Edexcel FM1 AS (Further Mechanics 1 AS) 2019 June

1. A lorry of mass 16000 kg moves along a straight horizontal road.

The lorry moves at a constant speed of \(25 \mathrm {~ms} ^ { - 1 }\)
In an initial model for the motion of the lorry, the resistance to the motion of the lorry is modelled as having constant magnitude 16000 N .

1. Show that the engine of the lorry is working at a rate of 400 kW . The model for the motion of the lorry along the same road is now refined so that when the speed of the lorry along the same road is \(V \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the lorry is modelled as having magnitude 640 V newtons. Assuming that the engine of the lorry is working at the same rate of 400 kW
2. use the refined model to find the speed of the lorry when it is accelerating at \(2.1 \mathrm {~ms} ^ { - 2 }\)

1. Two particles, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal plane. The particles are moving in opposite directions towards each other along the same straight line when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). In the collision the impulse of \(A\) on \(B\) has magnitude 5 mu .
   1. Find the coefficient of restitution between \(A\) and \(B\).
   2. Find the total loss in kinetic energy due to the collision.

1. A particle, \(P\), of mass \(m \mathrm {~kg}\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\) The total resistance to the motion of \(P\) is a force of magnitude \(\frac { 1 } { 5 } m g\)
   Use the work-energy principle to find the speed of \(P\) at the instant when it has moved a distance 8 m down the plane from the point of projection.

1. Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant.

Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
The coefficient of restitution between each pair of particles is \(e\).

1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. Given that the mass of \(P\) is \(k m\) and that there is a second collision,
2. write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.