L.41/42. Pre-Leaving Certificate Examination, Applied Mathematics. Marking Scheme. Ordinary Pg. 2. Higher Pg. 19.

Transcription

1 L.4/4 Pre-Leaving Certificate Examination, 04 Applied Mathematics Marking Scheme Ordinary Pg. Higher Pg. 9 Page of 44

2 exams Pre-Leaving Certificate Examination, 04 Applied Mathematics Ordinary Level Marking Scheme (00 marks) General Instructions. Penalties of three types are applied to students work as follows: Slips - numerical slips S( ) Blunders - mathematical errors B( ) Misreading - if not serious M( ) Serious blunder or omission or misreading which oversimplifies: - award the attempt mark only. Attempt marks are awarded as follows: 5 (Att. ), 0 (Att. ). Mark all answers, including excess answers and repeated answers whether cancelled or not, and award the marks for the best answers.. Mark scripts in red unless a student uses red. If a student uses red, mark the script in blue or black. 4. Number the grid on each script to 9 in numerical order, not the order of answering. 5. Scrutinise all pages of the answer book. 6. The marking scheme shows one correct solution to each question. In many cases, there are other equally valid methods. Six questions to be answered. All questions carry equal marks. (6 50m) 04 L.4/4_MS /44 Page of 44 exams

3 . Four points, A, B, C and D lie on a straight level road. A car, travelling with uniform acceleration, passes A, and then seconds later it passes B with a speed of 6 m s. Three seconds after the car passes B, it passes C which is 66 m from B. On passing C, the car immediately decelerates uniformly to rest at D. The total distance from A to D is 60 m. Find (i) the uniform acceleration of the car (0) For BC, u 6, s 66, t s ut at 66 (6)() () a a 9 8 a 9a 6 a 4ms (ii) the speed of the car at A (0) For AB, v6, a4, t vu at 6 u (4)() 6 u 8 u 8ms (iii) the speed of the car at C (5) For BC, u 6, a4, t vu at v (6) (4)() v 8 ms 04 L.4/4_MS /44 Page of 44 exams

4 (iv) AB, the distance from A to B (0) For AB, u 8, v6, a 4 v u as (6) (8) (4)s s 8s 9 s 4 m (v) the time from C to D, i.e. the time of the deceleration. (5) Distance from C to D For CD, u 8, v0, s 70 v u as (0) (8) a(70) a 40a a 56 5 Then vu at 0 (8) 5 6 t 56 t 8 t 5s 04 L.4/4_MS 4/44 Page 4 of 44 exams

5 . At noon, ship A is 40 km due north of ship B. A is moving at a constant speed of 0 km h in the direction east α south, 5 where tan. A 40 km α 0 km h B is moving due east at a constant speed of 5 km h. Find (i) the velocity of A in terms of i and B 5 km h j (0) va 0cos i 0sin j v 846i 770 j A (ii) the velocity of B in terms of i and j (5) v B 5i (iii) the velocity of A relative to B in terms of i and j (0) v AB v A v B vab 846i 7 70 j 5i v 46 i 770 j AB... (5 m) (iv) the shortest distance between A and B in the subsequent motion. (5) vab kmh v AB and 770 tan v AB 770 Further answers overleaf 04 L.4/4_MS 5/44 Page 5 of 44 exams

6 Let M be the point of closest approach on the path of A relative to B. Shortest distance: p BM 40cos 40cos km A 40 B p q M (v) the time, to the nearest minute, at which the ships are nearest to each other. Distance travelled by A relative to B q AM km Distance Time Speed hours 4 hours 9 minutes (0) 04 L.4/4_MS 6/44 Page 6 of 44 exams

7 . A particle is projected from a point P on horizontal ground with an initial speed of u m s at an angle of 45 to the horizontal. The particle strikes the ground 5 seconds later. P u 45 Wall Two seconds after being projected, the particle just clears the top of a vertical wall. Find (i) the value of u For the time of flight, t 5 and s 0 y u sin 45 5 (0) 5 0 u u 50 u 50 (5) (ii) the range of the particle on the horizontal ground (0) Range (50) cos m (iii) the greatest height reached by the particle (5) Time to greatest height 5 Greatest height (50)sin 45 (0) m Further answers overleaf 04 L.4/4_MS 7/44 Page 7 of 44 exams

