# Leaving Cert Applied Maths Higher Level 1979

## Question 1

i) How may a velocity-time graph be used to find the distance travelled in a given time?

ii) An athlete runs $100$ m in $12$ seconds. Starting from rest, he accelerates uniformly to a speed of $10$ m/s. and then continues at that speed. Calculate the acceleration.

iii) A body starting from rest travels in a straight line, first with uniform acceleration $a$ and then with uniform deceleration $b$. It comes to rest when it has covered a total distance $l$. If the overall time for journey is $T$, show that

$$T^2 = 2l \left(\frac{1}{a}+\frac{1}{b}\right)$$

## Question 2

A ship $A$ is travelling South-West at $10$ knots. Another ship $B$ is travelling at $20$ knots in a direction $30^\circ$ North of West.

i) Draw a diagram to show the velocity of $B$ relative to $A$.

ii) Calculate the magnitude of the relative velocity, correct to the nearest knot, and its direction correct to the nearest degree.

iii) By how much should $A$ increase its speed, without changing direction, so that $B$ would appear to $A$ to be travelling due North?

## Question 3

a) Explain the terms: coefficient of friction, angle of friction. What is the relationship between them?

b) A uniform ladder of weight $W$ rests with one end against a smooth vertical wall and the other end on rough ground which slopes away from the wall at an angle $\alpha$ to the horizontal (seed diagram). The ladder makes an angle $\theta$ with the wall.

i) Show that the reaction at the wall is $\frac{1}{2}W\tan\theta$.

ii) If the ladder is on the point of slipping prove that

$$\tan\theta = 2 \tan \left(\lambda-\alpha\right)$$

where $\lambda$ is the angle of friction.

## Question 4

a) A see-saw consist of a uniform plank freely pivoted at its mid-point on a support of vertical height $h$. It carries a mass $m$ at one end and a mass $2m$ at the other. The see-saw is released from rest with the mass $2m$ at its highest point. Find, in terms of $h$, the velocity with which the see-saw strikes the ground.

b) State the Principle of Archimedes.

c) A uniform road of length $l$ and relative density $s$ is freely hinged to the base of a tank containing a liquid to a depth $h$. The relative density of the liquid is $k$. As a result, the rod is inclined but not fully submerged. Derive an expression for the angle the rod makes with the horizontal.

## Question 5

A plane is inclined at an angle $\alpha$ to the horizontal. A particle is projected up the plane with a velocity $u$ at an angle $\theta$ to the plane. The plane of projection is vertical and contains the line of greatest slope.

i) Show that the time of flight is $\frac{2u\sin\theta}{g\cos\alpha}$.

ii) Prove that the range up the plane is a maximum when $\theta=\frac{1}{2}\left(\frac{\pi}{2}-\alpha\right)$.

iii) Prove that the particle will strike the plane horizontally if $\tan\theta = \frac{\sin\alpha\cos\alpha}{2-\cos^2\alpha}$.

## Question 6

a) A particle, moving at constant speed, is describing a horizontal circle on the inside surface of a smooth sphere of radius $r$. The centre of the circle is a distance $\frac{1}{2}r$ below the centre of the sphere. Prove that the speed of the particle is $\frac{1}{2}\sqrt{6gr}$.

b) A conical pendulum consists of a light elastic string with a mass $m$ attached to it which is rotating with uniform angular velocity $\omega$. The natural length of the string is $l$ and its elastic constant is $k$, i.e. a force $k$ produces unit extension. The extended length of the string is $l’$ and it makes an angle $\theta$ with the vertical. Prove that $k\left(l’-l\right)=ml’\omega^2$ and that

$$\cos\theta = \frac{g}{l\omega^2} – \frac{mg}{kl}$$

## Question 7

a) Two smooth spheres of masses $m$ and $2m$ collide directly when moving in opposite directions with speeds $u$ and $v$, respectively. The sphere of mass $2m$ is brought to rest by the impact. Prove that

$$e = \frac{2v-u}{u+v}$$

where $e$ is the coefficient of restitution.

b) A smooth sphere $A$ collides obliquely with another smooth sphere of equal mass which is at rest. Before impact the direction of motion of $A$ makes an angle $\alpha$ with the line of centres at impact (see diagram). After impact it makes an angle $\beta$ with that line. If the coefficient of restitution is $\frac{1}{2}$, prove that

$$\tan\beta = 4 \tan \alpha $$

## Question 8

a) Define simple harmonic motion. Using the usual notation, show that the equation $v^2 = \omega^2 \left(a^2 – x^2\right)$ represents simple harmonic motion.

b) A body is moving with simple harmonic motion, of amplitude $5$ m. When it is $4$ m from the mid-point of its path its speed is $6$ m/s. Find its speed when it is $2.5$ m from the mid-point.

c) A block rests on a rough platform which moves to and fro horizontally with simple harmonic motion. The amplitude is $0.75$ m and $20$ complete oscillations occur per minute. If the block remains at rest relative to the platform throughout the motion, find the least possible value the coefficient of friction can have.

## Question 9

a) State the theorem of parallel axes.

b) Prove that the moment of inertia of a uniform rod of mass $m$ and length $2l$ about a perpendicular axis through one of its end is $\frac{4}{3} ml^2$.

c) A uniform rod of length $6l$ is attached to the rim of a uniform disc of diameter $2l$. The rod is colinear with a diameter of the disc (see diagram). The disc and the rod are both of mass $m$.

i) Calculate the moment of inertia of the compound body about a perpendicular axis through the end $A$.

ii) If the compound body makes small oscillations in a vertical plane about a horizontal axis through $A$, show that the periodic time is $2\pi\sqrt{\frac{123l}{20g}}$.

## Question 10

a) Solve the differential equation

$$ x \frac{dy}{dx} = – y .$$

Hence or otherwise solve

$$x\frac{d^2y}{dx^2}=-\frac{dy}{dx}$$

where $y=0$ when $x=1$ and $y=3$ when $x=e$.

b) A body is moving in a straight line subject to a deceleration which is equal to $\frac{v^2}{10}$, where $v$ is the velocity. The initial velocity is $5$ m/s. In how many seconds will the velocity of the body be $2$ m/s and how far will it travel in that time?

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### Citation:

State Examinations Commission (2023). *State Examination Commission*. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2023). *Archive of Maths State Exams Papers*. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/