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2 1. A car is moving a long a straight horizontal road with constant acceleration a m s 2 (a > 0). At time t = 0 the car passes the point P moving with speed u m s 1 ln the next 4 s, the car travels 76 m and then in the following 6 s it travels a further 219 m. Find (i) the value of u, (ii) the value of a. (7),:;,.+ n,+ o~tk 10...tl\~ s :.'lb,, k 4 s l(; 1'l ~ : 77 (;U+1f ~+ z? :: U.l/.,,) 1,k11C Af 76:.4U + ~" U.+ ~::?~ &lr) S;.1.,~,... f... 1o_s lt::)? A:/() S;.U+ +L~+"l, 'l., z,):. {,((fo) t { A (D) 11~:..fol(_ ~~ 0._ Ut ~,.. ~ '29.~ L (.;fo) 3'1 ~ 1o.~ G:: 3 ~"' r 7.1 2(1,)}:1' LU =?'2 t'1 ~ J U ~ 2 ll P A O 2 2 4

3 2. Two particles P and Qare moving in opposite directions along the same horizontal straight line. Particle P has mass m and particle Q has mass km. The particles coll ide directly. mmediately before the collision, the speed of Pis u and the speed of Q is 2u. As a result of the co lli sion, the direction of motion of each particle is reversed and the speed of each pa1ticle is halved. (a) Fi nd the va lue of k. (4) (b) Find, in terms of m and u only, the magnitude of the impulse exerted on Q by Pin the col lision. +VL (2)» "? t..( Mt t.{ f f" M"t.( 'Z ;:_,... t ~ + ""'z. v '2.. f. (w..)+ l<.fl2...) =,ft <4.0 t Kfl1tt) (.{ Z.k... :: J.""./ k v.. "t ~ J(c.(,.?,/('{ :: 3 l<.,l (.;, t.<.) (;3) (b) / g, (J (x 7it 2~...,:r jr '''J~ 4 ll lllll lllll lllll lllll llllll llll lllll P A O 4 2 4

4 3. A block A of mass 9 kg is released from rest from a point P w hi ch is a height h metres above horizontal soft ground. The block falls and strikes another block B of mass 1.5 kg which is on the ground vertically below P. The speed of A immediately before it strikes B is s 1 The blocks are model led as particles. (a) Find the value of h. (2) mmediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude R newtons, (b) find the value of R. (t,,) V~u\2,,t :;'1.~ o '\ 2 (,.1z).5 4':. 11.t s s: 2.t,,.., (8) 7 7 (b) '\ u, 4.,.,z Lf t = ~C}) 4. D ~ (1o,~) V 1ll,~ V:: /3 V;. 6~\$ 1 V'2,::. u z 1 2A ~ (); /;?...+ 2A (o,12) 0.2~,.; 16 ~: ~1~e>"" /Z... (,., +,c,.,j v t{t s1ell...j._(f u.. :::.,.,,._ 1o.~, R z 1o.~(1tc,) 1o.~!9.i)R = 15:;i.s R::. 1~1s"" 1,,2. 6 ll lllll lllll lllll lllll llllll llll lllll P A O 6 2 4

5 4. Al0.6m C r 4111 Figure 1 B A diving board AB consists of a wooden plank of length 4 m and mass 30 kg. The plank is held at rest in a horizontal position by two supports at the points A and C, where AC = 0.6 m, as shown in Figure. The force on the plank at A acts vertically downwards and the force on the plank at C acts vertically upwards. A diver of mass 50 kg is standing on the board at the end B. The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium. (a) Find (i) the magnitude of the force acting on the plank at A, (ii) the magnitude of the force acting on the plank at C. (6) The support at A wi break if subjected to a force whose magnitude is greater than 5000 N. (b) Find, in kg, the greatest integer mass of a diver who can stand on the board at B without breaking the support at A. (3) (c) Explain how you have used the fact that the diver is modelled as a particle. ~' (1) fa) 1 R :: R ~ 3o_, t ~o<>i,4 rr1~ rr...,,,.,,\j/ (3 ng 4..J o... l 1 4P'\ 1,o""' 11 '\_':. RA + <10., \V RA 3Cj ~J l 10 ll lllll lllll lllll lllll llllll llll lllll lllll P46705A01024

