Edexcel M2 (Mechanics 2) 2018 June

1. A truck of mass 750 kg is moving with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 49 }\). The resistance to motion of the truck is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .
   1. Find the value of \(v\).
   On another occasion the truck is moving up the same straight road. The resistance to motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .
2. Find the acceleration of the truck at the instant when it is moving with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) lie on a smooth horizontal plane. A small ball of mass 0.2 kg is moving along the line \(A B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is at \(B\), the ball is given an impulse. Immediately after the impulse is given, the ball moves along the line \(B C\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(35 ^ { \circ }\) with the line \(A B\), as shown in Figure 1.

1. Find the magnitude of the impulse given to the ball.
2. Find the size of the angle between the direction of the impulse and the original direction of motion of the ball.

3. [The centre of mass of a semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h] \end{figure} Figure 2 shows the uniform lamina \(A B C D E\), such that \(A B D E\) is a square with sides of length \(2 a\) and \(B C D\) is a semicircle with diameter \(B D\).

1. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 20 a } { 3 ( 8 + \pi ) }\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
2. Find, to the nearest degree, the angle that \(D E\) makes with the downward vertical.

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), rests with its end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a light string attached to the rod at \(B\), as shown in Figure 3. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The coefficient of friction between the ground and the rod is \(\mu\).
Show that \(\mu = \frac { \cos \theta \sin \theta } { 2 - \cos ^ { 2 } \theta }\)

5. A particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal floor. Particle \(A\) collides directly with another particle \(B\) of mass \(2 m\) which is moving along the same straight line with speed \(u\) but in the opposite direction to \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 7 } { 5 } u\)
   2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). The first collision between \(A\) and \(B\) occurred at a distance \(x\) from the wall. The particles collide again at a distance \(y\) from the wall.
2. Find \(y\) in terms of \(x\).

1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$

1. Find
   1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
   2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).

7. A particle, of mass 0.3 kg , is projected from a point \(O\) on horizontal ground with speed \(u\). The particle is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = 2\), and moves freely under gravity. When the particle has moved a horizontal distance \(x\) from \(O\), its height above the ground is \(y\).

1. Show that $$y = 2 x - \frac { 5 g } { 2 u ^ { 2 } } x ^ { 2 }$$ The particle hits the ground at the point \(A\), where \(O A = 36 \mathrm {~m}\).
2. Find \(u\), the speed of projection.
3. Find the minimum kinetic energy of the particle as it moves between \(O\) and \(A\). The point \(B\) lies on the path of the particle. The direction of motion of the particle at \(B\) is perpendicular to the initial direction of motion of the particle.
4. Find the horizontal distance between \(O\) and \(B\).