Edexcel M2 (Mechanics 2) 2004 June

1. A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion of the lorry has magnitude 750 N and the lorry's engine is working at a rate of 36 kW .
   1. Find the acceleration of the lorry when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The lorry comes to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The magnitude of the resistance to motion from non-gravitational forces remains 750 N . The lorry moves up the hill at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate at which the lorry's engine is now working.
   (3)

2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] A ball has mass 0.2 kg . It is moving with velocity ( 30 i ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. The bat exerts an impulse of \(( - 4 \mathbf { i } + 4 \mathbf { j } )\) Ns on the ball. Find

1. the velocity of the ball immediately after the impact,
2. the angle through which the ball is deflected as a result of the impact,
3. the kinetic energy lost by the ball in the impact.

3. \begin{figure}[h] \end{figure} Figure 1 shows a decoration which is made by cutting the shape of a simple tree from a sheet of uniform card. The decoration consists of a triangle \(A B C\) and a rectangle \(P Q R S\). The points \(P\) and \(S\) lie on \(B C\) and \(M\) is the mid-point of both \(B C\) and \(P S\). The triangle \(A B C\) is isosceles with \(A B = A C , B C = 4 \mathrm {~cm} , A M = 6 \mathrm {~cm} , P S = 2 \mathrm {~cm}\) and \(P Q = 3 \mathrm {~cm}\).

1. Find the distance of the centre of mass of the decoration from \(B C\). The decoration is suspended from \(Q\) and hangs freely.
2. Find, in degrees to one decimal place, the angle between \(P Q\) and the vertical.

4. At time \(t\) seconds, the velocity of a particle \(P\) is \([ ( 4 t - 7 ) \mathbf { i } - 5 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is at the point with position vector \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\).

1. Find an expression for the position vector of \(P\) after \(t\) seconds, giving your answer in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { m }\). A second particle \(Q\) moves with constant velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( - 7 \mathrm { i } ) \mathrm { m }\).
2. Prove that \(P\) and \(Q\) collide.

5. Two small smooth spheres, \(P\) and \(Q\), of equal radius, have masses \(2 m\) and \(3 m\) respectively. The sphere \(P\) is moving with speed \(5 u\) on a smooth horizontal table when it collides directly with \(Q\), which is at rest on the table. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(2 ( 1 + e ) u\). After the collision, \(Q\) hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f , 0 < f \leqslant 1\).
2. Show that, when \(e = 0.4\), there is a second collision between \(P\) and \(Q\). Given that \(e = 0.8\) and there is a second collision between \(P\) and \(Q\),
3. find the range of possible values of \(f\).

6. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.6 . The other end \(B\) of the ladder rests against a smooth vertical wall. A builder of mass 10 m stands at the top of the ladder. To prevent the ladder from slipping, the builder's friend pushes the bottom of the ladder horizontally towards the wall with a force of magnitude \(P\). This force acts in a direction perpendicular to the wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 2 }\).

1. Show that the reaction of the wall on the ladder has magnitude 7 mg .
2. Find, in terms of \(m\) and \(g\), the range of values of \(P\) for which the ladder remains in equilibrium.

7. \begin{figure}[h] \end{figure} In a ski-jump competition, a skier of mass 80 kg moves from rest at a point \(A\) on a ski-slope. The skier's path is an arc \(A B\). The starting point \(A\) of the slope is 32.5 m above horizontal ground. The end \(B\) of the slope is 8.1 m above the ground. When the skier reaches \(B\), she is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and moving upwards at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Fig. 2. The distance along the slope from \(A\) to \(B\) is 60 m . The resistance to motion while she is on the slope is modelled as a force of constant magnitude \(R\) newtons. By using the work-energy principle,

1. find the value of \(R\). On reaching \(B\), the skier then moves through the air and reaches the ground at the point \(C\). The motion of the skier in moving from \(B\) to \(C\) is modelled as that of a particle moving freely under gravity.
2. Find the time for the skier to move from \(B\) to \(C\).
3. Find the horizontal distance from \(B\) to \(C\).
4. Find the speed of the skier immediately before she reaches \(C\). END