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Coimisiún na Scrúduithe Stáit
State Examinations Commission
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LEAVING CERTIFICATE EXAMINATION, 2006
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APPLIED MATHEMATICS − HIGHER LEVEL
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FRIDAY, 23 JUNE − AFTERNOON, 2.00 to 4.
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Six questions to be answered. All questions carry equal marks. Mathematics Tables may be obtained from the Superintendent. Take the value of g to be 9.8 m/s 2. Marks may be lost if necessary work is not clearly shown.
1. (a) A lift starts from rest. For the first part of its descent it travels with uniform acceleration f. It then travels with uniform retardation 3 f and comes to rest. The total distance travelled is d and the total time taken is t.
(i) Draw a speed-time graph for the motion.
(ii) Find d in terms of f and t.
(b) Two trains P and Q, each of length 79.5 m, moving in opposite directions along parallel lines, meet at o , when their speeds are 15 m/s and 10 m/s respectively. The acceleration of P is 0.3 m/s^2 and the acceleration of Q is 0.2 m/s^2. It takes the trains t seconds to pass each other.
(i) Find the distance travelled by each train in t seconds.
(ii) Hence, or otherwise, calculate the value of t.
(iii) How long does it take for 5
of the length of train Q to pass the
point o?
2. (a) Two aeroplanes A and B, moving horizontally, are travelling at 200 km/h relative to the ground. There is a wind blowing from the east at 60 km/h. The actual directions of flight of A and B are north-west and north-east respectively.
Find (i) the speed of aeroplane A in still air
(ii) the magnitude and direction of the velocity of A relative to B.
(b) A boy swims due west at a speed of 0.8 m/s.
A girl swims at 0.4 m/s in the direction
α °south of west.
At a certain instant the girl is 10 m, 60 ° north of west of the boy and 10 s later she is due north of the boy.
(i) Find the distance travelled by the boy and the girl in 10 s.
(ii) Hence, or otherwise, find the value of α.
(iii) Find the shortest distance between the boy and the girl in the subsequent motion.
α°
0.4 m/s
0.8 m/s
10 m
5. (a) A smooth sphere P, of mass 3 kg, moving with speed 6 m/s, collides directly with a smooth sphere Q, of mass 5 kg, which is moving in the same direction with speed 2 m/s. The coefficient of restitution for the collision is e.
(i) Find, in terms of e , the speed of each sphere after the collision. (ii) If the loss of kinetic energy due to the collision is k ( 1 − e^2 ), find the value of k.
(b) A smooth sphere A moving with speed u , collides with an identical smooth sphere B which is moving in a perpendicular direction with the same speed u.
The line of centres at the instant of impact is perpendicular to the direction of motion of sphere B.
The coefficient of restitution between the spheres is e.
(i) Find, in terms of e , the speed of each sphere after impact and hence, or otherwise, show that it is not possible for the two spheres to have the same speed after impact.
(ii) Prove that (^) ⎟ ⎠
tan
e
θ , where θ is the angle through which
sphere B is turned as a result of the impact.
6. (a) A particle moves with simple harmonic motion of period 3 π. At time t = 0,
the particle passes through the centre of the oscillation. It passes through a point distant 4 m from the centre of motion with a speed of 5 m/s away from the centre. Find, correct to two decimal places, (i) the maximum acceleration of the particle (ii) the time which elapses before it next passes through this point.
(b) A hollow cone with its vertex downwards and its axis vertical, revolves about its axis with a constant
angular velocity of 4 π rad/s.
A particle of mass m is placed on the inside rough surface of the cone. The particle remains at rest relative to the cone.
The coefficient of friction between the particle and
the cone is 4
The semi-vertical angle of the cone is 30 ° and the particle is a distance l m from the vertex of the cone.
Find the maximum value of l , correct to two places of decimals.
B
A
u
u
l
7. (a) A uniform rod [ pq ] of length 4 m is free to turn in a vertical plane about a hinge at p.
The mass of the rod is 20 kg.
The rod is supported in a horizontal position by a rope attached to q and to a point 3 m vertically above p.
Find (i) the tension in the rope
(ii) the magnitude and direction of the reaction at the hinge.
(b) One end of a uniform ladder rests on a rough horizontal floor and the other end rests against a rough vertical wall.
The coefficient of friction at each contact is μ.
The ladder makes an acute angle α with the wall.
If the ladder is on the point of slipping find α in terms of λ , the angle of
friction, where μ = tan λ.
8. (a) Prove that the moment of inertia of a uniform rod of mass m and length 2 l
about an axis through its centre perpendicular to the rod is 2 3
m l.
(b) A uniform rod of mass 3 m and length 1.2 metres can turn freely in a vertical plane about a horizontal axis through one end.
The rod oscillates through an angle of 120 °, as shown in the diagram.
(i) Find the angular velocity of the rod when the rod is vertical.
(ii) Find, in terms of m , the vertical thrust on the axis when the rod is vertical.
p (^) q
120 °
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