A boat has to travel by the shortest route to the point km and then return immediately to its starting point at the origin. The velocity of the water is km/hour and the boat has a speed of km/hour in still water.
If is the velocity of the boat on the outward journey,
i) find and and the time taken for the outward journey, leaving your answer in surd form.
ii) Find, also, the time taken for the whole journey.
A body of weight is supported by two vertical inextensible strings at and as in diagram where cm. The tensions in the strings are and and the string of tension makes an angle of with . The centre of gravity of the body is at , the centre of is and .
Express in terms of and and hence find the distance of from in terms of and .
A projectile is fired with initial velocity , where is along the horizontal. A plane passes through the point of projection and makes an angle with the horizontal.
i) If the projectile strikes the plane at right angles to after time , show that
ii) and deduce that .
iii) If , find in terms of and the range of the projectile alone .
a) State and prove the relationship between the coefficient of friction and the angle of friction .
b) The diagram shows a particle of weight on a rough plane making an angle with the horizontal. The particle is acted upon by a force whose lien of action makes an angle with the line of greatest slope. The particle is just one the point of moving up the plane.
i) Draw a diagram showing the forces acting on the particle
ii) and prove that
If the particle is just on the point of moving up the plane, deduce
iii) the forces acting up along the plane that would achieve this
iv) the horizontal force that would achieve it
v) the minimum force that would achieve it.
a) Two imperfectly elastic spheres of equal mass moving horizontally along the same straight line impinge and, as a result, one of them is brought to rest. Show that whatever be the value of the coefficient of restitution, , they must have been moving in opposite directions.
b) A sphere of mass kg moving with a speed m/s on a smooth horizontal table impinges on a smooth plane . This plane is inclined to the table at an angle and the line of intersection of it with the table is at right angles to the direction of motion of the sphere.
i) Write down the components of the velocity of perpendicular to the plane and parallel to the plane before impact and
ii) show that is the velocity of perpendicular to the plane after impact where is the coefficient of restitution between the sphere and the plane.
iii) Find the magnitude of the impulse due to the impact.
a) If a string whose elastic constant is is stretched a distance beyond its natural length, show that the work done is .
b) A particle of mass is on a rough horizontal plane is connected to a fixed point in the plane by a light string of elastic constant . Initially the string is just taut and the particle is projected along the plane directly away from with initial speed against constant resistance .
i) Find an expression for the distance travelled by the particle.
ii) Noting that the particle will just return to its point of projection if the potential energy at any point is equal to the work done up to that point in overcoming , show that
a) Establish the moment of inertia of a uniform rod about an axis through its centre perpendicular to the rod.
b) State the parallel axes theorem.
c) A thin uniform rod of length and of mass has a mass of attached at its mid-point. Find the positions of a point in the rod about which the rod (with attached mass) may oscillate as a compound pendulum, having period equal to that of a simple pendulum of length .
a) A particle is moving in a straight line such that its distance from a fixed point at time is given by
Show that the particle is movign with simple harmonic motion.
b) A particle is moving in a straight line with simple harmonic motion. When it is a point of distance m from the mean-centre, its speed is m/s and when it is at a point of distance m from the end-position on the same side of the mean-centre as , its acceleration is of magnitude m/s. If is the amplitude of the motion,
i) show that
and hence find the value of .
ii) Find also the period of the motion and the shortest time taken between and correct to two places of decimals.
a) Solve the differential equation
given that and when .
b) A car starts from rest. When it is at a distance from its starting point, its speed is and its acceleration is , show that
and find as accurately as the tables allow its speed when .
a) A vessel is in the form of a frustum of a right circular cone. It contains liquid to a depth and at that depth the area of the free surface of the liquid is of the area of the base. Find in simplest surd form the ratio of the thrust on the base due to the liquid to the weight of the liquid.
b) A piece of wood and a piece of metal weigh N and N, respectively. When combined together the compound body weighs N in water. Given that the specific gravity of the metal is , find the specific gravity of the wood.
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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/