ADVANCED SUBSIDIARY GCE UNIT 761/01 MATHEMATICS (MEI) Mechanics 1 MONDAY 1 MAY 007 Additional materials: Answer booklet (8 pages) Graph paper MEI Examination Formulae and Tables (MF) Morning Time: 1 hour 0 minutes INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions. 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 ms. 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. The total number of marks for this paper is 7. ADVICE TO CANDIDATES Read each question carefully and make sure you know what you have to do before starting your answer. 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. This document consists of 7 printed pages and 1 blank page. HN/ OCR 007 [M/10/69] OCR is an exempt Charity [Turn over
Section A (6 marks) 1 Fig. 1 shows four forces in equilibrium. 10N 60 0N PN QN Fig. 1 Find the value of P. [] (ii) Hence find the value of Q. [] A car passes a point A travelling at 10 m s 1. Its motion over the next seconds is modelled as follows. The car s speed increases uniformly from 10 m s 1 to 0 m s 1 over the first 10 s. Its speed then increases uniformly to 0 m s 1 over the next 1 s. The car then maintains this speed for a further 0 s at which time it reaches the point B. Sketch a speed-time graph to represent this motion. [] (ii) Calculate the distance from A to B. [] (iii) When it reaches the point B, the car is brought uniformly to rest in T seconds. The total distance from A is now 1700 m. Calculate the value of T. [] OCR 007 761/01 June 07
Fig. shows a system in equilibrium. The rod is firmly attached to the floor and also to an object, P. The light string is attached to P and passes over a smooth pulley with an object Q hanging freely from its other end. vertical light string 0kg P Q smooth pulley 1kg vertical light rod floor Fig. Why is the tension the same throughout the string? [1] (ii) Calculate the force in the rod, stating whether it is a tension or a thrust. [] Two trucks, A and B, each of mass 10 000 kg, are pulled along a straight, horizontal track by a constant, horizontal force of P N. The coupling between the trucks is light and horizontal. This situation and the resistances to motion of the trucks are shown in Fig.. 00N A 10000kg 00N B 10000kg PN Fig. The acceleration of the system is 0. m s in the direction of the pulling force of magnitude P. Calculate the value of P. [] Truck A is now subjected to an extra resistive force of 000 N while P does not change. (ii) Calculate the new acceleration of the trucks. [] (iii) Calculate the force in the coupling between the trucks. [] OCR 007 761/01 June 07 [Turn over
A block of weight 100 N is on a rough plane that is inclined at to the horizontal. The block is in equilibrium with a horizontal force of 0 N acting on it, as shown in Fig.. 0N Fig. Calculate the frictional force acting on the block. [] 6 A rock of mass 8 kg is acted on by just the two forces 80k N and ( i 16j 7k) N, where i and j are perpendicular unit vectors in a horizontal plane and k is a unit vector vertically upward. ( 1 - ) Show that the acceleration of the rock is - i + j -k ms. [] 8 The rock passes through the origin of position vectors, O, with velocity seconds later passes through the point A. (i j k) m s 1 and (ii) Find the position vector of A. [] (iii) Find the distance OA. [1] (iv) Find the angle that OA makes with the horizontal. [] OCR 007 761/01 June 07
Section B (6 marks) 7 Fig. 7 is a sketch of part of the velocity-time graph for the motion of an insect walking in a straight line. Its velocity, v ms 1, at time t seconds for the time interval t is given by v t t 8. v ms -1 O t seconds A not to scale Fig. 7 Write down the velocity of the insect when t 0. [1] (ii) Show that the insect is instantaneously at rest when t and when t. [] (iii) Determine the velocity of the insect when its acceleration is zero. Write down the coordinates of the point A shown in Fig. 7. [] (iv) Calculate the distance travelled by the insect from t 1 to t. [] (v) Write down the distance travelled by the insect in the time interval t. [1] (vi) How far does the insect walk in the time interval 1 t? [] OCR 007 761/01 June 07 [Turn over
6 8 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of m s 1 and at an angle q to the horizontal such that cos q 0.96 and sin q 0.8. Show that the height, y m, of the ball above the ground t seconds after projection is given by y 7t.9t. Show also that the horizontal distance, x m, travelled by this time is given by x t. [] (ii) Calculate the maximum height reached by the ball. [] (iii) Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times. [] (iv) Determine the following when t 1.. (A) The vertical component of the velocity of the ball. (B) Whether the ball is rising or falling. (You should give a reason for your answer.) (C) The speed of the ball. [] (v) Show that the equation of the trajectory of the ball is y 0.7x 76 ( 0 7x). Hence, or otherwise, find the range of the ball. [] OCR 007 761/01 June 07