8 (iv) the height of the vertical wall, correct to two decimal places. (0) When t, height of wall = ( 50)sin 45 () (0)() m 04 L.4/4_MS 8/44 Page 8 of 44 exams

9 4. (a) Two particles of masses kg and 5kg are connected by a taut, light, inextensible string which passes over a smooth, light pulley. The system is released from rest. Find (i) the common acceleration of the particles (5) T g a 5gT 5a g 8a g a 4 a 5ms kg 5 kg (ii) the tension in the string. (5) T a g T (5) (0) T 7 5 N Further answers overleaf 04 L.4/4_MS 9/44 Page 9 of 44 exams

10 (b) A particle, P, of mass kg sits on a rough plane inclined at an angle α to the horizontal, where tan. 4 The coefficient of friction P kg between P and the plane is. P is released from rest, and travels 4 m to the bottom of the plane. α Find (i) the acceleration of P down the plane. (5) R R 0sin 0cos6 g (0 m) R 6 R a 8 a 4 a a ms (ii) the time taken by P to reach the bottom of the plane. (5) s ut at 4 (0) t () t 4 t t s 04 L.4/4_MS 0/44 Page 0 of 44 exams

11 5. A smooth sphere A, of mass kg, collides directly with another smooth sphere B, of mass 8 kg, on a smooth horizontal table. A and B are moving in the same direction with speeds of 0 m s and m s, respectively. A 0 m s m s kg B 8 kg The coefficient of restitution for the collision is 8. Find (i) the speed of A and the speed of B after the collision (0) PCM: ()(0) (8)() () v (8) v... (0m) 6 v 8v v8v 6 NEL: vv (0 ) 8... (0m) v v v v Thus 9v 7 v ms and v ms... (0m) (ii) the loss in kinetic energy of A due to the collision (5) ()(0) 50 KE of A before collision KE of A after collision ()() Loss in KE of A J (iii) the magnitude of the impulse imparted to B due to the collision. (5) Impulse (8)() (8)() 8 Ns 04 L.4/4_MS /44 Page of 44 exams

12 6. (a) Particles of weight 5 N, 4 N, N and N are placed at the points (, q), (q, p), (p, 6) and (q +, q), respectively. The co-ordinates of the centre of gravity of the system are (p, q). Find (i) the value of q (ii) the value of p. (5) 5() 4( q) ( p) ( q) p 54 4 p 5 4q pq p7q 8 and 5( q) 4( p) (6) ( q) q 54 4q5q4 p q 4p6q... (0m)... (0m) then and p7q 8 p8q 6 q 44 q 4 p 8 8 p 6 p (b) A uniform lamina ABCDEF consists of a rectangle GBHE from which the two triangles GAF and CHD have been removed. E D H The co-ordinates of the points are A(6, 0), B(5, 0), C(5, ), D(, ), E(0, ), F(0, 6), G(0, 0) and H(5, ). F C Find the co-ordinates of the centre G A B of gravity of the lamina ABCDEF. (5) 04 L.4/4_MS /44 Page of 44 exams

13 Area Centre of gravity GBHE 5 80 (7 5,6) GAF (6)(6) 8 (,) CHD ()(9) 5 (4, 9) ABCDEF ( x, y ) 48 5 then and 8() 5(4) 485( x) x 485x 5 x 758 8() 5(9) 485( y) y 485y 9 5 y 6 Thus the co-ordinates of the centre of gravity are (758,6 ). 04 L.4/4_MS /44 Page of 44 exams

