My Journal

All things Mathematical
10 Dec 2018

Leaving Cert Applied Maths Higher Level 1972

/
Posted By
/
Comments0

Question 1

A racing car covers a journey of quicklatex.com-52db7b3b2bcad226db2b4f802b8dd96c_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds km from rest to rest. It accelerates uniformly in the first minute to reach its maximum speed of quicklatex.com-cbbca562b8a27132c102a81b44b1362d_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m/s, it holds this speed for a certain time and then slows uniformly to rest with a retardation of magnitude three times that of the acceleration.

i) Draw a rough velocity-time graph and find the distance travelled in the three stages of the journey and the total time taken.

ii) If the maximum speed over the final kilometre of the journey had been restricted to quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m/s, show that the time taken from rest to rest would have been at least quicklatex.com-c7b9db4ac0bed2458a7d5621c466572b_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds s longer than before, assuming the same rates of acceleration and deceleration as before.

[Video Solution]

Question 2

From a point quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds on horizontal ground an elastic particle is projected under gravity with a velocity of quicklatex.com-3f12db62b7c0b3163f5e237c508dcc2a_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m/s, where quicklatex.com-f453dfa2ac10361897075d9950ad4c04_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds and quicklatex.com-be4ffaf4d71dac859eb0823be8c6649f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds are unit vectors along the horizontal and upward drawn vertical, respectively.

i) Find the displacement quicklatex.com-9d6cb6705ba8afb57cbf6daf3ee743dd_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds at time quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds seconds afterwards, and in particular when the particle is at its highest point.

ii) If at this point [the highest point] the particle strikes a fixed vertical wall, where the coefficient of restitution is quicklatex.com-294859fdbb1fdda439ec2363e11fdfa1_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds, find how far from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds the particle strikes the ground.

[Video Solution]

Question 3

a) State the laws governing oblique, perfectly elastic collision between two spheres.

[Video Solution]

b) A small sphere collides obliquely with a similar sphere of equal mass at rest – both spheres being smooth and perfectly elastic.

i) Show that the paths of the two spheres after the collision are at right angles.

ii) Prove that there is no loss in kinetic energy.

[Video Solution]

Question 4

A wedge of mass quicklatex.com-fce5a805c0f6f0bd2d059819c56bf93a_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds rests on a smooth horizontal table with one of its plane faces inclined at quicklatex.com-9c3fb28be0c20ddb0b2c4bd59c303de8_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds to the horizontal. This plane face is smooth and on it is placed a particle of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds and the system is released from rest.

i) Draw separate diagrams showing the forces acting on the particle and on the wedge during the motion.

ii) By considering the acceleration of the particle in two components, one component down the plane and the other horizontal, show that the acceleration of the wedge is quicklatex.com-edd2af45100ab6594c23fee60a392947_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds and that of the particle is quicklatex.com-97de4b631207ba35a322e455f48bf277_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds.

[Video Solution]

Question 5

a) Using the usual notation, prove that

    quicklatex.com-a0e078deda67ccdad78235075b996463_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds

represents simple harmonic motion.

[Video Solution]

b) A light flexible elastic string of natural length quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m and elastic constant quicklatex.com-ed5decd492fdc6648538222379ceaf27_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds N/m has one end quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds tied to a point on a smooth horizontal table. To its other end is attached a particle of mass quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds kg, which is placed on the table at a point quicklatex.com-8859b07193d4c340b17f3853ff89fdaa_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds, distance quicklatex.com-3cd0bbd227f2e36fba8c2c9a410ea05f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m from quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds and released from rest. Show that in the first stage of motion the particle moves with simple harmonic motion of period quicklatex.com-8147c1ab19d7031eb9440efc8e2c584c_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds seconds and that the time taken to reach quicklatex.com-81c6b3e443a83c026f06b718b1dedcef_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds is quicklatex.com-e952c0e6fa6c5699d803146e9a5efa59_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds s.

[Video Solution]

Question 6

a) Show that a particle moving in a circle with constant speed is being accelerated towards the centre of the circle.

