Leaving Certificate Examination 1970 Honours Applied Mathematics
Question 1
A bullet of mass 
is fired with speed 
into a fixed block of wood and is brought to rest in a distance 
. Find the resistance to motion assuming it to be constant.
Another bullet also of mass is then fired with speed 
into another fixed block of thickness 
, which offers the same resistance as the first block. Find the speed with which the bullet emerges, and the time it takes to pass through the block.
Question 2
Prove that the formula 
represents the distance 
travelled in time 
by a body moving in a straight line with constant acceleration 
.
A train takes 
minutes to travel between two stations 
and 
which are 
meters apart. It starts from rest at and finishes at rest at , by travelling with uniform acceleration for the first minute and with uniform deceleration for the last 
minute. Find the train’s constant speed during the remainder of the journey.
If a second train, travelling with a constant speed of 
m/min, in the same direction passes as the first train leaves this station, find when overtaking occurs. (The lengths of the trains may be neglected).
Question 3
A particle is projected under gravity with an initial velocity 
at an angle 
to the horizontal. Find its position and the direction of motion after time 
in terms of in terms of , , 
and .
A particle is projected from the top of a cliff which is 
ft. above sea level and the angle of projection is 
to the horizontal. If the greatest height reached above the point of projection is 
ft, find the speed of projection and the time taken to reach this greatest height.
Find when and where the particles strikes the sea.
(Take to be 
ft/sec
).
Question 4
Prove that the bob of a simple pendulum moves in simple harmonic motion – stating any assumptions made.
The string of such a pendulum is 
ft. long and the bob is released from rest when at a distance 
ft from the equilibrium position. Calculate the time taken to travel halfway to the equilibrium position and the speed of the bob then.
(Take to be ft/sec).
Question 5
By deriving an expression for the necessary acceleration, prove that a particle of mass moving in a circle of radius 
with speed must have a force of magnitude 
pointing towards the centre acting on it.
A particle of mass 
lbs. moving on the inside smooth surface of a fixed spherical bowl of radius ft is describing a horizontal circle of radius 
ft. Find the constant speed of rotation and the reaction of the sphere on the particle.
(Take to be ft/sec).
Question 6
Show that the centre of gravity of a uniform triangular lamina coincides with the centre of gravity of three equal particles placed at the vertices of the triangle. Hence find the centre of gravity of a uniform trapezium 
, of weight 
, in which 
and 
.
A particle of weight 
is attached at 
and the system is suspended by a string attached to the midpoint 
of 
. If in the position of equilibrium is vertically above 
show that 
.
Question 7
A particle of mass 
lbs. is placed on a rough inclined plane. The least force acting up the plane which will prevent the particle slipping down the plane is lbs. weight. The least force acting up the plane which will make the particle slip upwards is lbs wt. Show that the coefficient of friction is and that the inclination of the plane is 
where 
.
Find the least force required to move the particle up the plane.
Question 8
Two equal uniform rods 
and 
each of weight are freely jointed at 
. The system is suspended freely from and a horizontal force 
is applied at the lowest point 
. If in the equilibrium position the inclination of to the downward vertical is 
, find the corresponding inclination of and the supporting force at .
Question 9
A small uniform cylinder of density 
, mass , total length 
and uniform cross section floats in a liquid of density 
with its axis vertical. Find the thrust on the cylinder when it is displaced vertically in the liquid, without being completely immersed, through a distance from the equilibrium position. Show that if it is released in this position, it will oscillate with simple harmonic motion of period 
.
Citation:
State Examinations Commission (2018). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
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