Edexcel M2 (Mechanics 2)

1. A particle \(P\) moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin \(O\) is given by

$$\mathbf { r } = \left( \frac { 3 } { 2 } t ^ { 2 } - 3 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - k t \right) \mathbf { j } ,$$ where \(k\) is a constant and \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find an expression for the velocity of \(P\) at time \(t\).
2. Given that \(P\) comes to rest instantaneously, find the value of \(k\).

2. Two smooth spheres \(P\) and \(Q\) of equal radius and of mass \(2 m\) and \(5 m\) respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, \(P\) and \(Q\) travel in opposite directions with speeds of \(3 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. Given that the coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\), find the speeds of \(P\) and \(Q\) before the collision.
(6 marks)

3. A car of mass 1200 kg experiences a resistance to motion, \(R\) newtons, which is proportional to its speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the power output of the car engine is 90 kW and the car is travelling along a horizontal road, its maximum speed is \(50 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(R = 36 v\). The car ascends a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 14 }\).
2. Find, correct to 3 significant figures, the maximum speed of the car up the hill assuming that the power output of the engine is unchanged.
   (6 marks)

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of mass 2 kg and length \(2 a\). The end \(A\) is attached by a smooth hinge to a fixed point on a vertical wall so that the rod can rotate freely in a vertical plane. A mass of 6 kg is placed at \(B\) and the rod is held in a horizontal position by a light string joining the midpoint of the rod to a point \(C\) on the wall, vertically above \(A\). The string is inclined at an angle of \(60 ^ { \circ }\) to the wall.

1. Show that the tension in the string is \(28 g\).
2. Find the magnitude and direction of the force exerted by the hinge on the rod, giving your answers correct to 3 significant figures.

5. A particle \(P\) moves in a straight line with an acceleration of \(( 6 t - 10 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) at time \(t\) seconds. Initially \(P\) is at \(O\), a fixed point on the line, and has velocity \(3 \mathrm {~ms} ^ { - 1 }\).

1. Find the values of \(t\) for which the velocity of \(P\) is zero.
2. Show that, during the first two seconds, \(P\) travels a distance of \(6 \frac { 26 } { 27 } \mathrm {~m}\).

6. \begin{figure}[h] \end{figure} A football player strikes a ball giving it an initial speed of \(14 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal as shown in Figure 2. At the instant he strikes the ball it is 0.6 m vertically above the point \(P\) on the ground. The trajectory of the ball is in a vertical plane containing \(P\) and \(M\), the middle of the goal-line. The distance between \(P\) and \(M\) is 12 m and the ground is horizontal. Given that the ball passes over the point \(M\) without bouncing,

1. find, to the nearest degree, the minimum value of \(\alpha\). Given that the crossbar of the goal is 2.4 m above \(M\) and that \(\tan \alpha = \frac { 4 } { 3 }\),
2. show that the ball passes 4.2 m vertically above the crossbar.

7. \begin{figure}[h] \end{figure} Figure 3 shows a hotel 'key' consisting of a rectangle \(O A B D\), where \(O A = 8 \mathrm {~cm}\) and \(O D = 4 \mathrm {~cm}\), joined to a semicircle whose diameter \(B C\) is 4 cm long. The thickness of the key is negligible and the same material is used throughout. The key is modelled as a uniform lamina.
Using this model,

1. find, correct to 3 significant figures, the distance of the centre of mass from
   1. OD ,
   2. \(O A\). A small circular hole of negligible diameter is made at the mid-point of \(B C\) so that the key can be hung on a smooth peg. When the key is freely suspended from the peg,
2. find, correct to 3 significant figures, the acute angle made by \(O A\) with the vertical.