Edexcel FM1 (Further Mechanics 1) 2021 June

1. A van of mass 900 kg is moving along a straight horizontal road.

At the instant when the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). When the engine of the van is working at a constant rate of 18 kW , the van is moving along the road at a constant speed \(V \mathrm {~ms} ^ { - 1 }\)

1. Find the value of \(V\). Later on, the van is moving up a straight road that is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 1 } { 21 }\) At the instant when the speed of the van is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). The engine of the van is again working at a constant rate of 18 kW .
2. Find the acceleration of the van at the instant when \(v = 15\)

1. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle \(A\) has mass \(5 m\) and particle \(B\) has mass \(3 m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)
Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,

1. find the set of possible values of \(e\). Given also that the kinetic energy of \(A\) immediately after the collision is \(16 \%\) of the kinetic energy of \(A\) immediately before the collision,
2. find
   1. the value of \(e\),
   2. the magnitude of the impulse received by \(A\) in the collision, giving your answer in terms of \(m\) and \(v\).

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

A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.5 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( u \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(u\) is a positive constant, and the velocity of \(Q\) is \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant when the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\).
The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 } { 5 }\)
As a result of the collision, the direction of motion of \(P\) is deflected through an angle of \(90 ^ { \circ }\) and the direction of motion of \(Q\) is deflected through an angle of \(\alpha ^ { \circ }\)

1. Find the value of \(u\)
2. Find the value of \(\alpha\)
3. State how you have used the fact that \(P\) and \(Q\) have equal radii.

1. A particle \(P\) has mass 0.5 kg . It is moving in the \(x y\) plane with velocity \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\lambda ( - \mathbf { i } + \mathbf { j } )\) Ns, where \(\lambda\) is a positive constant.

The angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse is \(\theta ^ { \circ }\) Immediately after receiving the impulse, \(P\) is moving with speed \(4 \sqrt { 10 } \mathrm {~ms} ^ { - 1 }\)
Find (i) the value of \(\lambda\)
(ii) the value of \(\theta\)

5. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) represent fixed vertical walls, with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards the wall \(A B\). Immediately before hitting the wall \(A B\) the ball is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to \(A B\). The ball hits the wall \(A B\) and then hits the wall \(B C\).
The coefficient of restitution between the ball and the wall \(A B\) is \(\frac { 1 } { 3 }\)
The coefficient of restitution between the ball and the wall \(B C\) is \(e\).
The floor and the walls are modelled as being smooth.
The ball is modelled as a particle.
The ball loses half of its kinetic energy in the impact with the wall \(A B\).

1. Find the exact value of \(\cos \theta\). The ball loses half of its remaining kinetic energy in the impact with the wall \(B C\).
2. Find the exact value of \(e\).

6. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\)
The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\)
By modelling \(P\) as a particle,

1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .

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 3 represents the plan view of part of a smooth horizontal floor, where \(A B\) is a fixed smooth vertical wall. The direction of \(\overrightarrow { A B }\) is in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\)
A small ball of mass 0.25 kg is moving on the floor when it strikes the wall \(A B\).
Immediately before its impact with the wall \(A B\), the velocity of the ball is \(( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
Immediately after its impact with the wall \(A B\), the velocity of the ball is \(\mathbf { v m s } ^ { - 1 }\)
The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\)
By modelling the ball as a particle,

1. show that \(\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }\)
2. Find the magnitude of the impulse received by the ball in the impact.