Edexcel M2 (Mechanics 2) 2014 January

1. A particle \(P\) of mass 2 kg is moving with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Find the magnitude of the impulse.
   2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied.
      
   3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
   $$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).
2. Find the acceleration of \(P\) when \(t = 3\)
3. Find the total distance travelled by \(P\) in the first 3 seconds of its motion.
4. Show that \(P\) never returns to \(O\).

1. A car has mass 550 kg . When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.
   1. (i) Find the value of \(R\).
      (ii) Find the value of \(P\).
   2. Find the acceleration of the car when it travels along the straight horizontal road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with the engine working at 50 kW .
      

4. \begin{figure}[h] \end{figure} A uniform lamina \(A B C D\) is formed by removing the isosceles triangle \(A D C\) of height \(h\) metres, where \(h < 2 \sqrt { 3 }\), from a uniform lamina \(A B C\) in the shape of an equilateral triangle of side 4 m , as shown in Figure 1. The centre of mass of \(A B C D\) is at \(D\).

1. Show that \(h = \sqrt { } 3\) The weight of the lamina \(A B C D\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(A B\) vertical. The horizontal force acts in the vertical plane containing the lamina.
2. Find \(F\) in terms of \(W\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(A C = \frac { 2 } { 3 } a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(B D = 2 a\).

1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac { 1 } { 2 } m g \sin \theta\)
2. Find the magnitude of the normal reaction on the rod at \(B\). The rod is in limiting equilibrium when \(\tan \theta = \frac { 4 } { 3 }\)
3. Find the coefficient of friction between the rod and the ground.

1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

\begin{figure}[h] \end{figure} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \(( 3 \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 } , v > 3\). The ball moves freely under gravity and passes through the point A before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac { 15 } { 49 } \mathrm {~s}\) after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac { 1 } { 2 } E\) joules.

1. Find the value of \(v\). At another point \(B\) on the path of the ball the kinetic energy is also \(\frac { 1 } { 2 } E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.
2. Find the value of \(T\).

7. Three particles \(A , B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5 u\) and \(4 u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3 u\). In the subsequent motion, \(A , B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find (i) the speed of \(A\) immediately after the collision,
   (ii) the speed of \(B\) immediately after the collision. Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
2. find the set of possible values of \(e\).