Edexcel A Level Further Maths M4 · Mechanics 4 June 2008 question paper 6680

[ In this question i and j are unit vectors due east and due north respectively.] A ship P is moving with velocity (5i – 4j) km h^-1 and a ship Q is moving with velocity (3i + 7j) km h^-1. Find the direction that ship Q appears to be moving in, to an observer on ship P, giving your answer as a bearing.

Two small smooth spheres A and B have equal radii. The mass of A is 2 m kg and the mass of B is m kg. The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of A is (2i – 2j) m s^-1 and the velocity of B is (–3i – j) m s^-1. Immediately after the collision the velocity of A is (i – 3j) m s^-1. Find the speed of B immediately after the collision.

At time t = 0, a particle of mass m is projected vertically downwards with speed U from a point above the ground. At time t the speed of the particle is v and the magnitude of the air resistance is modelled as being mkv, where k is a constant. Given that U < , find, in terms of k, U and g, the time taken for the particle to double its speed.

A small smooth ball B, moving on a horizontal plane, collides with a fixed vertical wall. Immediately before the collision the angle between the direction of motion of B and the

A light elastic spring has natural length l and modulus of elasticity mg. One end of the spring is fixed to a point O on a rough horizontal table. The other end is attached to a particle P of mass m which is at rest on the table with OP = l. At time t = 0 the particle is projected

magnitude 2mvω, where ω = . The coefficient of friction between P and the table is 0.5.

Show that, until P first comes to rest,

Find x in terms of t, l and ω.

Hence find, in terms of ω, the time taken for P to first come to instantaneous rest.

A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time t = 0, a boat leaves a point O on one bank and moves in a straight line across the river to a point P on the opposite bank. Its path OP is perpendicular to both banks and OP = 30 m, as shown in Figure 2. The speed of flow of the river, r m s^-1, at a point on OP which is at a distance x m from O, is modelled as

The speed of the boat relative to the water is constant at 5 m s^-1. At time t seconds the boat is at a distance x m from O and is moving with speed v m s^-1 in the direction OP.

Show that

Hence show that

Find the total time taken for the boat to cross the river from O to P.

A uniform rod AB, of length 2 a and mass kM where k is a constant, is free to rotate in a vertical plane about the fixed point A. One end of a light inextensible string of length 6 a is attached to the end B of the rod and passes over a small smooth pulley which is fixed at the point P. The line AP is horizontal and of length 2 a. The other end of the string is attached to a particle of mass M which hangs vertically below the point P, as shown in

Show that the potential energy of the system is

Find the value of k.

Determine the stability of each of the two positions of equilibrium.