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Maths Grinds
29 Nov 2018
##### Question 1

State the principle of the conservation of momentum.

A car weighing $2$ tons and moving at $60$ m.p.h. along a main road collides with a lorry of weight $10$ tons which emerges at $30$ m.p.h. from a crossroad at right angles to the main road. If the two vehicles lock, what will be their velocity after the collision?

##### Question 2

A helicopter flies from an aerodrome to a place due North, a distance $119$ kilometers from the aerodrome, and then returns. The speed of the helicopter in still air is $52$ meters/sec. and there is a wind of speed $20\sqrt{2}$ meters/sec. blowing from the North-East throughout the journey. Show that the actual speed of the helicopter on the outward journey is $28$ meters/sec. and calculate the total time for both journeys.

##### Question 3

(i) Two identical uniform rods $AB$ and $BC$ are rigidly connected at right angles at $B$. Find the centre of gravity of the compound body. The body is freely suspended from $A$. Find the inclination of $AB$ to the vertical.

(ii) A thin uniform wire is bent into the shape of a quadrilateral $PQRS$. The coordinates of the vertices are $P(3,0)$, $Q(8, 12)$, $R(5,16)$ and $S(0,4)$. Find the coordinates of the centre of gravity of the wire.

##### Question 4

A projectile is fired with an initial speed of $24$ ft./sec. to hit a target at a horizontal distance of $9$ ft. from the point of projection and at a vertical height of $4\frac{1}{2}$ ft. Find the two possible angles of projection and the ratio of the times of flight along the two paths. Find the speed of the projectile at impact in each case.

(Take $g$ to be $32$ ft./sec$^2$.).

##### Question 5

A train of mass $100$ tons and of constant $1120$ horse-power, is allowed to run with the engine turned off down a slope of inclination $\sin^{-1}\left(\frac{1}{30}\right)$ to the horizontal and reaches a maximum speed of $80$ m.p.h. Find the resistance to motion in tons weight. If the resistance to motion in all cases is directly proportional to the square of the speed, calculate the acceleration of the train on a level track when its speed is $60$ m.p.h.

(Take $g$ to be $32$ ft./sec$^2$.).

##### Question 6

A car $A$ moves along a straight road $PQ$ with constant acceleration of $5$ ft/sec$^2$. in the direction $PQ$ its velocity at $P$ is $10$ ft/sec. in the same direction. Three seconds after $A$ has left $P$ another car $B$ starts from $P$ with a velocity of $38$ ft/sec. and a uniform acceleration of $4$ ft/sec$^2$., both in the direction $PQ$. When and where will $B$ overtake $A$? Show that after passing $A$, $B$ will never be ahead by more that $32$ ft.

##### Question 7

A particle of mass $3$ lbs. is suspended from a fixed point $O$ by a light inelastic string of length $2$ feet. It is projected horizontally with speed $8$ ft./sec. from a point $2$ feet vertically below $O$. How high does it rise in the subsequent motion?

When it returns to $O$ it collides and coalesces with a stationary particle of mass $6$ lbs. How high does the combined mass rise in the subsequent motion?

Calculate the greatest tension in the string during the motion of the combined mass.

(Take $g$ to be $32$ ft./sec$^2$.).

##### Question 8

Define simple harmonic motion.

A particle is moving in a straight line with simple harmonic motion. When it is $5$ feet from the centre of its path its speed is $24$ ft./sec. and when it is $12$ feet away its speed is $10$ ft./sec. Find the period and amplitude of the motion. Find also the time taken for the particle to travel from the centre of its path to a point where its speed is half the maximum speed.

##### Question 9

Two solid uniform spheres each of radius $4$ cms. are connected by a light string and are completely immersed in a tank of water. The specific gravitates of the spheres are $\frac{1}{2}$ and $2\frac{1}{2}$ respectively. Find the tension in the string and the reaction between the bottom of the tank and heavier sphere.

##### 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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