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June 2005
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The position vector of a particle P, at time t seconds is r, where R = (ti + j + t2k)m.
Show that the acceleration of the particle is constant.
Find the cosine of the angle between the velocity and the position vector of the particle when t =1.
The velocity of another particle Q. relative to P is (i — j)ms-1. Given that when t=0, PQ = jm, find the
equation of the path of Q.
Find also the time at which Q is nearest to P.
A particle is moving with speed 20√5ms-1 from the top of a cliff so that its hits a target which is 200m
horizontally from the foot of the cliff at 200m vertically below the horizontal line through the point of
projection. Show that the two possible directions of projection are at right angles and find the
corresponding time of flight.
Using the angle of projection whose value lies in the first quadrant; find the maximum height above the point of projection attained by the particle, given your answer correct to two decimal places.
take g as 10ms-2
2.Two forces, F1 and F2, where F1 = (i + 2; + 3k)N, F2 = (2i + k)N.
Act at points whose position vectors are (2i + 5j + ck)m and (5i + cj + 2k)m respectively.
Given that the lines of action of F1 and F2 intersect at a point T, find
The value of the constant c and the position vector of T,
The magnitude of the resultant of F1 and F2,
The equation of the line of action of the resultant of F1 and F2.
i) A uniform bar AB of weight W newtons and length 2a metres rest with its upper end A against a rough vertical wall and lower end B against a rough horizontal ground. The coefficient of friction between the bar and the wall and between the bar and the ground is µ. Given that the bar is just about to slip when it is inclined at 45 to the horizontal, show that u =1/2
3.
A block of mass 3kg rest on a smooth plane inclined at an angle θ to the horizontal, where sin θ = 2/3
The block is connected by a light inextensible string parallel to the line of greatest slope of the plane. The
string passes over a smooth fixed pulley at the top of the plane to another particle B of mass 5kg hanging
freely. The system is released from rest. Find
the velocity of the system when B has descended through a distance of 2m, given that it does not hit the ground,
the magnitude of the force exerted on the pulley by the string.
(Take g as 10ms~2)
Three spheres A, B and C of equal radii but of masses 2m, 3m and 4m, respectively, rest in that order, in a straight line, on a smooth horizontal floor. A is projected with speed u so that it strikes B which
subsequently impinges on C, both impacts being direct. Given that the coefficient of restitution between
A and B is 2/3 and between B and an C is 1/2 ,find
The impulse experience by A in its collision with B,
the kinetic energy lost due to the collision of A with B
c. the speed in terms of u acquired by C.
i) A train is uniformly retarded from ( 25)/█(3@@) ms-1 to 25/█(9@) ms-1 of the acceleration is halved the
magnitude of the retardation. The train takes 450 seconds to cover a total distance of 2km. By sketching a velocity time graph or otherwise find the total distance travel at 25/█(9@) ms-1,
ii) A particle moves along a straight line so that at time t seconds its velocity v in ms-1 and its
displacement, s metres, from a fixed point on the line are given by v = 1+2s2 /2 ms-1 and s = 3t + 2. Find the acceleration of the particle when s = 3
7. i) The non-gravitational resistance to the motion of the car of mass 1000kg is proportional to the squere of the speed of the car. When the engine is working at 60KW, the maximum speed of the car on level road is 25m-1. When the car ascends a plane inclined at an angle θ to the horizontal, with its engine still working at 60KW, the maximum speed is 20ms_1. Find, correct to 3 decimal places, the value of sin θ.
(Take g as 10ms-1)
ii) A Ship A travelling due North at 30Kmh-1 observes another ship B on its radar screen. Ship B is 5km due East and appears to be travelling 40kmh-1 on a bearing of 210°. Find the actual velocity of the ship B and the shortest distance between the two ships, given your answere correct to two decimal places.
8. i) A man usually travels to Yaounde from Douala either by air(A), by bus(B) or by train(T). When he travels by air, bus or train, the probability that he vyill have an accident and — respectively. Given that the man is travelling to Yaounde, find the probability that
he will be involved in an accident,
he was travelling by air given that he is involved in an accident,
he will arrive safely, given that he is travelling by bus.
ii) 10% of a large consignment of mangoes is known to be bad. Three mangoes are chosen at random
from this consignment that
all will be bad,
none will be bad,
at least one will be bad
