Edexcel FM1 AS (Further Mechanics 1 AS) 2020 June

1. Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are at rest on a smooth horizontal plane. Particle \(P\) is given a horizontal impulse, of magnitude \(I\), in the direction \(P Q\). Particle \(P\) then collides directly with \(Q\). Immediately after this collision, \(P\) is at rest and \(Q\) has speed \(w\). The coefficient of restitution between the particles is \(e\).
   1. Find \(I\) in terms of \(m\) and \(w\).
   2. Show that \(e = \frac { 1 } { 4 }\)
   3. Find, in terms of \(m\) and \(w\), the total kinetic energy lost in the collision between \(P\) and \(Q\).

1. A car of mass 1000 kg moves along a straight horizontal road.

In all circumstances, when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(c v ^ { 2 } \mathrm {~N}\), where \(c\) is a constant. The maximum power that can be developed by the engine of the car is 50 kW .
At the instant when the speed of the car is \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the engine is working at its maximum power, the acceleration of the car is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Convert \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) into \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the acceleration of the car at the instant when the speed of the car is \(144 \mathrm { kmh } ^ { - 1 }\) and the engine is working at its maximum power. The maximum speed of the car when the engine is working at its maximum power is \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
3. Find, to the nearest whole number, the value of \(V\).

1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).

Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)

1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
2. Find the value of \(k\).

1. A small ball, of mass \(m\), is thrown vertically upwards with speed \(\sqrt { 8 g H }\) from a point \(O\) on a smooth horizontal floor. The ball moves towards a smooth horizontal ceiling that is a vertical distance \(H\) above \(O\). The coefficient of restitution between the ball and the ceiling is \(\frac { 1 } { 2 }\)
   In a model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to air resistance of constant magnitude \(\frac { 1 } { 2 } \mathrm { mg }\).
   Using this model,
   1. use the work-energy principle to find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the ceiling,
   2. find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the floor at \(O\) for the first time.
   In a simplified model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to no air resistance. Using this simplified model,
2. explain, without any detailed calculation, why the speed of the ball, immediately before it strikes the floor at \(O\) for the first time, would still be less than \(\sqrt { 8 g H }\)