Edexcel A Level Further Maths M4 · Mechanics 4 June 2017 question paper 6680

[ In this question the horizontal unit vectors i and j are due east and due north respectively.] A ship A has constant velocity (4i + 2j)kmh^–1 and a ship B has constant velocity (–i + 3j)km h^–1. At noon, the position vectors of the ships A and B with respect to a fixed origin O are (–2 i + j)km and (5 i – 2 j)km respectively. Find

the time at which the two ships are closest together,

the length of time for which ship A is within 2km of ship B.

Two smooth uniform spheres A and B have masses 3 m kg and m kg respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere A has velocity (5i – 2j)ms^–1 and sphere B has velocity (3i + 4j)ms^–1. When the spheres collide, the line joining their centres is parallel to j, as shown in Figure 1. The coefficient of restitution between the two spheres is e. The kinetic energy of sphere B immediately after the collision is 85% of its kinetic energy immediately before the collision. Find

the velocity of each sphere immediately after the collision,

the value of e.

A cyclist and her bicycle have a combined mass of 75 kg. The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force agnitude

cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude 3v² newtons. The cyclist starts from rest. At time t seconds, she has travelled a distance x metres from her starting point. Find

v in terms of x,

t in terms of v.

[ In this question, the unit vectors i and j are in a vertical plane, i being horizontal and j being vertically upwards.] A line of greatest slope of a fixed smooth plane is parallel to the vector (–4 i – 3 j). A particle P falls vertically and strikes the plane. Immediately before the impact, P has velocity –7jms^–1. Immediately after the impact, P has velocity (–ai + j)ms^–1, where a is a positive constant.

Show that a = 6

Find the coefficient of restitution between P and the plane.

A cyclist riding due north at a steady speed of 12 km h^–1 notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of 120° and also riding at a steady speed of 12 km h^–1, notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find

the wind speed,

the direction from which the wind is blowing, giving your answer as a bearing.

A particle P of mass 0.2kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8m and modulus of elasticity 7N. At time t = 0 the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of P is resisted by a force of magnitude 2v newtons, where v ms^–1 is the speed of P. At time t seconds, P is x metres below its equilibrium position.

Find x in terms of t.

Find the value of t when P first comes to instantaneous rest.

Figure 2 shows four uniform rods, each of mass m and length 2 a. The rods are freely hinged at their ends to form a rhombus ABCD. Point A is attached to a fixed point on a ceiling and the rhombus hangs freely with C vertically below A. A light elastic spring of natural length 2 a and modulus of elasticity 7 mg connects the points A and C. A particle of mass 3 m is attached to point C.

Show that, when AD is at an angle to the downward vertical, the potential energy V of the system is given by V = 28mga cos² – 48mga cosθ + constant

Given that θ > 0

find the value of θ for which the system is in equilibrium,

determine the stability of this position of equilibrium.