Edexcel FM1 AS (Further Mechanics 1 AS) 2018 June

1. A small ball of mass 0.3 kg is released from rest from a point 3.6 m above horizontal ground. The ball falls freely under gravity, hits the ground and rebounds vertically upwards.

In the first impact with the ground, the ball receives an impulse of magnitude 4.2 Ns . The ball is modelled as a particle.

1. Find the speed of the ball immediately after it first hits the ground.
2. Find the kinetic energy lost by the ball as a result of the impact with the ground.

2. \begin{figure}[h] \end{figure} Figure 1 shows a ramp inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\)
A parcel of mass 4 kg is projected, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point \(A\) on the ramp.
The parcel moves up a line of greatest slope of the ramp and first comes to instantaneous rest at the point \(B\), where \(A B = 2.5 \mathrm {~m}\).
The parcel is modelled as a particle.
The total resistance to the motion of the parcel from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.

1. Use the work-energy principle to show that \(R = 8.8\) After coming to instantaneous rest at \(B\), the parcel slides back down the ramp. The total resistance to the motion of the particle is modelled as a constant force of magnitude 8.8N.
2. Find the speed of the parcel at the instant it returns to \(A\).
3. Suggest two improvements that could be made to the model.

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1. A van of mass 750 kg is moving along a straight horizontal road. At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(\lambda \nu \mathrm { N }\), where \(\lambda\) is a constant.

The engine of the van is working at a constant rate of 18 kW .
At the instant when \(v = 15\), the acceleration of the van is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Show that \(\lambda = 50\) The van now moves up a straight road inclined at an angle to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\)
   At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v . When the engine of the van is working at a constant rate of 12 kW , the van is moving at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

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1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) is moving in the opposite direction to \(P\) along the same straight line.

The particles collide directly.
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. find \(x\) in terms of \(d\).