Edexcel M2 (Mechanics 2) 2020 June

1. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse \(\mathbf { I N }\) s, such that \(\mathbf { I } = a \mathbf { i } + b \mathbf { j }\)

Immediately after receiving the impulse, the particle is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a constant. Given that the magnitude of \(\mathbf { I }\) is \(\sqrt { 40 }\), find the two possible impulses.
(5)

1. A truck of weight 9000 N is travelling up a hill on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\)

When the truck travels up the hill with the engine working at \(3 P\) watts, the truck is moving at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Later on, the truck travels down the hill along the same road, with the engine working at \(P\) watts. At the instant when the speed of the truck is \(12 \mathrm {~ms} ^ { - 1 }\), the acceleration of the truck is \(\frac { g } { 20 }\) The resistance to motion of the truck from non-gravitational forces is a constant force of magnitude \(R\) newtons in all circumstances. Find (i) the value of \(P\),
(ii) the value of \(R\).
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3. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 25 kg and length 3 m , has end \(A\) resting on rough horizontal ground. The end \(B\) rests against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall.
The coefficient of friction between the rod and the ground is \(\frac { 4 } { 5 }\)
The coefficient of friction between the rod and the wall is \(\frac { 3 } { 5 }\)
The rod rests in limiting equilibrium.
The rod is at an angle of \(\theta\) to the ground, as shown in Figure 1. Find the exact value of \(\tan \theta\).
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4. \begin{figure}[h] \end{figure} The uniform lamina \(L\), shown shaded in Figure 2, is formed by removing the square \(P Q R V\), of side \(2 a\), and the square \(R S T U\), of side \(4 a\), from a uniform square lamina \(A B C D\), of side \(8 a\). The lines \(Q R U\) and \(V R S\) are straight. The side \(A D\) is parallel to \(P V\) and the side \(A B\) is parallel to \(P Q\). The distance between \(A D\) and \(P V\) is \(a\) and the distance between \(A B\) and \(P Q\) is \(a\). The centre of mass of \(L\) is at the point \(G\).

1. Show that the distance of \(G\) from the side \(A D\) is \(\frac { 42 } { 11 } a\) The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(C\). The lamina, with the attached particle, is freely suspended from \(B\) and hangs in equilibrium with \(B C\) making an angle of \(45 ^ { \circ }\) with the horizontal.
2. Find the value of \(k\).

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5. At time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 3\) When \(t = 0 , P\) is at the fixed point \(O\).
   The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

6. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp has length 15 m and is inclined at an angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 5 } { 12 }\). The line \(A B\) is a line of greatest slope of the ramp, where \(A\) is at the bottom of the ramp, and \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 6 kg is projected up the ramp with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in a straight line towards \(B\). The coefficient of friction between \(P\) and the ramp is 0.25

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). At the instant \(P\) reaches \(B\), the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). Immediately before hitting the ground at \(C\), the speed of \(P\) is \(w \mathrm {~ms} ^ { - 1 }\)
2. Use the work-energy principle to find
   1. the value of \(v\),
   2. the value of \(w\).
      
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7. Particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal surface. Particle \(A\) collides directly with particle \(B\) of mass \(m\), which is moving along the same straight line and in the same direction as \(A\). Immediately before the collision, the speed of \(B\) is \(u\).
As a result of the collision, the direction of motion of \(B\) is unchanged and the kinetic energy gained by \(B\) is \(\frac { 48 } { 25 } m u ^ { 2 }\)

1. Find the coefficient of restitution between \(A\) and \(B\).
   (8) After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(f\). Given that the speed of \(B\) immediately after first hitting the wall is equal to the speed of \(A\) immediately after its first collision with \(B\),
2. find the value of \(f\).

8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h] \end{figure} At time \(t = 0\), a small ball is projected from a fixed point \(O\) on horizontal ground. The ball is projected from \(O\) with velocity ( \(p \mathbf { i } + q \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(p\) and \(q\) are positive constants. The ball moves freely under gravity. At time \(t = 3\) seconds, the ball passes through the point \(A\) with velocity ( \(8 \mathbf { i } - 12 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 4.

1. Find the speed of the ball at the instant it is projected from \(O\). For an interval of \(T\) seconds the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the ball is such that \(v \leqslant 10\)
2. Find the value of \(T\). At the point \(B\) on the path of the ball, the direction of motion of the ball is perpendicular to the direction of motion of the ball at \(A\).
3. Find the vertical height of \(B\) above \(A\).