Edexcel M2 (Mechanics 2) 2023 October

1. At time \(t\) seconds, \(t > 0\), a particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\), where

$$\mathbf { r } = \left( t ^ { 4 } - 8 t ^ { 2 } \right) \mathbf { i } + \left( 6 t ^ { 2 } - 2 t ^ { \frac { 3 } { 2 } } \right) \mathbf { j }$$

1. Find the velocity of \(P\) when \(P\) is moving in a direction parallel to the vector \(\mathbf { j }\)
2. Find the acceleration of \(P\) when \(t = 4\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a template where

• PQUY is a uniform square lamina with sides of length \(4 a\)
• RSTU is a uniform square lamina with sides of length \(2 a\)
• VWXY is a uniform square lamina with sides of length \(2 a\)
• the three squares all lie in the same plane
• the mass per unit area of \(V W X Y\) is double the mass per unit area of \(P Q U Y\)
• the mass per unit area of \(R S T U\) is double the mass per unit area of \(P Q U Y\)
• the distance of the centre of mass of the template from \(P X\) is \(d\)
  1. Show that \(d = \frac { 5 } { 2 } a\)

The template is freely pivoted about \(Q\) and hangs in equilibrium with \(P Q\) at an angle of \(\theta\) to the downward vertical.

Find the value of \(\tan \theta\) The mass of the template is \(M\)
The template is still freely pivoted about \(Q\), but it is now held in equilibrium, with \(P Q\) vertical, by a horizontal force of magnitude \(F\) which acts on the template at \(X\). The line of action of the force lies in the same plane as the template.

Find \(F\) in terms of \(M\) and \(g\)

3. \begin{figure}[h] \end{figure} A particle \(Q\) of mass 0.25 kg is moving in a straight line on a smooth horizontal surface with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\). The impulse acts parallel to the horizontal surface and at \(60 ^ { \circ }\) to the original direction of motion of \(Q\). Immediately after receiving the impulse, the speed of \(Q\) is \(12 \mathrm {~ms} ^ { - 1 }\)
As a result of receiving the impulse, the direction of motion of \(Q\) is turned through \(\alpha ^ { \circ }\), as shown in Figure 2. Find the value of \(I\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]

\begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\)
Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 3 .
At time \(T _ { 1 }\) seconds, \(P\) is at its highest point above the ground.

1. Find the value of \(T _ { 1 }\) At time \(t = 0\), a particle \(Q\) is also projected from \(A\) but with velocity \(( 5 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(Q\) moves freely under gravity.
2. Find the vertical distance between \(Q\) and \(P\) at time \(T _ { 1 }\) seconds, giving your answer to 2 significant figures. At the instant when particle \(P\) reaches \(B\), particle \(Q\) is moving at \(\alpha ^ { \circ }\) below the horizontal.
3. Find the value of \(\alpha\). At time \(T _ { 2 }\) seconds, the direction of motion of \(Q\) is perpendicular to the initial direction of motion of \(Q\).
4. Find the value of \(T _ { 2 }\)

1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .

The total mass of the cyclist and her cycle is 80 kg .
The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .

1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
   The distance from \(A\) to \(B\) is 20 km .
   Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
   The cyclist starts from rest at \(A\).
   At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\)
   The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
3. Using this model, find the value of \(P\)

6. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to a fixed point on a vertical wall.
A particle of weight \(\frac { 1 } { 4 } W\) is attached to the rod at \(B\).
A light inelastic string of length \(5 a\) has one end attached to the rod at the point \(C\), where \(A C = 5 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is above \(A\) and \(A D = 5 a\), as shown in Figure 4. The rod rests in equilibrium.
The tension in the string is \(T\).

1. Show that \(T = \frac { 6 } { 5 } \mathrm {~W}\)
2. Find, in terms of \(W\), the magnitude of the force exerted on the rod by the hinge at \(A\).

1. Particle \(P\) has mass \(4 m\) and particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of each particle is reversed as a result of the collision.
The total kinetic energy of \(P\) and \(Q\) after the collision is half of the total kinetic energy of \(P\) and \(Q\) before the collision.

1. Show that \(y = \frac { 8 } { 3 } u\) The coefficient of restitution between \(P\) and \(Q\) is \(e\).
2. Find the value of \(e\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is no second collision between \(P\) and \(Q\),
3. find the range of possible values of \(f\). Given that \(f = \frac { 1 } { 4 }\)
4. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) as a result of its impact with the wall.