14 7. A uniform rod, [AB], of weight 40 N and length m, is smoothly hinged at end A to a vertical wall. The rod is held inclined at 0 below the horizontal by a force P applied at B. The force P is at right angles to the rod [AB]. A 0 P (i) Show on a diagram all the forces acting on the rod [AB]. B (5) X Y A 0 P 4g 40 0 B (5m) (ii) Find the magnitude of P. (5) Moments about A: Anticlockwise Moments Clockwise Moments P() 40(cos0 )... (5m, 5m) P 0 P 0 N 04 L.4/4_MS 4/44 Page 4 of 44 exams

15 (iii) Find the magnitude of the reaction at the hinge, A. (0) The rod is in equilibrium. : Y P 40...(5m) : P X thus Y Y 5 40 Y 5 P sin 60 P P 60 P P cos60 and X 0 5 The magnitude of the reaction force at hinge A is X Y (5 ) N 04 L.4/4_MS 5/44 Page 5 of 44 exams

16 8. (a) A particle describes a horizontal circle of radius r metres with uniform angular velocity ω radians per second. Its speed is m s and its acceleration is 4 m s. Find (i) the value of ω (5) Speed r...(5m) and Acceleration 4 r 4...(5m) then ( r) 4 4 rads...(5m) (ii) the value of r. (5) Then r() r 6m...(5m) (b) A particle of mass kg is attached by a light, inextensible string of length 75 cm to a fixed point, P. The particle moves in a horizontal circle, centre O, with constant speed such that the string makes an angle θ with the horizontal, 4 where tan. P O 75 cm θ Find (i) the value of r, the radius of the circular motion (0) 4 tan cos and 5 r cos 075 r sin 5 m 04 L.4/4_MS 6/44 Page 6 of 44 exams

17 (ii) the tension in the string P (0) Tsin g T T 7 5 N O r 075 T g (iii) the speed of the particle. (0) Circular motion: mv T cos r () v (75)(045) v v 75 v 84 ms 04 L.4/4_MS 7/44 Page 7 of 44 exams

18 9. (a) State the Principle of Archimedes. (0) Statement... (0m) A solid piece of metal has a weight of 800 N. When it is completely immersed in a liquid of relative density 5, the metal weighs 500 N. Find (i) the volume of the metal (0) B weight of displaced liquid 00 Vg 00 (500) V (0) V 00m (ii) the relative density of the metal. (0) Weight of metal Vg 800 (0 0)(0) 4000 s 4 (b) A solid cone has a base of radius 6 cm and a height of cm. The cone is made from plastic of relative density 0 8 and is completely immersed in a tank of water. The cone is held at rest with its axis vertical by a vertical string which is attached to the centre of its base and to the base of the water tank. Find the tension in the string. [Density of water = 000 kg m.] (0) and 6 cm B 000 (006) (0) (0) B 45 W 80 (006) (0) (0) W 66 T W B T T 086N cm 04 L.4/4_MS 8/44 Page 8 of 44 exams

19 exams Pre-Leaving Certificate Examination, 04 Applied Mathematics Higher Level Marking Scheme (00 marks) General Instructions. Penalties of three types are applied to students work as follows: Slips - numerical slips S( ) Blunders - mathematical errors B( ) Misreading - if not serious M( ) Serious blunder or omission or misreading which oversimplifies: - award the attempt mark only. Attempt marks are awarded as follows: 5 (Att. ). Mark all answers, including excess answers and repeated answers whether cancelled or not, and award the marks for the best answers.. Mark scripts in red unless a student uses red. If a student uses red, mark the script in blue or black. 4. Number the grid on each script to 0 in numerical order, not the order of answering. 5. Scrutinise all pages of the answer book. 6. The marking scheme shows one correct solution to each question. In many cases, there are other equally valid methods. Six questions to be answered. All questions carry equal marks. (6 50m) 04 L.4/4_MS 9/44 Page 9 of 44 exams

