Leaving Certificate Examination 1966 Honours Applied Mathematics
Question 1
(a) Represent any two velocities 
and 
by a vector diagram, and illustrate the vectors 
and 
.
(b) When the sun, S, is 
above the horizon, an aeroplane 
glides in for landing with the sun behind, moving with a speed of 
m.p.h. along a path sloping 
down from the horizontal, as in the diagram. Find the speed of the plane’s shadow on level ground (
).
Question 2
Write a short note on linear momentum.
A uniform cube of wood of mass 
lbs. hangs vertically by a long light string fixed in the centre of one face of the cube. A bullet of mass half an ounce moving horizontally with velocity 
strikes the centre of another face in a direction perpendicular to the face, embeds itself in the wood and causes the combined mass to swing. If in the first swing the centre of gravity of the combined mass rises 
inches, calculate the magnitude of .
Question 3
For a body moving in a straight line with constant acceleration 
, initial velocity 
and velocity at time 
, show that 
and 
.
A cage travels down a vertical mine-shaft 
yards deep in 
seconds, starting from rest. For the first quarter of the distance only the velocity undergoes a constant acceleration, and for the last quarter of the distance the velocity undergoes a constant retardation, the acceleration and retardation having equal magnitudes. If the velocity, , is uniform while traversing the central portion of the shaft, find the magnitude of .
Question 4
The co-ordinates 
of a moving particle at any time are given by:
, 
.
Deduce that its motion is circular.
A particle is moving along the circumference of a circle with constant speed. Show that that components of the motion of the particle along any two perpendicular diameters are Simple Harmonic Motions.
Question 5
A particle is projected horizontally from a height with velocity and falls under gravity. If are the co-ordinates of the particle at any time, and 
the co-ordinates of the point of projection, show that 
us the equation of its path, the x-axis being horizontal and the y-axis vertical.
A marble rolls off the top of a stairway horizontally and in a direction perpendicular to the edge of each step, with a speed of 
feet per second. Each step is in. high and 
in. wide. Which is the first step struck by the marble (assuming the stairway is sufficiently high)?
Question 6
Masses of 
lb., 
lbs., 
lbs., and 
lbs. are placed at points the co-ordinate of which are 
, 
, 
, and 
respectively. Locate the centre of gravity (or the centre of mass) of the system.
Two discs of radii 
and 
are placed flat on a horizontal plane with a distance 
between their centres. If 
is their centre of gravity and 
the centre of gravity of the large disc calculate the length of 
.
A square hole is punched out of a circular disc (radius 
), a diagonal of the square being a radius of the circle. Show that the centre of gravity of the remainder of the disc is at a distance 
from the centre of the circle.
Question 7
Explain “normal reaction” and “angle of friction”.
Deduce 
where 
is the angle of friction and 
the angle between the normal reaction and the reaction (resultant reaction).
A uniform rod 
feet long leans against the smooth edge of a table three feet in height with one end of the rod on a rough horizontal floor (as in diagram). If the rod is one the point of slipping when inclined at an angle of 
to the horizontal, find the coefficient of friction.
Question 8
Two particles 
and 
move with velocities and respectively. Use diagrams to illustrate clearly the velocity of relative to when and are
(i) parallel and in the same direction,
(ii) parallel and in opposite directions,
(iii) non-parallel.
A steamer going north-east at 
knots observes a cruiser which is miles south-east from the steamer and which is travelling in a direction east of north at 
knots.
Illustrate by diagram the cruiser’s course as i appears to the steamer. Find the velocity of the cruiser relative to the steamer, in magnitude and direction, and find also the shortest distance between the two ships.
Question 9
Establish an expression for the pressure at a depth 
in a liquid of density 
at rest under gravity. Hence show that the resultant thrust on a plane surface immersed vertically in the liquid is the product of the area of the plane surface and the depth of its centre of gravity.
Find the resultant horizontal thrust on a rectangular lockgate feet wide when the water stands at a depth of feet on one side and feet on the other.
(1 cu. ft. of water weighs 
lbs.).
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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