Edexcel M2 (Mechanics 2) 2018 June

1. A particle \(P\) of mass 0.7 kg is moving with velocity ( \(\mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse. Immediately after receiving the impulse, \(P\) is moving with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Find the impulse.
   2. Find, in degrees, the size of the angle between the direction of the impulse and the direction of motion of \(P\) immediately before receiving the impulse.
      (3)
      

2. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 6 kg and length 1.6 m , rests with its end \(A\) on rough horizontal ground. The rod is held in equilibrium at \(30 ^ { \circ }\) to the horizontal by a light string attached to the rod at \(B\). The string is at \(40 ^ { \circ }\) to the horizontal and lies in the same vertical plane as the rod, as shown in Figure 1. The tension in the string is \(T\) newtons. The coefficient of friction between the ground and the rod is \(\mu\).

1. Show that, to 3 significant figures, \(T = 27.1\)
2. Find the set of values of \(\mu\) for which equilibrium is possible. \includegraphics[max width=\textwidth, alt={}, center]{a3f28425-4acf-4878-b0e3-15b5bc8a92d7-07_27_40_2802_1893}

3. A cyclist and his bicycle, with a combined mass of 75 kg , move along a straight horizontal road. The cyclist is working at a constant rate of 180 W . There is a constant resistance to the motion of the cyclist and his bicycle of magnitude \(R\) newtons. At the instant when the speed of the cyclist is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). Later, the cyclist moves up a straight road with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 21 }\). The cyclist is working at a rate of 180 W and the resistance to the motion of the cyclist and his bicycle from non-gravitational forces is again the same constant force of magnitude \(R\) newtons.
2. Find the value of \(v\).

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) is an isosceles triangle with \(A B = B C , A C = 6 a\) and the distance from \(B\) to \(A C\) is \(3 a\). The uniform lamina \(M N C\) is an isosceles triangle with \(M N = N C\) and \(M C = 3 a\). Triangles \(A B C\) and \(M N C\) are similar and are made of the same material. The lamina \(L\) is formed by fixing triangle \(M N C\) on top of triangle \(A B C\), as shown in Figure 2.

1. Show that the distance of the centre of mass of \(L\) from \(A C\) is \(\frac { 9 } { 10 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.

5. A particle \(P\) of mass 0.3 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 ) , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + \left( 3 t ^ { 2 } - 8 t + 4 \right) \mathbf { j }$$

1. Find \(\mathbf { F }\) when \(t = 4\) At the instants when \(P\) is at the points \(A\) and \(B\), particle \(P\) is moving parallel to the vector i.
2. Find the distance \(A B\).
   

6. A particle \(P\) is projected from a fixed point \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. As \(P\) passes through the point \(B\) on its path, \(P\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\beta\) below the horizontal.

1. By considering energy, find the vertical distance between \(A\) and \(B\). Particle \(P\) takes 1.5 seconds to travel from \(A\) to \(B\).
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the length of time for which the speed of \(P\) is less than \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7. Three particles \(A\), \(B\) and \(C\) have masses \(2 m , 3 m\) and \(4 m\) respectively. The particles lie at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). The kinetic energy of \(A\) immediately after the collision is one ninth of the kinetic energy of \(A\) immediately before the collision. Given that the direction of motion of \(A\) is unchanged by the collision,

1. find the value of \(e\). After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). The coefficient of restitution between \(B\) and \(C\) is \(f\), where \(f < \frac { 3 } { 4 }\). The speed of \(B\) immediately after the collision with \(C\) is \(V\).
2. 1. Express \(V\) in terms of \(f\) and \(u\).
   2. Hence show that there will be a second collision between \(A\) and \(B\).