Edexcel FM1 (Further Mechanics 1) 2020 June

1. A particle \(P\) of mass 0.5 kg is moving with velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after receiving the impulse, \(P\) is moving with velocity \(( - \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Find the magnitude of \(\mathbf { J }\).
   The angle between the direction of the impulse and the direction of motion of \(P\) immediately before receiving the impulse is \(\alpha ^ { \circ }\)
2. Find the value of \(\alpha\)

1. A truck of mass 1200 kg is moving along a straight horizontal road.

At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck is modelled as a force of magnitude \(( 900 + 9 v ) \mathrm { N }\). The engine of the truck is working at a constant rate of 25 kW .

1. Find the deceleration of the truck at the instant when \(v = 25\) Later on, the truck is moving up a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck from non-gravitational forces is modelled as a force of magnitude ( \(900 + 9 v\) ) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(V\).

1. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\).

The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the direction of motion of each of the particles is unchanged by the collision.
   (8) After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2 m\), which is at rest on the surface. The coefficient of restitution between \(B\) and \(C\) is also \(e\).
2. Show that there will be a second collision between \(A\) and \(B\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where \(A B\) represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is \(( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Immediately after the impact the velocity of the ball is \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
The coefficient of restitution between the ball and the wall is \(e\).

1. Show that \(A B\) is parallel to \(( 2 \mathbf { i } + 3 \mathbf { j } )\).
2. Find the value of \(e\).

1. A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.2 kg .

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The kinetic energy of \(Q\) immediately after the collision is half the kinetic energy of \(Q\) immediately before the collision.

1. Find
   1. the velocity of \(P\) immediately after the collision,
   2. the velocity of \(Q\) immediately after the collision,
   3. the coefficient of restitution between \(P\) and \(Q\),
      carefully justifying your answers.
2. Find the size of the angle through which the direction of motion of \(P\) is deflected by the collision.

1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).

The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)
The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\)
By modelling the package as a particle,

1. show that \(k = \frac { 15 } { 26 }\)
2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).

7. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(C D\) represent fixed vertical walls, with \(A B\) parallel to \(C D\). A small ball is projected along the floor towards wall \(A B\). Immediately before hitting wall \(A B\), the ball is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to \(A B\), where \(0 < \alpha < \frac { \pi } { 2 }\) The ball hits wall \(A B\) and then hits wall \(C D\).
After the impact with wall \(C D\), the ball is moving at angle \(\frac { 1 } { 2 } \alpha\) to \(C D\).
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 2 } { 3 }\)
The coefficient of restitution between the ball and wall \(C D\) is also \(\frac { 2 } { 3 }\)
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

1. Show that \(\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }\)
2. Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.