6 P A O Turn over

7 5. Two forces, F 1 and F 2, act on a particle A. F 1 = (2i 3j) N and F 2 = (pi + qj) N, where p and q are constants. Given that the resultant of F 1 and F 2 is parallel to (i + 2j), (a) show that 2p q + 7 = 0 (5) Given that q = 11 and that the mass of A is 2 kg, and that F 1 and F 2 are the only forces acting on A, (b) find the magnitude of the acceleration of A. (,.) R== (2,3;)~ (p;+ij) f<: (ttp)~ + <i3)j (5). 1 r~ t. i::1~12,._ 'ltp 7 1 ~r,.flc(1o {r+1j) f.t.12p::: 't 3 2pi r/++5=o Zpi,+1=o (b) R:: (2tp), + <t3) J Zp11 + += o 2p:4'.:0 p::z R:=(4; + tis j)n F;:; """ ' 4:+i;:;: z~ C1\=f.. l~ + 4j).,~,'?.. ~ QED }q =Ji~ 4 2..;; ZR::: 'Z,.. / X P A O _

8 6. p... Leav~, Q Figure 2 Two cars, A and B, move on parallel straight horizontal tracks. nitially A and B are both at rest with A at the point P and B at the point Q, as shown in Figure 2. At time t = 0 seconds, A starts to move with constant acceleration a m s 2 for 3.5 s, reaching a speed of s 1. Car A then moves with constant speed s 1 (a) Find the value of a. (2) Car B also starts to move at time t = 0 seconds, in the same direction as car A. Car B moves with a constant acceleration of 3 m s 2. At time t = T seconds, B overtakes A. At this instant A is moving with constant speed. (b) On a diagram, sketch, on the same axes, a speedtime graph for the motion of A for the interval O ~ t ~ T and a speedtime graph for the motion of B for the interval O ~ t ~ T. (3) (c) Find the value of T (d) Find the distance of car B from the point Q when B overtakes A. (8) (1) (e) On a new diagram, sketch, on the same axes, an accelerationtime graph for the motion of A for the interval O ~ ~ T and an accelerationtime graph for the motion of B for the interval O ~ ~ T. (3) \.l. U1,11 t i4 ~ c2 f "'{3.,0 f. 3_,;).. / ' 4~~ZJ... 0 (~ ~ (.~' \~ i;,: i:y...; /)ii: ::r:.!;.. ;;<. ;< x> 16 11m P A O / /

9 Question 6 continued lj>) ~petd (~fj3t ~r,.. 1 ~ra bl ank \ O._._ 3. _> T'::>,r... ~ (\$) U=o 6\::. 3 +=T \h77 v~a~'l+ V=o+3T V:r3t ',' 11. Z CM'!> +r.vt f ~J +ht 6"1"\L Jr.s htl\u. :. y(14)(t+ T.~ ~) :: i"p 3T (f {) 14(2T3,~.1 = 31 2'2T4~ = 3T'Z 3T"" ~,g; +tllt~~o (3, + )CT +)=o [;_ 3 ~ J,, Re fuf(j bt.(;11'<.k,. WA.\ tiff/(,.rq,/e~tf ~. l s +Tl (,v A (_d) _Ors+~" _kc ffcl ~~){+"' '21. =, f3,tm Ai~a~+r~~ ( ~!>"' ) 4...,... 3.,.!.._ :,&., 13 + L ll lllll lllll lllll lllll llllll llll lllll lllll P46705A {,.ra 1 :::> fl'\(. ( s) 17 Turn over

10 7. Figure 3 A particle P of mass 4 kg is attached to one end of a light inextensible string. A particle Q of mass m kg is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a ve11ical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle a, where 3 tancx = and the second plane is inclined to the horizontal at an angle /J, where 4 4 tan /J =. Particle P is on the first plane and particle Q is on the second plane with the 3 string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between P and the plane is _!_. The second plane is smooth. The system is in limiting equilibrium. 4 Given that Pis on the point of slipping down the first plane, (a) find the value of m, (b) find the magnitude of the force exerted on the pulley by the string, ( c) find the direction of the force exerted on the pulley by the string. (10) (4) (1) P A O

11 , Question 7 continued (g\) ~1lr, k p t R;: 4J '"so< R :,.4, ( t) R~ ~_,Al ~ lrar ::.4_,Jr"o< Tr!<1t~=4j(f) ~f:j =1tj J ~ f_,n' p~,~q C:::> 1;_ ""'_j \$.," /3 r~_, f (' ~) rm::.21 (}) Pe>~{~"'+= (f,)'1:_!,)l= jjz j~ 22.vt/ Lc_) T),tcfro~=> /t.}(;5 "" "6'~ 4!/ 1,,(+h ~,b hrs+ ~ ~""J p r~\(.. ll lllll lllll lllll lllll llllll llll lllll lllll P46705A Turn ove r

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