Edexcel FM1 AS (Further Mechanics 1 AS) 2022 June

1. A car of mass 1200 kg moves up a straight road that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\)

The total resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. At the instant when the engine of the car is working at a rate of 32 kW and the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is \(0.5 \mathrm {~ms} ^ { - 2 }\) Find the value of \(R\)

1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are moving in opposite directions along the same straight line on a smooth horizontal plane when they collide directly.

Immediately before they collide, \(A\) is moving with speed \(2 u\) and \(B\) is moving with speed \(u\). The direction of motion of each particle is reversed by the collision.
In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is \(\frac { 9 m u } { 2 }\)

1. Find the value of the coefficient of restitution between \(A\) and \(B\).
2. Hence, write down the total loss in kinetic energy due to the collision, giving a reason for your answer.

1. A plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\), up a line of greatest slope of the plane. In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible. Using this model and the principle of conservation of mechanical energy,

1. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\). In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\frac { 3 } { 5 }\) Air resistance is still modelled as being negligible.
   Using this refined model and the work-energy principle,
2. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\).

1. A particle \(P\) of mass \(2 m \mathrm {~kg}\) is moving with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3 m \mathrm {~kg}\) which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   1. Show that \(v = \frac { 4 u ( 1 + e ) } { 5 }\)
   2. Show that \(\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }\)
   Given that the direction of motion of \(P\) is reversed by the collision,
2. find, in terms of \(u\) and \(e\), the speed of \(P\) immediately after the collision. After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 6 }\)
   Given that \(P\) and \(Q\) collide again,
3. find the full range of possible values of \(e\).