Leaving Cert Applied Maths Higher Level 1974
Special Thanks: A copy of this Leaving Cert Applied Maths Higher Level 1974 exam was kindly provided by Noel Cunningham.
Question 1
A sprinter runs a race with constant acceleration throughout. During the race he passes four posts
,
,
,
in a straight line such that
m. If the sprinter takes
seconds to run from
to
and
seconds to run from
to
, find how long, to the nearest tenth of a second, it takes him to run from
to
.
Question 2
Two straight roads cross at right angles at . On one road a car is travelling due north at
m/s. As the car passes through
a bus is
m from
and is travelling east towards
on the other road at
m/s. Find the velocity of the bus relative to the car and show the relative path on a diagram containing the initial positions of the car and bus. Calculate the least distance between the car and the bus in the subsequent motion and find when this occurs.
Question 3
is the cross-section of a smooth heavy wedge of mass
where
and
is a right angle. The wedge rests with its face
on a smooth horizontal table. A light inelastic string passes over a small smooth pulley fixed at the vertex
of the section of the wedge and supports particles of masses
, lying on the inclined plane
and
, hanging freely against the vertical face
. When the system is released from rest, the wedge moves with an acceleration
parallel to
and the mass
moves with an acceleration
relative to the wedge.
Show on separate diagrams the forces acting on each particle and on the wedge, in particular the action of the string on the wedge. Write down the five equations of motion of the particles and the wedge involving and
.
Question 4
a) Define Simple Harmonic Motion in a straight line and show that if the magnitude of the displacement from the equilibrium position after time is given by
, where
,
and
are constants, then the motion is simple harmonic.
b) A light flexible elastic string , of natural length
m and elastic constant
N/m, has one end
tied to a fixed point and has a particle of mass
kg attached to the end
. The particle is held
m below
and is then released from rest to fall under gravity.
i) By considering the forces acting on the particle when it has fallen a distance m, show that it moves with simple harmonic motion and that its acceleration is zero when
.
ii) Find the time taken to fall a distance 0.3m.
Question 5
a) Show that , where
is a constant, is the position vector of a particle moving in a circle of radius
. Prove that the velocity
is of magnitude
at right angles to
where
. Find the components of the acceleration
.
b) A particle of mass kg at
, is attached by two light inelastic strings
and
, each of length
m, to a fixed point
and to a ring of mass
kg at
, which is free to slide on a smooth fixed vertical wire
. The particle and strings rotate about
with constant angular speed
rad/s. Show on separate diagrams the forces acting on the particle and on the ring and calculate the tensions in the two strings.
Question 6
Two uniform heavy rods and
, each of length
and weight
, are freely joined together at
and hang freely from a fixed pivot at
. A force
acting in a horizontal direction is applied at
to the rod
and an equilibrium position is reached when
makes an angle of
with the vertical.
i) In separate diagrams show the forces acting on and on
.
ii) Find the horizontal and vertical components of the action in the hinge and show that
[sic].
iii) Prove that the magnitude of the reaction at is
.
iv) If makes an angle
with the vertical, prove that
.
Question 7
From a point on a plane, inclined at
to the horizontal, a particle is projected with speed
at
to the plane. The motion takes place in a vertical plane through a line of greatest slope up the plane from
.
i) Express the velocity and displacement
from
of the particle after time
in terms of
and
, where
and
are unit vectors along and perpendicular to the plane, respectively.
ii) Prove that the particle strikes the plane at and that the range on the inclined plane is
.
Question 8
a) State the laws governing the oblique collisions between smooth elastic spheres.
b) Two smooth spheres and
of masses
and
, respectively, collide obliquely and the coefficient of restitution for the collision is
. The velocity of
before impact is
and the velocity of
before impact is
, where
points along the line of centres at impact. Find the velocities of the spheres after the impact and show that the loss in kinetic energy is
.
Question 9
a) Prove that the moment of inertia of a uniform solid sphere of radius and of mass
about a diameter is
.
b) A symmetrical dumb-bell consists of two spheres joined by a narrow uniform rigid bar of mass and of length
so that the centres of the spheres are at a distance
apart. If the dumb-bell is freely pivoted at a point of the bar distance
from its centre so that it can perform small oscillations in a vertical plane,
i) prove that the moment of inertia of the body about the axis of rotation is and
ii) find the period of oscillation.
Question 10
A particle of mass kg falls vertically from rest under gravity in a medium which exerts a resisting force of magnitude
Newtons when the speed of the particle is
m/s.
i) Show that and find
.
iii) Find an expression for the distance travelled in time seconds.
Note:
I believe Question 6 iii) should read “Prove that the magnitude of the reaction at is
.”
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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/