20 . (a) An underground train has to travel a distance d from rest at one station to rest at the next station. The train has a maximum acceleration of a and a maximum deceleration of b. If the train makes the journey in the shortest possible time without having to travel at constant speed, show that the speed limit on the track, v, satisfies abd v. a b (5) If the train makes the journey without having to travel at constant speed, then it decelerates immediately after accelerating. Let s and s be the distances travelled while accelerating and decelerating, respectively. Let v be the maximum speed attained. Then ss d... Period of acceleration: Let t be the time for acceleration. Then v (0) ( a)( t) v t a and s (0)( t) ( a)( t) v v s a a a Likewise, v s b From, v v d a b b a v d ab abd v a b abd v a b To make the journey without a period of constant speed, we need v v abd v a b 04 L.4/4_MS 0/44 Page 0 of 44 exams

21 (b) A stone is dropped from the top of a tower which is located on horizontal ground through the base of the tower. One second later, another stone is thrown vertically downwards from the same point with a speed of 5 m s. (i) If the two stones reach the ground simultaneously, find the height of the tower. (5) Let h m be the height of the tower. If the first stone takes t s to reach the ground, then the second stone will take ( ) t s. st stone: nd stone: then h(0)( t) ( g)( t) h gt h(5)( t) g( t ) h5t5 gt t h 5t 5 gt gt g 5t5 gt gt g gt 5t5gt g 0 5gt 5 g 5 t 0 t 94s The height of the tower is h (49)( 94) 8 44 m (ii) If a third stone is thrown downwards from the same point half a second after the second stone, what initial speed must it have to reach the ground at the same time as the first two stones? (0) Let v ms be the initial speed of the third stone. It travels for s while covering a distance of 8 44 m. Thus 844 ( v)(044) (98)(0 44) v v v 9 75 The initial speed is 9 75 m s. 04 L.4/4_MS /44 Page of 44 exams

22 . (a) Two straight roads cross at right angles at O. On one road, a car is travelling due north at 4 m s. As the car passes through O, a bus 4 m s is 50 m from O and travelling east m s towards O at a speed of m s. O (i) Find the velocity of the bus relative to the car. (0) vc 4 50 m j vb i v C then v BC v B v 4 C i 4j thus v B v BC 4 5 ms and 4 tan 4 tan 5 v 4 BC (ii) Calculate the shortest distance between the car and the bus and the time at which this occurs. (0) If p is the shortest distance, p sin 50 4 p 50 5 p 40 m and q cos 50 q 50 5 q 0 m then relative distance time relative speed 0 5 6s v B 50 C BC q p 04 L.4/4_MS /44 Page of 44 exams

23 (iii) Find the length of time during which the car and the bus are within 4 m of one another. x x 8 x 9 x Distance travelled 9 8 m Time 6s. 5 x (5) (b) A ship P is travelling in the direction 8 north of east. To an observer on another ship Q, which is travelling 8 south of east at km h, P appears to be travelling in the direction 67 north of west. (i) Find the actual speed of P, correct to one decimal place. (5) b 75 a 56 Diagram with angles:... (0m) Let a v P. Then a sin 49 sin75 a 58 km h v P (ii) Find the magnitude of the velocity of P relative to Q, correct to one decimal place. Let b v PQ. Then b sin56 sin75 b8 km h v PQ... (5) (5) Further answers overleaf 04 L.4/4_MS /44 Page of 44 exams

24 (iii) Q reduces its speed, without changing direction, so that P appears to be travelling due north. Find the reduced speed of Q, correct to one decimal place. (5) Let a be the new speed of Q. a 58 sin5 sin7 a 4 km h v Q... (5) v PQ 7 v Q a v P L.4/4_MS 4/44 Page 4 of 44 exams

25 . (a) A particle is projected from a point on a horizontal plane so that it just clears a vertical a wall of height at a horizontal distance of a from the point of projection and strikes the plane at a horizontal distance a beyond the wall. (i) Express the range of the particle on the horizontal plane in terms of a. (5) a a a From the diagram, the range is aa 4a (ii) If the angle of projection measured to the horizontal is value of k. tan, find the k Let be the angle of projection. TOF: sy 0 usint gt 0 u sin t g R 4a : u sincos 4a g ga u sin cos u sin 4aucos g (0) Path contains the point ucos t a, t u a cos a a, : a usin t gt Further answers overleaf 04 L.4/4_MS 5/44 Page 5 of 44 exams

