Edexcel M2 (Mechanics 2) 2008 January

1. A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude \(R\) newtons. Modelling the parcel as a particle, find
   1. the kinetic energy lost by the parcel in coming to rest,
   2. the value of \(R\).
      
   3. At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { p }\) metres, with respect to a fixed origin \(O\), where
   $$\mathbf { p } = \left( 3 t ^ { 2 } - 6 t + 4 \right) \mathbf { i } + \left( 3 t ^ { 3 } - 4 t \right) \mathbf { j } .$$ Find
2. the velocity of \(P\) at time \(t\) seconds,
3. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\). When \(t = 1\), the particle \(P\) receives an impulse of \(( 2 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Given that the mass of \(P\) is 0.5 kg ,
4. find the velocity of \(P\) immediately after the impulse.

3. A car of mass 1000 kg is moving at a constant speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N .

1. Show that \(\sin \theta = \frac { 1 } { 14 }\). When the car is travelling up the road at \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car comes to rest, without braking, having moved a distance \(y\) metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N .
2. Find the value of \(y\).

4. \begin{figure}[h] \end{figure} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(A B C\), with \(\angle A B C = 90 ^ { \circ } , A B = 12 \mathrm {~cm}\) and \(B C = 21 \mathrm {~cm}\). The point \(O\) is 5 cm from \(A B\) and 5 cm from \(B C\), as shown in Figure 1.

1. Find the distance of the centre of mass of \(S\) from
   1. \(A B\),
   2. \(B C\). The set square is freely suspended from \(C\) and hangs in equilibrium.
2. Find, to the nearest degree, the angle between \(C B\) and the vertical.

5. \begin{figure}[h] \end{figure} A ladder \(A B\), of mass \(m\) and length \(4 a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3 m\) is fixed on the ladder at the point \(C\), where \(A C = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground.

6. \begin{figure}[h] \end{figure} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector 47.5j metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \(( 2 u \mathbf { i } + 5 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30 \mathbf { i }\) metres, as shown in Figure 3.

1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s .
2. Find the value of \(u\).
3. Find the speed of \(P\) at \(B\).

1. A particle \(P\) of mass \(2 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3 m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 2 }\).
   1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 8 } { 5 } u\).
   2. Find the total kinetic energy lost in the collision.
   After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).
2. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\).