Edexcel M4 (Mechanics 4) 2015 June

1. Particles \(P\) and \(Q\) move in a plane with constant velocities. At time \(t = 0\) the position vectors of \(P\) and \(Q\), relative to a fixed point \(O\) in the plane, are \(( 16 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m }\) and \(( - 5 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\) respectively. The velocity of \(P\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

Find the shortest distance between \(P\) and \(Q\) in the subsequent motion.

1. When a woman walks due North at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due East. When she runs due South at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

Assuming that the velocity of the wind relative to the earth is constant, find

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

3. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) with equal radii have masses \(m\) and \(2 m\) respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, \(A\) has speed \(3 u\) with its direction of motion at an angle \(\theta\) to the line of centres, and \(B\) has speed \(u\) with its direction of motion at an angle \(\theta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 1 } { 8 }\) Immediately after the collision, the speed of \(A\) is twice the speed of \(B\).
Find the size of the angle \(\theta\).

4. A car of mass 900 kg is moving along a straight horizontal road with the engine of the car working at a constant rate of 22.5 kW . At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 } ( 0 < v < 30 )\) and the total resistance to the motion of the car has magnitude \(25 v\) newtons.

1. Show that when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is $$\frac { 900 - v ^ { 2 } } { 36 v } \mathrm {~m} \mathrm {~s} ^ { - 2 }$$ The time taken for the car to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
2. Show that $$T = 18 \ln \frac { 8 } { 5 }$$
3. Find the distance travelled by the car as it accelerates from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 1.5 kg is attached to the midpoint of a light elastic spring \(A B\), of natural length 2 m and modulus of elasticity 12 N . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal floor. The end \(B\) is held at a point on the floor where \(A B = 6 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the floor at the point \(O\), where \(A O = 3 \mathrm {~m}\), as shown in Figure 2. The end \(B\) is now moved along the floor in such a way that \(A O B\) remains a straight line and at time \(t\) seconds, \(t \geqslant 0\), $$A B = \left( 6 + \frac { 1 } { 4 } \sin 2 t \right) \mathrm { m }$$ At time \(t\) seconds, \(A P = ( 3 + x ) \mathrm { m }\).

1. Show that, for \(t \geqslant 0\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 16 x = 2 \sin 2 t$$ The general solution of this differential equation is $$x = C \cos 4 t + D \sin 4 t + \frac { 1 } { 6 } \sin 2 t$$ where \(C\) and \(D\) are constants.
2. Find the time at which \(P\) first comes to instantaneous rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-20_705_1104_116_420} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}

6. A smooth wire, with ends \(A\) and \(B\), is in the shape of a semicircle of radius \(r\). The line \(A B\) is horizontal and the midpoint of \(A B\) is \(O\). The wire is fixed in a vertical plane. A small ring \(R\) of mass \(2 m\) is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at \(A\) and is attached to a particle of mass \(\sqrt { 6 } m\). The other string passes through a small smooth ring fixed at \(B\) and is attached to a second particle of mass \(\sqrt { 6 } \mathrm {~m}\). The particles hang freely under gravity, as shown in Figure 3. The angle between the radius \(O R\) and the downward vertical is \(2 \theta\), where \(- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }\)

1. Show that the potential energy of the system is $$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
2. Find the values of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of the position of equilibrium for which \(\theta > 0\)

7. \begin{figure}[h] \end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where \(A B\) and \(B C\) are smooth vertical walls. The angle between \(A B\) and \(B C\) is \(120 ^ { \circ }\). A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is \(\frac { 1 } { 2 }\)

1. Show that the speed of the ball immediately after it has hit \(A B\) is \(\frac { \sqrt { 7 } } { 4 } u\). The speed of the ball immediately after it has hit \(B C\) is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find \(w\) in terms of \(u\).