i) How may a velocity-time graph be used to find the distance travelled in a given time?
ii) An athlete runs m in seconds. Starting from rest, he accelerates uniformly to a speed of 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 and then with uniform deceleration . It comes to rest when it has covered a total distance . If the overall time for journey is , show that
A ship is travelling South-West at knots. Another ship is travelling at knots in a direction North of West.
i) Draw a diagram to show the velocity of relative to .
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 increase its speed, without changing direction, so that would appear to to be travelling due North?
a) Explain the terms: coefficient of friction, angle of friction. What is the relationship between them?
b) A uniform ladder of weight 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 to the horizontal (seed diagram). The ladder makes an angle with the wall.
i) Show that the reaction at the wall is .
ii) If the ladder is on the point of slipping prove that
where is the angle of friction.
a) A see-saw consist of a uniform plank freely pivoted at its mid-point on a support of vertical height . It carries a mass at one end and a mass at the other. The see-saw is released from rest with the mass at its highest point. Find, in terms of , the velocity with which the see-saw strikes the ground.
b) State the Principle of Archimedes.
c) A uniform road of length and relative density is freely hinged to the base of a tank containing a liquid to a depth . The relative density of the liquid is . As a result, the rod is inclined but not fully submerged. Derive an expression for the angle the rod makes with the horizontal.
A plane is inclined at an angle to the horizontal. A particle is projected up the plane with a velocity at an angle to the plane. The plane of projection is vertical and contains the line of greatest slope.
i) Show that the time of flight is .
ii) Prove that the range up the plane is a maximum when .
iii) Prove that the particle will strike the plane horizontally if .
a) A particle, moving at constant speed, is describing a horizontal circle on the inside surface of a smooth sphere of radius . The centre of the circle is a distance below the centre of the sphere. Prove that the speed of the particle is .
b) A conical pendulum consists of a light elastic string with a mass attached to it which is rotating with uniform angular velocity . The natural length of the string is and its elastic constant is , i.e. a force produces unit extension. The extended length of the string is and it makes an angle with the vertical. Prove that and that
a) Two smooth spheres of masses and collide directly when moving in opposite directions with speeds and , respectively. The sphere of mass is brought to rest by the impact. Prove that
where is the coefficient of restitution.
b) A smooth sphere collides obliquely with another smooth sphere of equal mass which is at rest. Before impact the direction of motion of makes an angle with the line of centres at impact (see diagram). After impact it makes an angle with that line. If the coefficient of restitution is , prove that
a) Define simple harmonic motion. Using the usual notation, show that the equation represents simple harmonic motion.
b) A body is moving with simple harmonic motion, of amplitude m. When it is m from the mid-point of its path its speed is m/s. Find its speed when it is 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 m and 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.
a) State the theorem of parallel axes.
b) Prove that the moment of inertia of a uniform rod of mass and length about a perpendicular axis through one of its end is .
c) A uniform rod of length is attached to the rim of a uniform disc of diameter . The rod is colinear with a diameter of the disc (see diagram). The disc and the rod are both of mass .
i) Calculate the moment of inertia of the compound body about a perpendicular axis through the end .
ii) If the compound body makes small oscillations in a vertical plane about a horizontal axis through , show that the periodic time is .
a) Solve the differential equation
Hence or otherwise solve
where when and when .
b) A body is moving in a straight line subject to a deceleration which is equal to , where is the velocity. The initial velocity is m/s. In how many seconds will the velocity of the body be m/s and how far will it travel in that time?
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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/