[Video Solution]

b) A small ring of mass quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds kg is threaded on a smooth light flexible inelastic string of length quicklatex.com-598fb5a16d3def68572898357672eea3_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m. The ends of the string are attached to two fixed points, distance quicklatex.com-d87274adcc3e9b397c1d81918125c810_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m apart in the same vertical line, and the ring describes with constant speed a horizontal circle whose centre is the lower fixed point. Find the constant speed of rotation and show that the tension in the string is quicklatex.com-924888cd81054557ddfb6ec384c79837_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds N.

[Video Solution]

Question 7

A uniform rod of length quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m and weight quicklatex.com-4fbb80c3d7c3c9bdfaf780287e19f597_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds N is smoothly hinged at one end to a rough horizontal floor. The rod rests on the smooth curved surface of a hemisphere whose plane face is on the floor. The rod is in equilibrium inclined at quicklatex.com-9c3fb28be0c20ddb0b2c4bd59c303de8_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds to the horizontal and the hemisphere, of weight quicklatex.com-7260fe4f20eb47e11e44b79da133e21c_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds N and radius quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m, is in limiting equilibrium.

i) Show in separate diagrams the forces acting on the rod and on the hemisphere.

ii) Find the reaction between the rod and the hemisphere,

iii) and prove that the coefficient of friction between the latter and the floor is quicklatex.com-93ba715eab64d4939af4318b719cde8b_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds.

[Video Solution]

Question 8

a) Prove that the moment of inertia of a uniform rod quicklatex.com-819964b5440e1d7a1941259fada7d9c6_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds and length quicklatex.com-0e440a33f92d8dd692a299e70ffc00e6_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds about an axis through quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds, perpendicular to the rod, is quicklatex.com-c9e1e21d9be2dcb521f901e10584d648_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds.

[Video Solution]

b i) Such a rod [from part (a)] is free to rotate in a vertical plane about a fixed horizontal axis at quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds, with a particle of mass quicklatex.com-fce5a805c0f6f0bd2d059819c56bf93a_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds attached to the rod at quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds. The system is released from rest with the rod vertical and the end quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds above quicklatex.com-84fdf06eaf2b99889abd1c8acdc13d5f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds. Show that the angular speed of the rod when it is next vertical is quicklatex.com-313058a344915195606de8ef37105d1b_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds.

ii) At this point the particle falls off. Find the height to which the end quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds subsequently rises.

[Video Solution]

Question 9

A particle of mass quicklatex.com-941d689ad0e65c5d5733efadca408734_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds kg is projected vertically upwards from ground level with a speed of quicklatex.com-d2a5ac082f8383090a608c4c7c1c9961_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m/s. In addition to the weight of the particle, there is the resistance force of the air of magnitude quicklatex.com-a33bafaeaa97757b3494e0c166058a47_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds newtons when the speed if quicklatex.com-8bafe7d16e660869fc203e8bffd77abc_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds metres per second. Show that the equation of motion during the upward journey is

    quicklatex.com-d916509718ede2b2a49a343fec8322b6_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds

Prove that the maximum height is quicklatex.com-b4f941db5bcaaf0f85b6e8de59b01f6b_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds m, and that the time taken to reach it is approximately quicklatex.com-41a58f48e28275f942c9e409f8055af9_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds s.

(Note: quicklatex.com-ba3280f6361a0ec3494c9c383d955133_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds)

[Video Solution]

Question 10

a) State the condition for the equilibrium of a floating body.

[Video Solution]

b) A uniform rectangular board quicklatex.com-532a8f530b02adee3e441c75769e908c_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds of weight quicklatex.com-054ef36c6e54f65ccdd586c0869b4fb0_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds, floats with the diagonal quicklatex.com-dbc51c69debe1b39c6347aca12bad038_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds on the surface of the of the water, the lowest corner quicklatex.com-91dc9a245a0f25a550d41a6feb58273f_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds being attached to the bottom of the vessel by a light inelastic string.

i) Show in a diagram the forces acting on the board and prove that the specific gravity of the board is quicklatex.com-c1202576eb6006ba366f3dcac824fc44_l3 | Leaving Cert Applied Maths Higher Level 1972 | Maths Grinds.

ii) Find the tension in string.

[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/

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.