A cyclist has a maximum acceleration of m/s, a maximum speed of m/s and a maximum deceleration of m/s. The cyclist wishes to travel a distance from rest to rest in the shortest time. Find the time taken in the two cases
(i) m and
Draw a rough velocity-time graph for each case and explain why m is a critical distance.
A perfectly elastic particle falls vertically with speed on to a smooth plane inclined at an angle to the horizontal, and rebounds, hopping down the plane.
i) Write down its displacement from the point of contact after time in terms of and , where is drawn directly down the plane.
ii) Show that the length of the first hop is and that
iii) the length of the second hop is double this.
A ship is steaming with velocity m/s where where and are point East and North, respectively. At midday a second ship has a position km with respect to .
i) Find the minimum speed must have if it is to intercept .
ii) If the maximum speed of is in fact m/s, show that it can steer in either of two directions to intercept and
iii) calculate the two times of interception.
The diagram shows a light inelastic string with one end connected to a fixed point of a ceiling, passing under a heavy movable pulley of mass kg and then over a fixed pulley attached to the ceiling. To the other end of the string is attached a particle of mass kg hanging freely.
i) Show in separate diagrams the forces acting on the particle and on the pulley when they are released from rest.
ii) Show that the acceleration of is double that of ,
iii) and calculate the acceleration of and
iv) the tension in the string. (Neglect the inertia of both pulleys).
A particle of mass kg hangs freely from the end of a light inextensible string of length m which is attached at the other end to a fixed point . The particle is then projected horizontally with speed m/s.
i) Show in a diagram the forces acting on the particle when is inclined at an angle to the downward vertical assuming that it has a speed m/s at that point.
ii) Use conservation of energy to determine and
iii) express the tension in the string in terms of .
iv) Find where the particle comes to instantaneous rest and
v) show that the tension in the string is then N.
A light string is attached to fixed points at its ends and , so that is vertically below . Particles of weights N and N are attached to the string at and , respectively, and a horizontal force newtons is applied to the particle at so that the string is in equilibrium in a vertical plane through with and . Show in a separate diagrams the forces acting on the two particles and prove that N.
An equilateral triangle is is formed from three uniform rods, each of length and weight , freely jointed at their ends. The triangle is freely suspended by a string attached to the midpoint of so that it hangs symmetrically under gravity with vertically below . Show in separate diagrams the forces acting on and , and calculate the horizontal and vertical components of the reactions at .
i) Prove that the moments of inertia of a uniform circular disc of radius m and mass kg about an axis through its centre and perpendicular to the disc is kg m.
ii) Such a disc can rotate freely about the axis 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 of mass kg which hangs vertically. If the system is released from rest, show that the speed of is m/s after it has descended a distance m.
An engine pulls a train along a level track against a resistance which at any time is times the momentum. The engine works at a constant power , where is the total mass of the train and engine and , are constants. Show that the equation of motion of the train is
and find the time taken to increase speed from to .
a) State the principle of Archimedes.
b) A solid hemisphere of radius is help submerged in a liquid of density with its plane face horizontal and uppermost at a distance 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.
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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/