Edexcel M2 (Mechanics 2)

1. A bullet of mass 25 g is fired directly at a fixed wooden block of thickness 4 cm and passes through it. When the bullet hits the block, it is travelling horizontally at \(200 \mathrm {~ms} ^ { - 1 }\). The block exerts a constant resistive force of 8000 N on the bullet.
   1. Find the work done by the block on the bullet.
   By using the Work-Energy principle,
2. show that the bullet emerges from the block with speed \(120 \mathrm {~ms} ^ { - 1 }\).

2. A car is travelling along a straight horizontal road against resistances to motion which are constant and total 2000 N . When the engine of the car is working at a rate of \(H\) kilowatts, the maximum speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(H\). The car driver wishes to overtake another vehicle so she increases the rate of working of the engine by \(20 \%\) and this results in an initial acceleration of \(0.32 \mathrm {~ms} ^ { - 2 }\). Assuming that the resistances to motion remain constant,
2. find the mass of the car.
   (4 marks)

3. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) placed with edge \(B C\) along the line of greatest slope of a plane inclined at an angle \(\theta\) to the horizontal. The lengths \(A C\) and \(B C\) are 15 cm and 9 cm respectively and \(\angle A B C\) is a right angle.

1. Find the distance of the centre of mass of the lamina from
   1. \(\quad A B\),
   2. \(B C\). Assuming that the plane is rough enough to prevent the lamina from slipping,
2. find in degrees, correct to 1 decimal place, the maximum value of \(\theta\) for which the lamina remains in equilibrium.
   (4 marks)

4. The velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds is given by \(\mathbf { v } = 3 t \mathbf { i } - t ^ { 2 } \mathbf { j }\).

1. Find the magnitude of the acceleration of \(P\) when \(t = 2\). When \(t = 0\), the displacement of \(P\) from a fixed origin \(O\) is \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
2. Show that the displacement of \(P\) from \(O\) when \(t = 6\) is given by \(k ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(k\) is an integer which you should find.
   (6 marks)

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length \(2 a\) and mass 8 kg is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium inclined at an angle of \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(F\) newtons acting horizontally at \(B\) which is below the level of \(A\) as shown in Figure 2.

1. Find, correct to 3 significant figures, the value of \(F\).
2. Show that the magnitude of the reaction at the hinge is 133 N , correct to 3 significant figures, and find to the nearest degree the acute angle which the reaction makes with the vertical.

6. A particle \(P\) is projected from a point \(A\) on horizontal ground with speed \(u\) at an angle of elevation \(\alpha\) and moves freely under gravity. \(P\) hits the ground at the point \(B\).

1. Show that \(A B = \frac { u ^ { 2 } } { g } \sin 2 \alpha\). An archer fires an arrow with an initial speed of \(45 \mathrm {~ms} ^ { - 1 }\) at a target which is level with the point of projection and at a distance of 80 m . Given that the arrow hits the target,
2. find in degrees, correct to 1 decimal place, the two possible angles of projection.
3. Write down, with a reason, which of the two possible angles of projection would give the shortest time of flight.
   (2 marks)
4. Show that the minimum time of flight is 1.8 seconds, correct to 1 decimal place.
   (2 marks)

7. A smooth sphere \(A\) of mass \(4 m\) is moving on a smooth horizontal plane with speed \(u\). It collides directly with a stationary smooth sphere \(B\) of mass \(5 m\) and with the same radius as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).

1. Show that after the collision the speed of \(B\) is 4 times greater than the speed of \(A\).
   (7 marks)
   Sphere \(B\) subsequently hits a smooth vertical wall at right angles. After rebounding from the wall, \(B\) collides with \(A\) again and as a result of this collision, \(B\) comes to rest. Given that the coefficient of restitution between \(B\) and the wall is \(e\),
2. find \(e\). END