# Leaving Certificate Examination 1937 Honours Applied Mathematics

##### Question 1

A body of mass $10$ units moves in a straight line so that its distance $s$ from a fixed point in the line of motion at time $t$ is given by the equation : $s=a+bt+ct^2$.

Derive formulae for its velocity, acceleration and momentum at any time.

Given that when $t=0$ the velocity, acceleration and momentum at any time.

Given that when $t=0$ the velocity in $2$ units and that the distances travelled when $t=1$, and $t=2$ are $9.5$ and $23.5$ respectively, determine the values $a$, $b$ and $c$ in the above equation.

##### Question 2

A mass weighing $\frac{1}{4}$ ton drops without rebound from a height of $16$ ft. on to the top of a pile weighing $12$ cwt., and the pile is driven $8$ inches into the ground by the blow.

Calculate in ft.lbs. the energy lost in the impact, and the average resistance offered by the ground to the motion of the pile.

##### Question 3

Calculate the effective h.p. delivered by the engine of a $2$ ton motor vehicle in each of the following cases :-

(a) The vehicle is ascending an incline of $1$ in $28$ with a uniform speed of $30$ miles per hour;

(b) the vehicle accelerates uniformly from a speed of $45$ miles per hour to a speed of $60$ miles per hour while travelling a distance of $550$ yards along a level road.

The frictional resistances to motion may be taken as the same in both cases and equal to a force of $60$ lbs. wt.

##### Question 4

State the laws of statical friction.

A uniform ladder $26$ feet long and weighing $96$ lbs. is on the point of slipping when placed with its upper end against a smooth vertical wall and its lower end on a rough horizontal surface at a distance of $10$ feet from the wall.

Calculate the value of the frictional force between the foot of the ladder and the ground and also the value of the coefficient of friction.

##### Question 5

A point $A$ describes a circular path of radius $r$ with a uniform velocity. $P$ is the foot of the perpendicular from $A$ to a fixed diameter of the circle. Show that the velocity and acceleration of $P$ at any instant are equal to $rw\cos wt$ and $-w^2x$ respectively, where $x$ is the displacement of $P$ fron [sic] the centre of the circle, $t$ is the time for this displacement and $w$ is the angular velocity of $A$.

(b) Show that for small oscillations the bob of a simple pendulum may be regarded as having a similar motion of that of $P$ and that the time of oscillation is $2\pi\sqrt{\frac{l}{g}}$ where $l$ is the length of the pendulum.

##### Question 6

Through what vertical height will the bob of a conical pendulum drop if its speed falls from $60$ to $30$ revolutions per minute?

##### Question 7

A train is travelling at the rate of $45$ miles per hour round a circular track of $660$ yards radius. The horizontal distance between the rails is $64$ inches. How much must the outer rail be raised [above the inner one so that there may be no side thrust on the rails ?

##### Question 8

The force acting on a body moving in a straight line is given at any instant by the formula $$F=\frac{320}{x^2}$$ where $F$ is in lbs.-st. and $x$ is the distance in feet of the body at that instant from a fixed point in the line. If the motion is resisted by a constant force of $3$ lbs.-wt., find the kinetic energy stored in the body in moving from $x=10$ to $x=5$.

##### Question 9

A particle of mass $5$ lbs. is placed on a smooth plane inclined at $30^\circ$ to the horizon and is connected by a light string passing over the top of the plane to a particle weighing $3$ lbs. which hangs vertically.

Find (a) the common acceleration;

(b) the velocity after $3$ seconds of the centre of inertia of the particles.

##### Question 10

A body is projected up a smooth incline of $30^\circ$ with an initial velocity of $96$ feet per second. If the length of the incline is $160$ feet find the greatest height attained by the body and the range on a horizontal plane passing through the foot of the incline.

**Citation:**

**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/

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