Edexcel M2 (Mechanics 2) 2022 June

1. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 4\) At time \(T\) seconds, \(T \geqslant 0 , P\) is moving in the direction of ( \(2 \mathbf { i } + \mathbf { j }\) )
2. Find the value of \(T\)

2. \begin{figure}[h] \end{figure} The point \(A\) lies on a smooth horizontal floor between two fixed smooth parallel vertical walls \(W X\) and \(Y Z\), as shown in the plan view in Figure 1.
The distance between \(W X\) and \(Y Z\) is \(3 d\).
The distance of \(A\) from \(Y Z\) is \(d\).
A particle is projected from \(A\) along the floor with speed \(u\) towards \(Y Z\) in a direction perpendicular to \(Y Z\). The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\)
The time taken for the particle to move from \(A\), bounce off each wall once and return to A for the first time is \(T _ { 1 }\)

1. Find \(T _ { 1 }\) in terms of \(d\) and \(u\). The ball returns to \(A\) for the first time after bouncing off each wall once. The further time taken for the particle to move from \(A\), bounce off each wall once and return to \(A\) for the second time is \(T _ { 2 }\)
2. Find \(T _ { 2 }\) in terms of \(d\) and \(u\).

3. A particle \(P\) of mass 0.5 kg is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(P\) receives an impulse of magnitude \(\sqrt { \frac { 5 } { 2 } } \mathrm { Ns }\) Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) Given that \(\lambda\) is a constant, find the two possible values of \(\lambda\)

4. A truck of mass 900 kg is moving along a straight horizontal road with the engine of the truck working at a constant rate of \(P\) watts. The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the speed of the truck is \(15 \mathrm {~ms} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Later the same truck is moving down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to the motion of the truck is again modelled as a constant force of magnitude \(R\) newtons. The engine of the truck is again working at a constant rate of \(P\) watts.
At the instant when the speed of the truck is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find the value of \(R\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 4 m and weight 50 N .
The rod has its end \(A\) on rough horizontal ground. The rod is held in equilibrium at an angle \(\alpha\) to the ground by a light inextensible cable attached to the rod at \(B\), as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is \(T\) newtons. Given that \(\sin \alpha = \frac { 3 } { 5 }\)

1. show that \(T = 20\) Given also that the rod is in limiting equilibrium,
2. find the value of the coefficient of friction between the rod and the ground.

6. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface so that the particles collide directly.
The mass of \(P\) is \(k m\) and the mass of \(Q\) is \(m\).
Immediately before the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\). Immediately after the collision, \(P\) and \(Q\) are moving in the same direction, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(2 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\)
The magnitude of the impulse received by \(Q\) in the collision is \(5 m v\)

1. Find (i) \(y\) in terms of \(v\)
   (ii) \(x\) in terms of \(v\)
   (iii) the value of \(k\)
2. Find, in terms of \(m\) and \(v\), the total kinetic energy lost in the collision between \(P\) and \(Q\).

7. \begin{figure}[h] \end{figure} The template shown in Figure 3 is formed by joining together three separate laminas. All three laminas lie in the same plane.

• PQUV is a uniform square lamina with sides of length \(3 a\)
• URST is a uniform square lamina with sides of length \(6 a\)
• \(Q R U\) is a uniform triangular lamina with \(U Q = 3 a , U R = 6 a\) and angle \(Q U R = 90 ^ { \circ }\)

The mass per unit area of \(P Q U V\) is \(k\), where \(k\) is a constant.
The mass per unit area of URST is \(k\).
The mass per unit area of \(Q R U\) is \(2 k\).
The distance of the centre of mass of the template from \(Q T\) is \(d\).

1. Show that \(d = \frac { 29 } { 14 } a\) The template is freely suspended from the point \(Q\) and hangs in equilibrium with \(Q R\) at \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\)
   

8. \begin{figure}[h] \end{figure} Figure 4 shows a rough ramp fixed to horizontal ground.
The ramp is inclined at angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 1 } { 6 }\)
The point \(A\) is on the ground at the bottom of the ramp.
The point \(B\) is at the top of the ramp.
The line \(A B\) is a line of greatest slope of the ramp and \(A B = 4 \mathrm {~m}\).
A particle \(P\) of mass 3 kg is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) directly towards \(B\).
The coefficient of friction between the particle and the ramp is \(\frac { 3 } { 4 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that at the instant \(P\) reaches the point \(B\), the speed of \(P\) is \(5 \mathrm {~ms} ^ { - 1 }\)
2. use the work-energy principle to find the value of \(U\). The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the ground at the point \(C\).
3. Find the horizontal distance from \(B\) to \(C\).