Edexcel FM1 (Further Mechanics 1) 2024 June

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

A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse exerted on \(A\) by \(B\) in the collision.
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.

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

A particle \(P\) of mass \(m\) is at rest at a point on the plane. The particle is projected up the plane with speed \(\sqrt { 2 a g }\)
The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance \(d\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on \(P\) as it moves up the plane is \(\frac { 4 m g } { 35 }\) Air resistance is assumed to be negligible.
   Using the work-energy principle,
2. find \(d\) in terms of \(a\).

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of \(P \mathrm {~kW}\).
Using the model,

1. find the value of \(P\). The car now travels in a straight line up a road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 49 }\)
   • In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude \(20 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car
   At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at \(10 \mathrm {~ms} ^ { - 1 }\), the acceleration of the car is \(a \mathrm {~ms} ^ { - 2 }\) Using the refined model,
2. find the value of \(a\). Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the refined model,
3. find the value of \(U\).

1. A particle \(A\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.

The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)

1. Show that the speed of \(B\) immediately after the collision is \(2 u ( 1 + e )\). After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
2. Show that there will be a second collision between \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\)
   Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
3. the magnitude of the impulse received by \(B\) in its collision with the wall,
4. the loss in kinetic energy of \(B\) due to its collision with the wall.

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\). One end of the string is attached to a fixed point \(A\) on a horizontal ceiling. The other end is attached to a particle \(P\) of mass \(m\).

The particle \(P\) hangs in equilibrium at the point \(E\), where \(A E = 3 a\).
The particle \(P\) is then projected vertically downwards from \(E\) with speed \(\frac { 3 } { 2 } \sqrt { a g }\)
Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when \(P\) first comes to instantaneous rest. Give your answer in the form kmga, where \(k\) is a constant to be found.

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

A particle \(P\) is moving with velocity ( \(4 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The coefficient of restitution between \(P\) and the wall is \(e\).

1. Find the value of \(e\). After the collision, \(P\) goes on to hit a second smooth vertical wall, which is parallel to \(\mathbf { i }\).
   The coefficient of restitution between \(P\) and this second wall is \(\frac { 1 } { 3 }\)
   The angle through which the direction of motion of \(P\) has been deflected by its collision with this second wall is \(\alpha ^ { \circ }\).
2. Find the value of \(\alpha\), giving your answer to the nearest whole number.

7. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
2. Show that \(e > \frac { 1 } { 3 }\)
3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)