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June 2008
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The position vector of a particle P, at time t seconds is r, where r = (ti +j + t2 k)m Show that the acceleration of the particle is constant.
Find the cosine of the angle between the velocity vector and the position 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 with speed 20√5 ms-1 from the top of a cliff so that its hits a target which is 200 m
horizontally from the foot of the cliff and 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 times of flight.
Using the angle of projection whose value lies the first quadrant, find the maximum height above the
point of projection attained by the particle, giving your answer correct to two decimal places.
(Takes g as 10ms-2)
3. Two forces, F1 and F2,where
F1 = (i + 2j + 3k)N and F2 = (2i + k)N
Act at points with position vectors (2i 4- 5j + ck)m and (Si + cj + 2k)m respectively.
Given that the lines of action of F1 and F2 intersect at 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.
A uniform bar AB of weight W newtons and length 2a meters rests with its upper end A against a
rough vertical wall and the lower end B on 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 µ = ½
A block of mass 3 kg rests on a smooth plane inclined at an angle 8 to the horizontal, where
sinθ= 2/3 . The block is connected by a light inextensible string parallel to a line of greatest slope
of the plane. The string passes over a smooth fixed pulley at the plane to another particle B of
mass 5 kg hanging freely. The system is released from the rest.
Find
The velocity of the system when B has descended through a distance of 2 m, 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)
5. 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 as to strike 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 C is ½ , find.
The impulse experienced by A in its collision with B,
The kinetic energy lost due to the collision of A with B,
The speed in terms of µ acquired by c .
6. A train is uniformly retarded from 25/3 ms-1 to 25/9 ms-1.The magnitude of the acceleration is half
the magnitude of the retardation. The train takes 450 seconds to cover a total distance of 2 km.
By sketching a velocity time graph or otherwise find the distance travelled at 25/9ms-1.
A particle moves along a straight line so that at time t seconds its velocity v ms-1 and its
displacement, s metres, from a fixed on the line are given by
v =( 1 + 2S2 /S) and s = 3t + 2
Find the acceleration of the particle when s = 3.
7.
i) The non-gravitational resistance to the motion of a car of mass 1000 kg is proportional to the
square of the speed of the car. When the engine is working at 60 kW, the maximum speed of the
car on a level road is 25ms-1. When the car ascends a plane inclined at an angle 6 to the
horizontal, with its engine still working at 60 kW, the maximum speed is 20ms-1. Find, correct to
3 decimal places, the value of sin θ.
(Take g as 10ms-2)
8. A ship A travelling due North at 30 Kmh~1 observes another ship B on its radar screen. Ship B is 5 km due East and appears to be travelling 40kmh-1 on a bearing of 210°. Find the actual
velocity of ship B and the shortest distance between the two ships, giving your answer correct to 2 decimal places.
A man usually travels to Yaounde from Douala by air (34), by bus (#) or by train (T). When he
travels by air, bus or train, the probability that he will have an accident is 1/5, 3/5, and 1/10 respectively.
Given that the man is travelling to Yaounde, find the probability that
Fie will be involved in accident
Fie was travelling by air given that he is involved in an accident,
Fie will arrive safety, given that he is travelling by bus
10% of a large consignment of mangoes is known to be bad. Three mangoes are chosen at
random from this consignment.
Calculate the probability that
i.All will be bad
ii.None will be had
iii.At least one will be
