Edexcel FM1 (Further Mechanics 1) Specimen

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 2 \mathbf { i } - \mathbf { j } )\) Ns.

Show that the kinetic energy gained by \(P\) as a result of the impulse is 12 J .

1. A parcel of mass 5 kg is projected with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined ramp.
   The ramp is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\)
   The parcel is projected from the point \(A\) on the ramp and comes to instantaneous rest at the point \(B\) on the ramp, where \(A B = 14 \mathrm {~m}\).

The coefficient of friction between the parcel and the ramp is \(\mu\).
In a model of the parcel's motion, the parcel is treated as a particle.

1. Use the work-energy principle to find the value of \(\mu\).
2. Suggest one way in which the model could be refined to make it more realistic.

1. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface.

Initially the particle is at rest at a point \(O\) between two fixed parallel vertical walls.
The point \(O\) is equidistant from the two walls and the walls are 4 m apart.
At time \(t = 0\) the particle is projected from \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is \(\frac { 3 } { 4 }\)
The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u\) Ns.

1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 7\) seconds.
2. Find the value of \(u\).

4. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are perpendicular vertical walls. The floor and the walls are modelled as smooth.
A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle.
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 1 } { \sqrt { 3 } }\)
The coefficient of restitution between the ball and wall \(B C\) is \(\sqrt { \frac { 2 } { 5 } }\)

1. Show that, using this model, the final kinetic energy of the ball is \(35 \%\) of the initial kinetic energy of the ball.
2. In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?

1. A car of mass 600 kg is moving along a straight horizontal road.

At the instant 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 \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .

1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
   At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(w\).

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

A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(3 m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The coefficient of restitution between the spheres is \(\frac { 1 } { 4 }\)

1. Find the velocity of \(B\) immediately after the collision.
2. Find, to the nearest degree, the size of the angle through which the direction of motion of \(B\) is deflected as a result of the collision.

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(3 m g\).

The other end of the string is attached to a fixed point \(O\) on a ceiling.
The particle hangs freely in equilibrium at a distance \(d\) vertically below \(O\).

1. Show that \(d = \frac { 4 } { 3 } a\). The point \(A\) is vertically below \(O\) such that \(O A = 2 a\).
   The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).
2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest.
3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).
4. Find, in terms of \(a\), the distance \(O B\).

1. A particle \(P\) of mass \(2 m\) and a particle \(Q\) of mass \(5 m\) are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The direction of motion of \(Q\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 3 }\)
2. show that the kinetic energy lost in the collision is \(\frac { 40 m u ^ { 2 } } { 7 }\).
3. Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if \(e > \frac { 1 } { 3 }\)