Edexcel M4 (Mechanics 4) 2011 June

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\). Find, in terms of \(m\), the total kinetic energy lost in the collision.

2. \begin{figure}[h] \end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at \(B\), 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner \(C\). The ball is kicked so that it travels along the floor, bounces off the first wall at the point \(X\) and hits the second wall at the point \(Y\). The point \(Y\) is 7.5 m from the corner \(C\).
The coefficient of restitution between the ball and the first wall is \(\frac { 3 } { 4 }\).
Modelling the ball as a particle, find the distance \(C X\).

1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A coastguard patrol boat \(C\) is moving with constant velocity \(( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Another ship \(S\) is moving with constant velocity \(( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find, in terms of \(u\), the velocity of \(C\) relative to \(S\). At noon, \(S\) is 10 km due west of \(C\).
   If \(C\) is to intercept \(S\),
2. 1. find the value of \(u\).
   2. Using this value of \(u\), find the time at which \(C\) would intercept \(S\). If instead, at noon, \(C\) is moving with velocity \(( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and continues at this constant velocity,
3. find the distance of closest approach of \(C\) to \(S\).

1. A hiker walking due east at a steady speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320, and also walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due north.

Find

1. the direction from which the wind is blowing,
2. the wind speed.

5. A particle \(Q\) of mass 6 kg is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(Q\) from the origin \(O\) is \(x\) metres and the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves under the action of a retarding force of magnitude ( \(a + b v ^ { 2 }\) ) N, where \(a\) and \(b\) are positive constants. At time \(t = 0 , Q\) is at \(O\) and moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The particle \(Q\) comes to instantaneous rest at the point \(X\).

1. Show that the distance \(O X\) is $$\frac { 3 } { b } \ln \left( 1 + \frac { b U ^ { 2 } } { a } \right) \mathrm { m }$$ Given that \(a = 12\) and \(b = 3\),
2. find, in terms of \(U\), the time taken to move from \(O\) to \(X\).

1. A particle \(P\) of mass 4 kg moves along a horizontal straight line under the action of a force directed towards a fixed point \(O\) on the line. At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and the force towards \(O\) has magnitude \(9 x\) newtons. The particle \(P\) is also subject to air resistance, which has magnitude \(12 v\) newtons when \(P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Show that the equation of motion of \(P\) is
   $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 0$$ It is given that the solution of this differential equation is of the form $$x = \mathrm { e } ^ { - \lambda t } ( A t + B )$$ When \(t = 0\) the particle is released from rest at the point \(R\), where \(O R = 4 \mathrm {~m}\). Find,
2. the values of the constants \(\lambda , A\) and \(B\),
3. the greatest speed of \(P\) in the subsequent motion.

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The rod \(A B\) has length \(2 a\) and mass \(4 m\), and the rod \(B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium.