Edexcel FM1 AS (Further Mechanics 1 AS) 2023 June

1. Two particles, \(P\) and \(Q\), of masses \(3 m\) and \(2 m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.

Immediately before the collision, \(P\) is moving with speed \(2 u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\frac { 9 m u } { 2 }\)

1. Find the speed of \(P\) immediately after the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that the speed of \(Q\) immediately before the collision is \(u\),
2. find the value of \(e\).

1. A racing car of mass 750 kg is moving along a straight horizontal road at a constant speed of \(U \mathbf { k m ~ h } ^ { - \mathbf { 1 } }\). The engine of the racing car is working at a constant rate of 60 kW .

The resistance to the motion of the racing car is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. Using the model,

1. find the value of \(U\) Later on, the racing car is accelerating up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 5 } { 49 }\). The engine of the racing car is working at a constant rate of 60 kW . The total resistance to the motion of the racing car from non-gravitational forces is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. At the instant when the acceleration of the racing car is \(2 \mathrm {~ms} ^ { - 2 }\), the speed of the racing car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model,
2. find the value of \(V\)

1. A stone of mass 0.5 kg is projected vertically upwards with a speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(A\). The point \(A\) is 2.5 m above horizontal ground.

The speed of the stone as it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the stone from the instant it is projected from \(A\) until the instant it hits the ground is modelled as that of a particle moving freely under gravity.

1. Use the model and the principle of conservation of mechanical energy to find the value of \(U\). In reality, the stone will be subject to air resistance as it moves from \(A\) to the ground.
2. State how this would affect your answer to part (a). The ground is soft and the stone sinks a vertical distance \(d \mathrm {~cm}\) into the ground. The resistive force exerted on the stone by the ground is modelled as a constant force of magnitude 2000 N and the stone is modelled as a particle.
3. Use the model and the work-energy principle to find the value of \(d\), giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} Three particles, \(P , Q\) and \(R\), lie at rest on a smooth horizontal plane. The particles are in a straight line with \(Q\) between \(P\) and \(R\), as shown in Figure 1 . Particle \(P\) is projected towards \(Q\) with speed \(u\). At the same time, \(R\) is projected with speed \(\frac { 1 } { 2 } u\) away from \(Q\), in the direction \(Q R\). Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(2 m\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision between \(P\) and \(Q\) is $$\frac { u ( 1 + e ) } { 3 }$$ It is given that \(e > \frac { 1 } { 2 }\)
2. Determine whether there is a collision between \(Q\) and \(R\).
3. Determine the direction of motion of \(P\) immediately after the collision between \(P\) and \(Q\).
4. Find, in terms of \(m , u\) and \(e\), the total kinetic energy lost in the collision between \(P\) and \(Q\), simplifying your answer.
5. Explain how using \(e = 1\) could be used to check your answer to part (d).