Edexcel M1 (Mechanics 1) 2012 June

1. Two particles \(A\) and \(B\), of mass \(5 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, the speed of \(A\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the speed of \(B\) immediately after the collision.
   In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is 3.3 N s .
2. Find the value of \(m\).

2. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find

1. the magnitude of the reaction of the support at \(Q\) on the rod,
2. the distance \(A G\).

3. \begin{figure}[h] \end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the box,
2. the coefficient of friction between the box and the plane.

1. A car is moving on a straight horizontal road. At time \(t = 0\), the car is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at the point \(A\). The car maintains the speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s . The car then moves with constant deceleration \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reducing its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 60 s . The car then moves with constant acceleration until it is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\).
   1. Sketch a speed-time graph to represent the motion of the car from \(A\) to \(B\).
   2. Find the time for which the car is decelerating.
   Given that the distance from \(A\) to \(B\) is 1960 m ,
2. find the time taken for the car to move from \(A\) to \(B\).

1. A particle \(P\) is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is 17.5 m above horizontal ground. The particle \(P\) moves freely under gravity until it reaches the ground with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Show that \(u = 21\)
   At time \(t\) seconds after projection, \(P\) is 19 m above \(A\).
2. Find the possible values of \(t\). The ground is soft and, after \(P\) reaches the ground, \(P\) sinks vertically downwards into the ground before coming to rest. The mass of \(P\) is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on \(P\).
3. Find the vertical distance that \(P\) sinks into the ground before coming to rest.

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
3. Find the distance of \(S\) from \(B\) when \(t = 3\)
4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time \(t = 0\), a constant force \(\mathbf { F }\) of magnitude 4 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 3. The system moves under the action of this force until \(t = 6 \mathrm {~s}\). During the motion, the resistance to the motion of \(P\) has constant magnitude 1 N and the resistance to the motion of \(Q\) has constant magnitude 2 N . Find

1. the acceleration of the particles as the system moves under the action of \(\mathbf { F }\),
2. the speed of the particles at \(t = 6 \mathrm {~s}\),
3. the tension in the rod as the system moves under the action of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , \mathbf { F }\) is removed and the system decelerates to rest. The resistances to motion are unchanged. Find
4. the distance moved by \(P\) as the system decelerates,
5. the thrust in the rod as the system decelerates.