Edexcel M2 (Mechanics 2) 2017 October

1. A small ball \(B\) of mass 0.2 kg is hit by a bat. Immediately before being hit, \(B\) has velocity \(( 10 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being hit, \(B\) has velocity \(( 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the magnitude of the impulse exerted on \(B\) by the bat.
   (4)

2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .

1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find, in kW , the rate at which the van's engine is now working.
   "

3. A particle \(P\) of mass 4 kg moves from point \(A\) to point \(B\) down a line of greatest slope of a fixed rough plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal and \(A B = 12 \mathrm {~m}\). The coefficient of friction between \(P\) and the plane is 0.5

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that the speed of \(P\) at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. use the work-energy principle to find the speed of \(P\) at \(A\).

4. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The end \(A\) rests against a rough vertical wall. One end of a light inextensible string is attached to the rod at the point \(C\), where \(A C = 2 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is vertically above \(A\), with the string perpendicular to the rod. A particle of mass \(m\) is attached to the rod at the end \(B\). The rod is in equilibrium in a vertical plane which is perpendicular to the wall. The rod is inclined at \(60 ^ { \circ }\) to the wall, as shown in Figure 1. Find, in terms of \(m\) and \(g\),

1. the tension in the string,
2. the magnitude of the horizontal component of the force exerted by the wall on the rod. The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
3. find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) is in the shape of an equilateral triangle with sides of length \(4 a\). The midpoint of \(B C\) is \(D\). The point \(E\) lies on \(A D\) with \(D E = \frac { 3 a } { 2 }\). A circular hole, with centre \(E\) and radius \(a\), is made in the lamina \(A B C\) to form the lamina \(L\), shown shaded in Figure 2.

1. Find the distance of the centre of mass of \(L\) from \(D\). The lamina \(L\) is freely suspended from the point \(B\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the acute angle between \(A D\) and the downward vertical.

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) is
   \(( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. When \(t = 3 , v = 2\)
   1. Find \(v\) in terms of \(t\).
      (4)
   The particle first comes to instantaneous rest at the point \(A\) and then comes to instantaneous rest again at the point \(B\).
2. Find the distance \(A B\).
   

7. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.

1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
4. Find the horizontal distance from \(A\) to \(B\).

8. A particle \(A\) of mass \(3 m\) lies at rest on a smooth horizontal floor. A particle \(B\) of mass \(2 m\) is moving in a straight line on the floor with speed \(u\) when it collides directly with \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). As a result of the collision the direction of motion of \(B\) is reversed.

1. Find an expression, in terms of \(u\) and \(e\), for
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. The particle \(A\) subsequently strikes a smooth vertical wall. The wall is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(A\) and the wall is \(\frac { 1 } { 7 }\) There is a second collision between \(A\) and \(B\).
2. Show that \(\frac { 2 } { 3 } < e < \frac { 16 } { 19 }\)