u = 28 m s -1 ** GIVEN ANSWER A1 2. (b) 33.6 = 28t t 2 (a) 4.9t 2 28t = 0 t = 28 ± 282 4x4.9x M1 = 4 s or (1.7 s or 1.71 s) A1 A1 3 2 (5) 8 v v + 1 CLM: 3x3-2x2 = 3v + 2(v+1) v P = 0.6 m s -1 ; v Q = 1.6 m s -1 M1A1 (A1 ft) (5) (b) 3(v 3) OR 2(v + 1 2) ft = 7.2 Ns = 7.2 Ns A1 8 GCE Mechanics M1 (6677) June

48 Question Number 3. Scheme 4cosα + F = Wsinα R = 4sinα + Wcosα F = 0.5R cosα = 0.8 or sinα= 0.6 R = 20N ** GIVEN ANSWER W = 22N Marks B1 B1 A1 OR Rsinα = 4 + Fcosα Rcosα + Fsinα = W F = 0.5R B1 cosα = 0.8 or sinα= 0.6 B1 R = 20N ** GIVEN ANSWER W = 22N A1 (9) (9) 9 4. (a) 5 V B1 shape B1 figs (b) ( 1 x4x5) + 60 x 5 2 = 310 A1 (c) (5 + V ) x 20 = ( ) 2 V = 4 M1 A2 ft D (2) (5) (d) = 0.05 ms-2 (2) 12 GCE Mechanics M1 (6677) June

49 Question Number Scheme Marks 5. (a) P 2 m 2 m 2 m Q 2 m R X 40g 20g X Mg 6. (i) EITHER M(R), 8X + 2X = 40g x g x 4 M1 A2 solving for X, X = 32g = 314 or 310 N (ii) ( ) X + X = 40g + 20g + Mg (or another moments equation) M1 A2 solving for M, M = 4 (i) OR M(P), 6X = 40g x g x 4 + Mg x 8 M1 A2 solving for X, X = 32g = 314 or 310 N ( ) X + X = 40g + 20g + Mg (or another moments equation) M1 A2 (ii) solving for M, M = 4 (b) Masses concentrated at a point or weights act at a point (a) R = 0.3g cosα M1 = 0.24g = 2.35 (3sf)=2.4 (2sf) A1 (b) mg T = 1.4m T 0.3g sinα F = 0.3 x 1.4 M1 A2 F = 0.5R M1 Eliminating R and T DM1 m = 0.4 A1 (c) v = 1.4 x 0.5 B1 0.3g sinα F = 0.3a a = 9.8 A1 0 = t M1 t = s or s (1/14 A0) A1 B1 (10) (1) 11 (2) (8) (6) 16 GCE Mechanics M1 (6677) June

50 Question Number Scheme Marks 7. (a) tanθ= 3 4 ; bearing is 37o (nearest degree) M1; A1 (b) (i) p = (i + j) + t(2i 3j) (ii) q = ( 2j) + t(3i + 4j) A1 (iii) PQ = q p = ( i 3j) + t(i + 7j) (c) (i) 1+ t = 0 M1 t = 1 or 3pm A1 (ii) 1+ t = ( 3+ 7t) M1 t = 1 2 or 2.30 pm A1 (2) (5) (4) 11 GCE Mechanics M1 (6677) June