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20Dec2018

Leaving Cert Applied Maths Higher Level 1973

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Question 1

A cyclist has a maximum acceleration of 2 m/s^2, a maximum speed of 15 m/s and a maximum deceleration of 4 m/s^2. The cyclsit wishes to travel a distance s from rest to rest in the shorest time. Find the time taken in the two cases

(i) s=105 m and

(ii) s=54 m.

Draw a rough velocity-time graph for each case and explain why 84\frac{3}{8} m is a critical distance.

[Video Solution]

Question 2

A perfectly elastic particle falls vertically with speed u on to a smooth plane inclined at an angle \alpha to the horizontal, and rebounds, hopping down the plane.

i) Write down its displacement \vec{r} from the point of contact after time t in terms of \vec{i} and \vec{j}, where \vec{i} is drawn directly down the plane.

ii) Show that the length of the first hop is \frac{4u^2 \sin \alpha}{g} and that

iii) the length of the second hop is double this.

[Video Solution]

Question 3

A ship A is steaming with velocity v m/s where \vec{v}=5\sqrt{3}\vec{i}+5\vec{j} where \vec{i} and \vec{j} are point East and North, respectively. At midday a second ship B has a position 10\vec{i} km with respect to A.

i) Find the minimum speed B must have if it is to intercept A.

ii) If the maximum speed of B is in fact 6 m/s, show that it can steer in either of two directions to intercept A and

iii) calculate the two times of interception.

[Video Solution]

Question 4

The diagram shows a light inelastic string with one end connected to a fixed point a of a ceiling, passing under a heavy movable pulley b of mass 10 kg and then over a fixed pulley c attached to the ceiling. To the other end d of the string is attached a particle of mass 10 kg hanging freely.

i) Show in separate diagrams the forces acting on the particle and on the pulley b when they are released from rest.

ii) Show that the acceleration of d is double that of b,

iii) and calculate the acceleration of d and

iv) the tension in the string. (Neglect the inertia of both pulleys).

[Video Solution]

Question 5

A particle of mass 10 kg hangs freely from the end b of a light inextensible string of length 0.2 m which is attached at the other end to a fixed point o. The particle is then projected horizontally with speed 1.4 m/s.

i) Show in a diagram the forces acting on the particle when ob is inclined at an angle \theta to the downward vertical assuming that it has a speed v m/s at that point.

ii) Use conservation of energy to determine v and

iii) express the tension in the string in terms of \theta.

iv) Find where the particle comes to instantaneous rest and

v) show that the tension in the string is then 49 N.

[Video Solution]

Question 6

A light string rstu is attached to fixed points at its ends r and u, so that u is vertically below r. Particles of weights 50 N and 100 N are attached to the string at s and t, respectively, and a horizontal force x newtons is applied to the particle at s so that the string is in equilibrium in a vertical plane through ru with \angle sru = \angle rut = 45^\circ and \angle rst = 105^\circ. Show in a separate diagrams the forces acting on the two particles and prove that x=\left(250+100\sqrt{3}\right) N.

[Video Solution]

Question 7

An equilateral triangle is pqr is formed from three uniform rods, each of length 2a and weight W, freely jointed at their ends. The triangle is freely suspended by a string attached to the midpoint o of pq so that it hangs symmetrically under gravity with r vertically below o. Show in separate diagrams the forces acting on pq and pr, and calculate the horizontal and vertical components of the reactions at r.

[Video Solution]

Question 8

i) Prove that the moments of inertia of a uniform circular disc of radius 0.4 m and mass 5 kg about an axis oq through its centre o and perpendicular to the disc is 0.4 kg m^2.

[Video Solution]

ii) Such a disc can rotate freely about the axis oq which is fixed horizontally. A light inextensible string is wound around the rim of the disc with one end attached to it, and to the other end is tied a particle p of mass 2 kg which hangs vertically. If the system is released from rest, show that the speed of p is 2.8 m/s after it has descended a distance 0.9 m.

[Video Solution]

Question 9

An engine pulls a train along a level track against a resistance which at any time is K times the momentum. The engine works at a constant power 25KMU_0^2, where M is the total mass of the train and engine and K, U_0 are constants. Show that the equation of motion of the train is

    \[v\frac{dv}{dt}=K\left(25U_0^2-v^2\right),\]

and find the time taken to increase speed from U_0 to 4U_0.

[Video Solution]

Question 10

a) State the principle of Archimedes.

[Video Solution]

b) A solid hemisphere of radius a is help submerged in a liquid of density \rho with its plane face horizontal and uppermost at a distance 2a below the free surface of the liquid. Calculate the magnitude, direction and line of action of

i) the force exerted by the liquid on the plane face,

ii) the total force exerted by the liquid on the surface of the solid.

[Video Solution]


Latest PSI Licence:

“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.

Important Exception to the above Licence:

The State Examination Commission is the copyright holder which is providing the material under the above license (as per current directives and regulations from the relevant government bodies). However the State Examination Commission as an Irish examination body is able to use copyrighted material in its exams without infringing copyright but this right is not extended to third parties when those exams are re-used.

(For example: the State Examination Commission may include in their exam a copyrighted poem and this action does not require the permission of the poet but the poet’s permission must be sought when the exam is re-used by someone other than the State Examination Commission.)

Also, all derived and related work (such as video solutions, lessons, notes etc) are the copyrighted material of Stephen Easley-Walsh (unless stated otherwise). And that the above licence is for only the exam itself and nothing further.

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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