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M
Coimisiún na Scrúduithe Stáit
State Examinations Commission
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LEAVING CERTIFICATE EXAMINATION, 2005
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APPLIED MATHEMATICS − HIGHER LEVEL
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FRIDAY, 24 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) Car A and car B travel in the same direction along a horizontal straight road. Each car is travelling at a uniform speed of 20 m/s. Car A is at a distance of d metres in front of car B. At a certain instant car A starts to brake with a constant retardation of 6 m/s^2. 0.5 s later car B starts to brake with a constant retardation of 3 m/s 2.
Find (i) the distance travelled by car A before it comes to rest
(ii) the minimum value of d for car B not to collide with car A.
(b) A mass of 8 kg falls freely from rest. After 5 s the mass penetrates sand. The sand offers a constant resistance and brings the mass to rest in 0.01 s.
Find (i) the constant resistance of the sand
(ii) the distance the mass penetrates into the sand.
2. (a) A woman can swim at u m/s in still water. She swims across a river of width d metres. The river flows with a constant speed of v m/s parallel to the straight banks, where v < u. Crossing the river in the shortest time takes the woman 10 seconds.
Find, in terms of u and v , the time it takes the woman to cross the river by the shortest path.
(b) Two straight roads intersect at an angle of 45°. Car A is moving towards the intersection with a uniform speed of p m/s. Car B is moving towards the intersection with a uniform speed of 8 m/s. The velocity of car A relative to car B is i j
r r − 2 − 10 , where i j
r r and are unit perpendicular vectors in the east and north directions, respectively. At a certain instant car A is 220 2 m from the intersection and car B is 136 m from the intersection.
(i) Find the value of p.
(ii) How far is car A from the intersection at the instant when the cars are nearest to each other? Give your answer correct to the nearest metre.
B
A
5. (a) Three identical smooth spheres P, Q and R, lie at rest on a smooth horizontal table with their centres in a straight line. Q is between P and R. Sphere P is projected towards Q with speed 2 m/s. Sphere P collides directly with Q and then Q collides directly with R.
The coefficient of restitution for all of the collisions is
Show that P strikes Q a second time.
(b) A smooth sphere A, of mass m , moving with speed u , collides with an identical smooth sphere B moving with speed u. The direction of motion of A, before impact, makes an angle 45 ° with the line of centres at impact. The direction of motion of B, before impact, makes an angle 45 ° with the line of centres at impact. The coefficient of restitution between the spheres is e.
(i) Find, in terms of e and u , the speed of each sphere after the collision.
(ii) If 2
e = , show that after the collision the angle between the directions
of motion of the two spheres is (^)
tan 1.
6. (a) A conical pendulum consists of a light inelastic string [ pq ], fixed at the end p, with a particle attached to the other end q. The particle moves uniformly in a horizontal circle whose centre o is vertically below p. If po = h , find the period of the motion in terms of h.
(b) A light elastic string of natural length a and elastic constant k is fixed at one end to a point o on a smooth horizontal table. A particle of mass m is attached to the other end of the string. Initially the particle is held at rest on the table at a distance 2 a from o , and is then released.
Show that the time taken for the particle to reach o is 1 2
m k
45 ° ˚^45 ° ˚ u u
A B
p
o q
h
7. (a) A particle of weight 100 N lies on a plane. The plane is inclined at 30° to the horizontal. A horizontal force F is applied to the particle. The coefficient of friction between the particle
and the inclined plane is 5
Find the least value of F that will move the particle up the plane.
(b) Two uniform rods, [ ab ] and [ bc ], of equal length, are smoothly jointed at b. They rest in a vertical plane with a and c on
rough horizontal ground and |∠ abc | =^2 θ^.
The weight of the rod [ ab ] is 2 W and the weight of the rod [ bc ] is W.
The coefficient of friction at both a and c is μ.
Find the least value of μ , in terms of θ , necessary for equilibrium.
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
ml.
(b) A uniform rod [ pq ], of mass 9 m and length 2 l , has a particle of mass 2 m attached at q. The system is free to rotate about a smooth horizontal axis through p. The rod is held in a horizontal position and is then given an
initial angular velocity
g l
downwards.
The diagram shows the rod [ pq ] when it makes
an angle θ with the horizontal.
(i) Show that when the rod makes an angle θ below its initial horizontal
position, its angular velocity is ( 15 13sin ) 10
g l
(ii) Hence, or otherwise, show that the rod performs complete revolutions about p.
p
a
F
c
b
2 θ
q
