Edexcel FM1 AS (Further Mechanics 1 AS) 2024 June

1. A particle \(A\) has mass \(2 m\) and a particle \(B\) has mass \(3 m\). The particles are moving in opposite directions along the same straight line and collide directly.

Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(0.5 u\) and the speed of \(B\) is \(w\). Given that the direction of motion of each particle is reversed by the collision,

1. find \(w\) in terms of \(u\)
2. find the coefficient of restitution between the particles,
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(A\) in the collision.

1. A lorry has mass 5000 kg .

In all circumstances, when the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion of the lorry from non-gravitational forces is modelled as having magnitude \(490 v\) newtons. The lorry moves along a straight horizontal road at \(12 \mathrm {~ms} ^ { - 1 }\), with its engine working at a constant rate of 84 kW . Using the model,

1. find the acceleration of the lorry. Another straight road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 14 }\)
   With its engine again working at a constant rate of 84 kW , the lorry can maintain a constant speed of \(V \mathrm {~ms} ^ { - 1 }\) up the road. Using the model,
2. find the value of \(V\).

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the end elevation of a building which sits on horizontal ground. The side of the building is vertical and has height \(h\). A small stone of mass \(m\) is at rest on the roof of the building at the point \(A\). The stone slides from rest down a line of greatest slope of the roof and reaches the edge \(B\) of the roof with speed \(\sqrt { 2 g h }\) The stone then moves under gravity before hitting the ground with speed \(W\).
In a model of the motion of the stone from \(\boldsymbol { B }\) to the ground

• the stone is modelled as a particle
• air resistance is ignored

Using the principle of conservation of mechanical energy and the model,

1. find \(W\) in terms of \(g\) and \(h\). In a model of the motion of the stone from \(\boldsymbol { A }\) to \(\boldsymbol { B }\)
   • the stone is modelled as a particle of mass \(m\)
   • air resistance is ignored
   • the roof of the building is modelled as a rough plane inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
   • the coefficient of friction between the stone and the roof is \(\frac { 1 } { 3 }\)
   • \(A B = d\)
   Using this model,
2. find, in terms of \(m\) and \(g\), the magnitude of the frictional force acting on the stone as it slides down the roof,
3. use the work-energy principle to find \(d\) in terms of \(h\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4 m\) are at rest on a smooth horizontal plane, as shown in Figure 2. Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.
The coefficient of restitution between the particles is \(e\), where \(e > \frac { 1 } { 4 }\) As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\frac { u } { 5 } ( 4 e - 1 )\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\). The coefficient of restitution between \(P\) and the wall is \(f\), where \(f > 0\)
   Given that \(e = \frac { 3 } { 4 }\)
2. find, in terms of \(m , u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. After its impact with the wall, \(P\) goes on to collide with \(Q\) again.
3. Find the complete range of possible values of \(f\).