Edexcel M2 (Mechanics 2) 2009 January

1. A car of mass 1500 kg is moving up a straight road, which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to the motion of the car from non-gravitational forces is constant and is modelled as a single constant force of magnitude 650 N . The car's engine is working at a rate of 30 kW .

Find the acceleration of the car at the instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.

1. Find the magnitude of the frictional force of the ground on the ladder.
2. Find, to the nearest degree, the value of \(\beta\).
3. State how you have used the modelling assumption that Reece is a particle.

1. A block of mass 10 kg is pulled along a straight horizontal road by a constant horizontal force of magnitude 70 N in the direction of the road. The block moves in a straight line passing through two points \(A\) and \(B\) on the road, where \(A B = 50 \mathrm {~m}\). The block is modelled as a particle and the road is modelled as a rough plane. The coefficient of friction between the block and the road is \(\frac { 4 } { 7 }\).
   1. Calculate the work done against friction in moving the block from \(A\) to \(B\).
   The block passes through \(A\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed of the block at \(B\).

4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6
\frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when

1. \(t = 6\),
2. \(t = 10\).

5. \begin{figure}[h] \end{figure} A uniform lamina \(A B C D\) is made by joining a uniform triangular lamina \(A B D\) to a uniform semi-circular lamina \(D B C\), of the same material, along the edge \(B D\), as shown in Figure 2. Triangle \(A B D\) is right-angled at \(D\) and \(A D = 18 \mathrm {~cm}\). The semi-circle has diameter \(B D\) and \(B D = 12 \mathrm {~cm}\).

1. Show that, to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(A D\) is 4.69 cm . Given that the centre of mass of a uniform semicircular lamina, radius \(r\), is at a distance \(\frac { 4 r } { 3 \pi }\) from the centre of the bounding diameter,
2. find, in cm to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(B D\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
3. Find, to the nearest degree, the angle which \(B D\) makes with the vertical.

6. \begin{figure}[h] \end{figure} A cricket ball is hit from a point \(A\) with velocity of \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), at an angle \(\alpha\) above the horizontal. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertically upwards. The point \(A\) is 0.9 m vertically above the point \(O\), which is on horizontal ground. The ball takes 3 seconds to travel from \(A\) to \(B\), where \(B\) is on the ground and \(O B = 57.6 \mathrm {~m}\), as shown in Figure 3. By modelling the motion of the cricket ball as that of a particle moving freely under gravity,

1. find the value of \(p\),
2. show that \(q = 14.4\),
3. find the initial speed of the cricket ball,
4. find the exact value of \(\tan \alpha\).
5. Find the length of time for which the cricket ball is at least 4 m above the ground.
6. State an additional physical factor which may be taken into account in a refinement of the above model to make it more realistic.

1. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal table. It collides directly with another particle \(Q\) of mass \(2 m\) which is moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 1 } { 5 } ( 9 e + 4 ) u\).
   The speed of \(P\) immediately after the collision is \(\frac { 1 } { 2 } u\).
2. Show that \(e = \frac { 1 } { 4 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision \(Q\) hits a smooth fixed vertical wall which is at right-angles to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\).
3. Show that \(P\) is a distance \(\frac { 3 } { 5 } d\) from the wall at the instant when \(Q\) hits the wall. Particle \(Q\) rebounds from the wall and moves so as to collide directly with particle \(P\) at the point \(B\). Given that the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 5 }\),
4. find, in terms of \(d\), the distance of the point \(B\) from the wall.