Edexcel M2 (Mechanics 2)

1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).

Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)

2. \begin{figure}[h] \end{figure} A uniform ladder \(A B\) has one end \(A\) on smooth horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is modelled as a uniform rod of mass \(m\) and length 4a. The ladder is kept in equilibrium by a horizontal force \(F\) acting at a point \(C\) of the ladder where \(A C = a\). The force \(F\) and the ladder lie in a vertical plane perpendicular to the wall. The ladder is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = 2\), as shown in Fig. 1. Find \(F\) in terms of \(m\) and \(g\).
(6 marks)

3. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles triangle \(A B C\), where \(A B = A C\). The mid-point of \(B C\) is \(M , A M = 30 \mathrm {~cm}\) and \(B M = 40 \mathrm {~cm}\). The mid-points of \(A C\) and \(A B\) are \(D\) and \(E\) respectively. The triangular portion \(A D E\) is removed leaving a uniform plane lamina \(B C D E\) as shown in Fig. 2.

1. Show that the centre of mass of the lamina \(B C D E\) is \(6 \frac { 2 } { 3 } \mathrm {~cm}\) from \(B C\).
   (6 marks)
   The lamina \(B C D E\) is freely suspended from \(D\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle which \(D E\) makes with the vertical.
   (3 marks)

4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate at which the cyclist is working.
   (6 marks)

5. A smooth sphere \(S\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal plane. The sphere \(S\) collides with another smooth sphere \(T\), of equal radius to \(S\) but of mass \(k m\), moving in the same straight line and in the same direction with speed \(\lambda u , 0 < \lambda < \frac { 1 } { 2 }\). The coefficient of restitution between \(S\) and \(T\) is \(e\). Given that \(S\) is brought to rest by the impact,

1. show that \(e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }\).
2. Deduce that \(k > 1\).

6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
2. find the speed of \(P\) immediately after the impulse has been applied.

7. \begin{figure}[h] \end{figure} A shot is projected upwards from the top of a cliff with a velocity of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. It strikes the ground 52.5 m vertically below the level of the point of projection, as shown in Fig. 3. The motion of the shot is modelled as that of a particle moving freely under gravity. Find, to 3 significant figures,

1. the horizontal distance from the point of projection at which the shot strikes the ground,
2. the speed of the shot as it strikes the ground.

8. A particle \(P\) is projected up a line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is projected from the point \(O\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to instantaneous rest at the point \(A\). By Using the Work-Energy principle, or otherwise,

1. find, to 3 significant figures, the length \(O A\).
2. Show that \(P\) will slide back down the plane.
3. Find, to 3 significant figures, the speed of \(P\) when it returns to \(O\).