26 thus a g a a u sin u u cos cos ga sin cos a a tan cos ga a atan atan 4 tan 4 tan by tan. hence k (b) A plane is inclined at an angle β to the horizontal. A body is projected up the plane with velocity u m s at an angle of 0 to the inclined plane. The plane of projection is vertical and contains the line of greatest slope. The particle strikes the plane at right angles. (i) Find the value of c and the value of d if c tan. (5) d u 0 04 L.4/4_MS 6/44 Page 6 of 44 exams

27 At the same time, sy 0 and vx 0. sy 0 ut cos g t 0 u t g cos u At t : g cos vx 0 u u g sin 0 g cos tan thus c and d (ii) If the range of the body on the inclined plane is 0 m, find the value of u. (0) Then u 7 7u TOF. g g and Range 0 u 7u 7u g 0 g 7 4g u u 0 4g 8g u u 80g u 80g u 8 04 L.4/4_MS 7/44 Page 7 of 44 exams

28 4. (a) A light inextensible string is attached at one end to a particle A, of mass 5 kg, hanging freely. The string passes over a smooth fixed pulley and its other end is attached to another particle B, of mass kg, which is held at a height of 5 m above the horizontal plane. Initially the string is slack. The system is released from rest. After B has fallen through m, the string becomes taut. Find (i) the speed of B directly after the string becomes taut (0) First metre: u 0 v u as v? v 0 g() a g v g s v g Let v be the velocity of each particle after the jerk. Momentum before Momentum of B ()( g ) g Momentum after Momentum of A + Momentum of B (5)( v) ()( v) 8v PCM: Momentum after Momentum before 8v g g v m s 8 (ii) the nearest that B gets to the horizontal plane. (5) 5 kg mass : Newton s nd Law 5aT 5g a 5 a 04 L.4/4_MS 8/44 Page 8 of 44 exams

29 kg mass : Newton s nd Law a T ag T 5 Adding and, 8a g g a 4 5g g u v u as 8 v 0 g g 0 s 8 4 g gs 8g a 4 64 s? 6 s 64 9 s 0565m 6 Nearest B gets to the plane = 44 m T g a (b) A wedge of mass 8 kg sits on a smooth horizontal plane. A particle of mass 4 kg sits on a face of the wedge which is inclined at 0 to the horizontal. The coefficient of friction between the particle and the wedge is. The system is released from rest. 0 4 kg 8 kg Find the acceleration of the wedge. (5) 4 kg particle b N N a 4 4g Further answers overleaf 04 L.4/4_MS 9/44 Page 9 of 44 exams

30 Forces Accelerations N N 4g sin0 g 0 4g 4g cos0 g 0 a a a b Wedge : Newton s nd Law a 4 g N a g N : Newton s nd Law N 8a N 4 an N : ( ) N a a N : a a g ( ) a ( ) g a (4 ) a(4 6) g (6 ) a(4 6) g (8 ) a( ) g a g m s 8 or a 0 8 m s 0 N cos0 N N N a 0 0 N N N N N 4 mg N sin0 N N N 4 04 L.4/4_MS 0/44 Page 0 of 44 exams

31 5. (a) Two smooth spheres, P and Q, of equal mass, are travelling directly towards each other with speeds u and u, respectively. (i) Determine if it is possible for the sphere Q to be at rest after the collision. (5) P Q u u m m v v PCM: mvmv mu mu vv u NEL: vv eu : vv u : vv eu v (e ) u (e) u v For Q to be at rest after the collision, v 0. Thus (e) u 0 e 0 e As 0e, it is possible for Q to be at rest after the collision. 04 L.4/4_MS /44 Page of 44 exams

