Edexcel M4 (Mechanics 4) 2014 June

1. A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 45 ^ { \circ }\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of \(45 ^ { \circ }\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m , u\) and \(e\), the magnitude of the impulse of the plane on the ball.
   
2. A ship \(A\) is travelling at a constant speed of \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(050 ^ { \circ }\). Another ship \(B\) is travelling at a constant speed of \(v \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course to intercept \(A\). At 1400 hours \(B\) is 20 km from \(A\) and the bearing of \(A\) from \(B\) is \(290 ^ { \circ }\).
   1. Find the least possible value of \(v\).
   Given that \(v = 32\),
3. find the time at which \(B\) intercepts \(A\).

1. A small ball of mass \(m\) is projected vertically upwards from a point \(O\) with speed \(U\). The ball is subject to air resistance of magnitude \(m k v\), where \(v\) is the speed of the ball and \(k\) is a positive constant.

Find, in terms of \(U , g\) and \(k\), the maximum height above \(O\) reached by the ball.

4. A smooth uniform sphere \(S\) is moving on a smooth horizontal plane when it collides obliquely with an identical sphere \(T\) which is at rest on the plane. Immediately before the collision \(S\) is moving with speed \(U\) in a direction which makes an angle of \(60 ^ { \circ }\) with the line joining the centres of the spheres. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\) and \(U\) where necessary,
   1. the speed and direction of motion of \(S\) immediately after the collision,
   2. the speed and direction of motion of \(T\) immediately after the collision. The angle through which the direction of motion of \(S\) is deflected is \(\delta ^ { \circ }\).
2. Find
   1. the value of \(e\) for which \(\delta\) takes the largest possible value,
   2. the value of \(\delta\) in this case.

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 l\) and mass \(12 m\), has its end \(A\) smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end \(B\) of the rod. The string passes over a small smooth pulley which is fixed at the point \(C\), where \(A C\) is horizontal and \(A C = 2 l\). A particle of mass \(m\) is attached to the other end of the string and the particle hangs vertically below \(C\). The angle \(B A C\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.

1. Show that the potential energy of the system is $$4 m g l \left( \sin \frac { \theta } { 2 } - 3 \sin \theta \right) + \mathrm { constant }$$
2. Find the value of \(\theta\) when the system is in equilibrium and determine the stability of this equilibrium position.

6. \begin{figure}[h] \end{figure} A railway truck of mass \(M\) approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring \(P Q\), which is fixed at the end \(P\). The spring has a natural length \(a\) and modulus of elasticity \(M n ^ { 2 } a\), where \(n\) is a postive constant. At time \(t = 0\), the spring has length \(a\) and the truck strikes the end \(Q\) with speed \(U\). A resistive force whose magnitude is \(M k v\), where \(v\) is the speed of the truck at time \(t\), and \(k\) is a positive constant, also opposes the motion of the truck. At time \(t\), the truck is in contact with the buffer and the compression of the buffer is \(x\).

1. Show that, while the truck is compressing the buffer $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + n ^ { 2 } x = 0$$ It is given that \(k = \frac { 5 n } { 2 }\)
2. Find \(x\) in terms of \(U , n\) and \(t\).
3. Find, in terms of \(U\) and \(n\), the greatest value of \(x\).