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M
Coimisiún na Scrúduithe Stáit
State Examinations Commission
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LEAVING CERTIFICATE EXAMINATION, 2003
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APPLIED MATHEMATICS − ORDINARY 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 10 m/s 2.
i
r
and j
r
are unit perpendicular vectors in the horizontal and vertical directions, respectively,
or eastwards and northwards, respectively, as appropriate to the question.
Marks may be lost if necessary work is not clearly shown or you do not indicate where a
calculator has been used.
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1. A car travels from p to q on a straight level road. It passes p with a speed of 4 m/s and accelerates uniformly to its maximum speed of 8 m/s in 4 seconds. The car maintains this speed of 8 m/s for 6 seconds before decelerating uniformly to rest at q. The car takes 12 seconds to travel from p to q.
(i) Draw a speed-time graph of the motion of the car from p to q. (ii) Find the uniform acceleration of the car. (iii) Find the uniform deceleration of the car. (iv) Find pq , the distance from p to q.
Another car travels the same distance from p to q in the same time of 12 seconds. This car starts from rest at p and accelerates uniformly to its maximum speed of v m/s and then immediately decelerates uniformly to rest at q.
(v) Find v , the maximum speed of this car, giving your answer as a fraction.
2. The velocity of ship A is i j
r r
3 − 4 m/s and the velocity of ship B is i j
r r
− 2 + 8 m/s.
(i) Find the velocity of ship A relative to ship B in terms of i
r
and j
r
(ii) Find the magnitude and direction of the velocity of ship A relative to ship B, giving the direction to the nearest degree.
At a certain instant, ship B is 26 km due east of ship A.
(iii) Show, on a diagram, the positions of ship A and ship B at this instant and show, also, the direction in which ship A is travelling relative to ship B.
(iv) Calculate the shortest distance between the ships, to the nearest km.
3. A particle is projected from a point p on level horizontal ground with an initial speed of 50 m/s at an angle β to the horizontal,
where tan β =
(i) Find the initial velocity of the particle in terms of i
r
and j
r
After 4 seconds in flight, the particle hits a target which is above the ground.
(ii) Show that the distance from the point p to the target is 40 17 m.
(iii) How far below the highest point reached by the particle is the target?
(iv) Find, correct to the nearest m/s, the speed with which the particle hits the target.
6. (a) Particles of weight 2 N, 3 N, 4 N and 1 N are placed at the points (−2, 1), (−1, −1), (2, 2) and ( x , y ), respectively. The centre of gravity of the four particles is at the origin.
Find the value of x and the value of y.
(b) Two uniform rods, [ rp ] and [ rq ], are rigidly jointed at r. The rods are of equal length. Each rod has a mass of M kg. | po | = | oq | = | or | = 0.2 m.
(i) Give a reason why the centre of gravity of the two rods lies on the line or.
(ii) Find the distance of the centre of gravity of the two rods from o.
The diagram shows an optician's advertising sign.
The sign consists of the two rods, [ rp ] and [ rq ], described above, now rigidly jointed at p and q to two uniform discs, each of radius 0.2 m and mass M kg.
ab is a horizontal line going through the centre of each disc and the points p and q. The distance from r to ab is 0.2 m.
(iii) Find the distance of the centre of gravity of the sign from the line ab.
7. A uniform beam, [ ab ], of mass 20 kg and length 8 m, is placed with its end a on rough horizontal ground and end b against a rough vertical wall.
The coefficient of friction at a is μ and at b is also μ.
The beam is on the point of slipping when inclined at an angle of 45° to the horizontal.
(i) Show on a diagram all the forces acting on the beam.
(ii) Write down the two equations that arise from resolving the forces horizontally and vertically.
(iii) Write down the equation that arises from taking moments about the point a.
(iv) Use the three equations from parts (ii) and (iii) to show that
8. (a) A vehicle of mass 1000 kg rounds a bend which is in the shape of an arc of a circle of radius 25 m. The coefficient of friction between the tyres and the road is 0.8.
(i) Show on a diagram the three forces acting on the vehicle. (ii) Calculate the maximum speed with which the vehicle can round the bend without slipping. Give your answer correct to two places of decimals.
(b) A smooth particle, of mass 2 kg, describes a horizontal circle of radius 0.5 metres on a smooth horizontal table with constant angular velocity 3 radians per second. The particle is connected by means of a light inelastic string to a fixed point o which is vertically above the centre of the circle. The length of the string is 1 metre. The inclination of the string to the vertical is α.
(i) Find α. (ii) Find the tension in the string. (iii) Show that the normal reaction between the particle and the table
is 20 − 9 3 N.
9. A solid sphere of volume V m^3 has a relative density of 1.2. The sphere is immersed in water in a tank. The sphere rests on the bottom of the tank.
(i) Show, on a diagram, all the forces acting on the sphere. (ii) Find, in terms of V , the normal reaction between the bottom of the tank and the sphere.
The sphere is now taken out of the tank of water and placed in a tank of liquid whose relative density is s , where s > 1.2. The sphere is held immersed in the liquid by a light inelastic string tied to the sphere and to the bottom of the tank.
(iii) Explain why the sphere must be tied by a string to the bottom of the tank, so as to remain immersed in the liquid.
(iv) Find the value of s , given that the tension in the string is 1000 V newtons.
[Density of water = 1000 kg/m^3 .]