32 7 (ii) If the fraction of the kinetic energy lost during the collision is, find the 40 value of e, the coefficient of restitution. (5) Adding and, v ( e) u ( eu ) v Then: 5 KE before impact mu m( u) mu KE after impact e mu mu e e 4 4 mu (8e ) mu ( 9 e ) Loss in KE mu mu ( 9 e ) 4 4 (0 9 ) 4 mu e 9 ( ) 4 mu e 9 mu ( e ) Fraction lost 4 5 mu 9 7. ( e ) ( e ) 4e e e 4 04 L.4/4_MS /44 Page of 44 exams

33 (b) A smooth sphere A, of mass m, collides obliquely with a smooth sphere B, of mass m, which is at rest. Before the collision, A has a velocity of u in a direction which makes an angle of 0 with the line of centres. If A is deflected through an angle of 90 by the collision, find the value of e, the coefficient of restitution. (0) A B u 0 u u m m x x u y 0 u tan0 x u x u PCM i : u u m my m u u y u u u y u y NEL i u u u : e u u e u u e e = 04 L.4/4_MS /44 Page of 44 exams

34 6. (a) A particle is performing simple harmonic motion of amplitude 08 m about a fixed point O. A and B are two points on the path of the particle such that OA = 0 6 m and OB = 0 4 m. The particle takes seconds to travel from A to B. Find, correct to two decimal places, the periodic time of the motion if (i) A and B are on the same side of O (5) xacos( t ) x06, t 0: 06 08cos cos0 75 cos x04, t : 04 08cos( ) cos( ) 0 5 cos Period: T T s (ii) A and B are on opposite sides of O. (0) xacos( t ) x04, t : 04 08cos( ) cos( ) 0 5 cos ( 0 5) Period: T T s 04 L.4/4_MS 4/44 Page 4 of 44 exams

35 (b) A particle P is moving on the inner surface of a smooth hemispherical bowl with centre O and radius a. The particle is describing a horizontal circle, centre C, with angular speed a g. Find (i) the magnitude of the force exerted on P by the surface of the bowl (0) g a The circular motion force is g m r m r a mgr a : Nsin mg mgr Circular : N cos a mgr Also, from the triangle, a r cos a r mgr, : N a a N mg a a r mg a N r O h C N sin O h C N N cos (ii) the depth of C below O. (5) : mg sin mg sin 0 then h sin a h a h = a 04 L.4/4_MS 5/44 Page 5 of 44 exams

36 7. Two uniform rods, AB and AC, each of length a and weight W, are smoothly jointed at A. The end C is freely hinged to a point on a rough horizontal plane. A The end B rests on the plane and is on the point of slipping. Both rods are in a vertical plane. C θ B The coefficient of friction between B and the plane is μ and the angle between AC and the horizontal is θ. (i) Show, on separate diagrams, all the forces acting on the structure ABC and on the rod AB. Structure ABC: (0) A Y W W N C X N B k k k k Rod AB: A V U a W N a N B 04 L.4/4_MS 6/44 Page 6 of 44 exams

37 (ii) Prove that. (5) tan For structure ABC, moments about C: W. kw. k N.4k N W For rod AB, moments about A: W. acos N.asin N.acos W Ntan N, : W W tan W tan tan (iii) Find, in terms of W and μ, the magnitude and direction of the reaction force at the end B resting on the plane. (5) The total reaction, R, at B is given by R N N N N N ( ) N W The direction of this reaction force makes an angle with the horizontal, where N tan N i.e. tan R N N (iv) Find, in terms of W and θ, the magnitude of the reaction at the joint A. (0) Rod AB : V W N V W W V 0 : U N U W W or U tan which is the magnitude of the reaction at the joint A. 04 L.4/4_MS 7/44 Page 7 of 44 exams

38 8. (a) Prove that the moment of inertia of a uniform rod of mass m and length l about an axis through its centre perpendicular to the rod is ml. (0) Standard Proof Moment of mass element Moment of body Integral Deduce (b) A uniform rod AB of mass m and length l is free to rotate in a vertical plane about a horizontal axis through A. A particle of mass m is attached to the rod at B. The system is released from rest with B vertically above A. (i) Find the angular velocity when the system is next vertical. (0) For the system, I Irod IB 4 Irod ml IB mb ( m)( l) 8ml then 4 I ml 8ml 8 ml Position : B above A KE 0 PE PErod PEB mg( l) ( m) g(4 l) 0mgl Position : B below A KE I 8 ml zero PE for rod B A zero PE for mass B l l l 04 L.4/4_MS 8/44 Page 8 of 44 exams