761 Mark Scheme June 007 Q1 0 P cos 60 = 0 A1 P = 80 A1 cao For any resolution in an equation involving P. Allow for P = 0 cos 60 or P = 0 cos 0 or P = 0 sin 60 or P = 0 sin 0 Correct equation (ii) Q+ Pcos 0 = 10 Q = 0( ) = 0.7179 so 0.7 ( s. f.) A1 Resolve vert. All forces present. Allow sin cos No extra forces. Allow wrong signs. cao Q (ii) (iii) Q Straight lines connecting (0, 10), (10, 0), B1 Axes with labels (words or letter). Scales indicated. Accept no arrows. (, 0) and (, 0) B1 Use of straight line segments and horiz section B1 All correct with salient points clearly indicated 0.(10 + 0) 10 + 0.(0 + 0) 1 + 0 0 Attempt at area(s) or use of appropriate uvast Evidence of attempt to find whole area = 00 + + 800 = 1 A1 cao 0. 0 T = 1700 1 Equating triangle area to 1700 their (ii) so 0T = 17 and T = 8.7 F1 (1700 their (ii))/0. Do not award for ve answer. String light and pulley smooth E1 Accept pulley smooth alone 8 1 (ii) g (9) N thrust Three forces in equilibrium. Allow sign errors. B1 for 1g (17) N used as a tension A1 g (9) N thrust. Accept ± g (9). Ignore diagram. [Award SC for ± g (9) N without thrust and SC if it is] 1
761 Mark Scheme June 007 Q (ii) (iii) Q NL. Allow F = mga. Allow wrong or zero resistance. No extra forces. Allow sign errors. If done as 1 equn need m = 0 000. If A and B analysed separately, must have equns with T. P 800 = 0000 0. A1 NL correct. P = 800 A1 New accn 800 800 = 0000a F = ma. Finding new accn. No extra forces. Allow 00 N but not 00 N omitted. Allow sign errors. a = 0.1 A1 FT their P T 00 = 10000 0.1 T = 00 so 00 N A1 cao NL with new a. Mass 10000. All forces present for A or B except allow 00 N omitted on A. No extra forces Take F +ve up the plane F + 0cos = 100sin Resolve // plane (or horiz or vert). All forces present. At least one resolved. Allow sin cos and sign errors. Allow 100g used. B1 Either ± 0 cos or ± 100sin or equivalent seen F =.91 so.6 N ( s. f.) A1 Accept ±.91 or ± 90.17... even if inconsistent or wrong signs used. up the plane A1.6 N up the plane (specified or from diagram) or equiv all obtained from consistent and correct working. 7
761 Mark Scheme June 007 Q6 (ii) (iii) (iv) ( ) ( + i 16j+ 7k + 80k) = 8a Use of NL. All forces present. 1 a= i+ j k m s E1 Need at least the k term clearly derived 8 1 r = ( i j+ k) + 0. 16 i+ j k 8 Use of appropriate uvast or integration (twice) A1 Correct substitution (or limits if integrated) = i + k A1 + = so m B1 FT their (ii) even if it not a displacement. Allow surd form arctan Accept arctan. FT their (ii) even if not a displacement. Condone sign errors. (May use arcsin/ or equivalent. FT their (ii) and (iii) even if not displacement. Condone sign errors) =.10 so.1 ( s. f.) A1 cao 1 8
761 Mark Scheme June 007 Q7 (ii) (iii) (iv) (v) (vi) 8 m s 1 (in the negative direction) B1 Allow ± and no direction indicated ( t+ )( t ) = 0 Equating v to zero and solving or subst so t = or A1 If subst used then both must be clearly shown a = t Differentiating A1 Correct a = 0 when t = 1 F1 v (1) = 1 8 = 9 so 9 m s 1 in the negative direction A1 Accept 9 but not 9 without comment (1, 9) B1 FT 1 ( t t ) 8 dx Attempt at integration. Ignore limits. t = t 8t 1 A1 Correct integration. Ignore limits. 6 1 = 16 1 8 Attempt to sub correct limits and subtract = 18 A1 Limits correctly evaluated. Award if 18 seen but no need to evaluate distance is 18 m A1 Award even if 18 not seen. Do not award for 18. cao 18= 6m F1 Award for their (iv). t ( t t 8 ) dx = t 8t attempted or, otherwise, complete method seen. 1 80 1 = 0 = A1 Correct substitution 1 1 1 so + 18= 1 m A1 Award for + their (positive) (iv) 1 1 17
761 Mark Scheme June 007 Q8 (ii) (iii) (iv) (A) (B) (C) y = sinθt+ 0. ( 9.8) t Use of s ut at ± 9.8, ± 10, u = and derivation of.9 not clear. = 7t.9t E1 Shown including deriv of.9. Accept sinθ t = 7t WW x = cosθ t = 0.96t = t B1 Accept 0.96t or cosθt seen WW 1 = +.Accept sin, cos, 0.96, 0.8, 0 = 7 19.6s Accept sequence of uvast. Accept u= but not. Allow u v and ± 9.8 and ± 10 s =. so. m A1 +ve answer obtained by correct manipulation. Need 7t.9t = 1. Equate y to their (ii)/ or equivalent. so.9t 7t+ 1. = 0 Correct sub into quad formula of their term quadratic being solved (i.e. allow manipulation errors before using the formula). t = 0.0909. and 1.1961 A1 Both. cao. [Award A1 for two correct roots WW] need (1.19-0.0909 ) = 1.01... so. m ( s.f.) B1 FT their roots (only if both positive) y = 7 9.8t Attempt at y. Accept sign errors and u = but not y (1.) = 7 9.8 1. =. m s -1 A1 Falling as velocity is negative E1 Reason must be clear. FT their y even if not a velocity Could use an argument involving time. Speed is + (.) Use of Pythag and or 7 with their y =.67 so.6 m s 1 ( s. f.) A1 cao
761 Mark Scheme June 007 (v) y = 7t.9 t, x = t Elimination of t 7x x Elimination correct. Condone wrong notation with so y =.9 A1 interpretation correct for the problem. 7x x 0.7x y =.9 = ( 0 7x ) 76 76 E1 If not wrong accept as long as = 76 seen. Condone wrong notation with interpretation correct for the problem. either Need y = 0 m A1 Accept x = 0 not mentioned. Condone 0 0 X 7. or B1 Time of flight 10 7 s B1 Range 0 0 7 m. Condone 0 X. so x = 0 or 0 7 so 0 7 7 19 6