Edexcel FM1 (Further Mechanics 1) 2022 June

1. A particle \(A\) of mass \(3 m\) and a particle \(B\) of mass \(m\) are moving along the same straight line on a smooth horizontal surface. The particles are moving in opposite directions towards each other when they collide directly.

Immediately before the collision, the speed of \(A\) is \(k u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\). The magnitude of the impulse received by \(B\) in the collision is \(\frac { 3 } { 2 } m u\).

1. Find \(v\) in terms of \(u\) only.
2. Find the two possible values of \(k\).

2. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. Find the speed of \(P\) immediately after receiving the impulse.

4. \begin{figure}[h] \end{figure} Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3 m\) and the mass of \(B\) is \(4 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres and \(B\) is moving with speed \(2 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90 ^ { \circ }\) by the collision, as shown in Figure 3.

1. Find the size of the angle through which the direction of motion of \(A\) is turned as a result of the collision.
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.

1. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
   The mass of \(P\) is \(3 m\) and the mass of \(Q\) is \(4 m\).
   Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
   The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 7 } ( 9 e + 2 )\)
   After the collision with \(P\), particle \(Q\) collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(Q\).
   The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 2 }\)
2. Find the complete range of possible values of \(e\) for which there is a second collision between \(P\) and \(Q\).

6. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(A\) is held on the plane with the distance \(A P\) greater than 3 m .
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between \(A\) and the plane is \(\mu\)
Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,

1. find the potential energy lost by the whole system as a result of \(B\) falling 3 m . Given that the speed of \(B\) at the instant it hits the ground is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and ignoring air resistance,
2. use the work-energy principle to find the value of \(\mu\) After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
   Block \(A\) comes to instantaneous rest after moving a total distance of ( \(3 + d\) ) m from its point of release. Ignoring air resistance,
3. use the work-energy principle to find the value of \(d\)
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1. A spring of natural length \(a\) has one end attached to a fixed point \(A\). The other end of the spring is attached to a package \(P\) of mass \(m\).
   The package \(P\) is held at rest at the point \(B\), which is vertically below \(A\) such that \(A B = 3 a\).
   After being released from rest at \(B\), the package \(P\) first comes to instantaneous rest at \(A\). Air resistance is modelled as being negligible.

By modelling the spring as being light and modelling \(P\) as a particle,

1. show that the modulus of elasticity of the spring is \(2 m g\)
2. 1. Show that \(P\) attains its maximum speed when the extension of the spring is \(\frac { 1 } { 2 } a\)
   2. Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of \(a\) and \(g\). In reality, the spring is not light.
3. State one way in which this would affect your energy equation in part (b).

8. \begin{figure}[h] \end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where \(R S\) and \(S T\) are smooth fixed vertical walls. The vector \(\overrightarrow { R S }\) is in the direction of \(\mathbf { i }\) and the vector \(\overrightarrow { S T }\) is in the direction of \(( 2 \mathbf { i } + \mathbf { j } )\). A small ball \(B\) is projected across the floor towards \(R S\). Immediately before the impact with \(R S\), the velocity of \(B\) is \(( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The ball bounces off \(R S\) and then hits \(S T\). The ball is modelled as a particle.
Given that the coefficient of restitution between \(B\) and \(R S\) is \(e\),

1. find the full range of possible values of \(e\). It is now given that \(e = \frac { 1 } { 4 }\) and that the coefficient of restitution between \(B\) and \(S T\) is \(\frac { 1 } { 2 }\)
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after its impact with \(S T\).