39 4 ml PE 0 PCE: KE PE KE PE 4 ml 00 0mgl... (5m, 5m) 7l 5g 5g 7l 5g rad s 7l At this point the mass at B falls off. (ii) Find the height of B when it next comes to rest. (0) For the rod after the mass falls off, 4 I ml (Let the position of zero PE be when B is vertically below A.) Position : B below A. KE I 4 5g ml 7l 0 7 mgl PE 0 Position : rod comes to rest. Let the height of the centre of mass above the zero PE level be h. KE 0 PE mgh PCE: KE PE KE PE 0 0mgh mgl 0 h 0 7 l 7 y B x zero PE for rod l A B Further answers overleaf l l h 04 L.4/4_MS 9/44 Page 9 of 44 exams

40 From the diagram, then 6 y x l 7 0 x l l l 7 7 hence, at the highest point B is 6 l above the level of A L.4/4_MS 40/44 Page 40 of 44 exams

41 9. (a) A bucket, in the shape of a frustum, has a height of 50 cm, a base of radius 0 cm and a top of radius 0 cm. 0 cm Find, in terms of π and g, 50 cm (i) the thrust on the base if the bucket is filled with water (5) Let T be the thrust on the base. T gha 0 cm 000() g(05) (0 ) 0 g N (ii) the thrust on the curved side if water is poured into the bucket to a depth of 5 cm. (5) Height of top of water is 5 cm 0 5 m Volume of water: V (05)(05) (05)(0) (0) 6 m 4800 Weight of water: W Vg g g N W T T Further answers overleaf 04 L.4/4_MS 4/44 Page 4 of 44 exams

42 Thrust on base: T gha 000() g (0 5) (0 ) 0 g N Let T be the vertical thrust on the curved side. Then TT W T 7g0 g T 7 g N (b) A block of wood, of volume V and relative density s, floats in water. A smaller block of metal, of volume V and relative density 5s, is placed on top of the wood, such that the upper surface of the wooden block is in the free surface of the water. Prove that V V 5s. s Let B be the buoyancy, W the weight of the W block of wood and W the weight of the B block of metal. Let V be the volume of the wood and V be the volume of the metal. W B 000() Vg 000gV (0) and weight W W 000( svg ) 000(5 svg ) 000sgV 5000sgV In equilibrium: W W B 000sgV 5000sgV 000gV sv 5sV V 5sV V sv 5 sv ( s) V 5s V s V 04 L.4/4_MS 4/44 Page 4 of 44 exams

43 0. (a) Solve the differential equation dy x y( x) dx given that y = when x =. (0) dy x y( x) dx dy x d x y x dy d x y x ln y ln x x c I.C.: y, x ln 0 c c ln Unique solution: ln y ln x x ln ln y ln x ln e x ln ln y ln x e x x x y xe x e (b) A particle of mass m moves with a velocity v m s in a straight line through a medium in which the resistance to motion is mkv, where k is a constant. No other force acts on the body. The velocity of the particle falls from 5 m s to 7 5 m s in a time of t seconds. Show that the distance travelled in this time is 0t m. (0) R mkv Newton s nd Law: d x m mkv dt d x kv dt dv kv dt Further answers overleaf 04 L.4/4_MS 4/44 Page 4 of 44 exams

44 v dv kdt v dv k dt v kt c v kt c I.C.: 5, 0 c 450 Unique solution: kt v 450 also, t t when v 75: kt kt kt k 50t then dv v kv dx v dv kdx v dv k dx kx d v kx d I.C.: 5, 0 v v x d 5 When v 75, kx 75 5 kx 5 5 x 50t 5 50t x 5 x 0t 04 L.4/4_MS 44/44 Page 44 of 44 exams