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2008. M
Coimisiún na Scrúduithe Stáit
State Examinations Commission
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LEAVING CERTIFICATE EXAMINATION, 2008
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APPLIED MATHEMATICS − HIGHER LEVEL
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FRIDAY, 20 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 ball is thrown vertically upwards with an initial velocity of 39.2 m/s.
Find (i) the time taken to reach the maximum height
(ii) the distance travelled in 5 seconds.
(b) Two particles P and Q, each having constant acceleration, are moving in the same direction along parallel lines. When P passes Q the speeds are 23 m/s and 5.5 m/s, respectively. Two minutes later Q passes P, and Q is then moving at 65.5 m/s.
Find (i) the acceleration of P and the acceleration of Q
(ii) the speed of P when Q overtakes it
(iii) the distance P is ahead of Q when they are moving with equal speeds.
2. (a) Two straight roads cross at right angles. A woman C, is walking towards the intersection with a uniform speed of 1.5 m/s. Another woman D is moving towards the intersection with a uniform speed of 2 m/s.
C is 100 m away from the intersection as D passes the intersection.
Find (i) the velocity of C relative D
(ii) the distance of C from the intersection when they are nearest together.
(b) On a particular day the velocity of the wind, in terms of i
r and j
r , is x i
r
r , where x ∈ N.
i
r and j
r are unit vectors in the directions East and North respectively.
To a man travelling due East the wind appears to come from a direction
North α° West where tan α = 2.
When he travels due North at the same speed as before, the wind appears to
come from a direction North β° West where
tan β =.
Find the actual direction of the wind.
C
D
5. (a) Three identical smooth spheres lie at rest on a smooth horizontal table with their centres in a straight line. The first sphere is given a speed 2 m/s and it collides directly with the second sphere. The second sphere then collides directly with the third sphere. The coefficient of restitution for each collision is e , where e < 1.
(i) Find, in terms of e , the speed of each sphere after two collisions have taken place. (ii) Show that there will be at least one more collision.
(b) A smooth sphere A moving with speed u , collides with an identical smooth sphere B which is at rest.
The direction of motion of A, before impact, makes an angle of 45° with the line of centres at the instant of impact.
The coefficient of restitution between the spheres is e.
Show that the direction of motion of A is deflected through an angle α where
e
e −
tan α.
6. (a) A particle of mass 5 kg is suspended from a fixed point by a light elastic string which hangs vertically. The elastic constant of the string is 500 N/m. The mass is pulled down a vertical distance of 20 cm from the equilibrium position and is then released from rest.
(i) Show that the particle moves with simple harmonic motion. (ii) Find the speed and acceleration of the mass 0.1 seconds after it is released from rest.
(b) A and B are two fixed pegs, A is 4 m vertically above B. A mass m kg, connected to A and B by two light inextensible strings of equal length, is describing a horizontal circle with uniform angular velocity ω.
For what value of ω will the tension in the upper string be double the tension in the lower string?
4 m
A
B
A B
u
7. (a) One end of a uniform ladder, of weight W , rests against a rough vertical wall, and the other end rests on rough horizontal ground. The coefficient of friction at the ground is 4
and at the wall is 2
The ladder makes an angle α with the horizontal and
is in a vertical plane which is perpendicular to the wall.
The ladder is on the point of slipping. Findtan α.
(b) Two equal uniform rods AB and BC smoothly jointed at B are in equilibrium with the end C resting on a rough horizontal surface. The end A is held above the surface.
The rod AB is horizontal and the rod BC is inclined at an angle of 45° to the horizontal.
If C is on the point of slipping find the coefficient of friction.
8. (a) Prove that the moment of inertia of a uniform circular disc, of mass m and
radius r , about an axis through its centre perpendicular to its plane is 2 2
m r.
(b) Masses of 4 kg and 6 kg are suspended from the ends of a light inextensible string which passes over a pulley. The axis of rotation of the pulley is horizontal, perpendicular to the pulley, and passes through the centre of the pulley. The moment of inertia of the pulley is 0.08 kg m^2 and its radius is 20 cm. The particles are released from rest and move vertically. When each mass has acquired a speed of 1 m/s, find
(i) the common acceleration of the masses
(ii) the tensions in the vertical portions of the string.
4 kg 6 kg
∟^ α
A B
C
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