Edexcel M1 (Mechanics 1) 2005 January

1. A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the speed of \(Q\) immediately after the impact.
   2. State whether or not the direction of motion of \(Q\) is changed by the collision.
   3. Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer.
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{975a5462-0ba3-448a-8a5a-fc90da640d57-03_521_858_306_552}
   \end{figure} A plank \(A B\) has mass 40 kg and length 3 m . A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate
2. the tension in the rope at \(C\),
3. the distance \(C B\).

3. \begin{figure}[h] \end{figure} A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 s . The speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race. Calculate

1. the distance covered by the sprinter in the first 20 s of the race,
2. the value of \(u\),
3. the deceleration of the sprinter in the last 5 s of the race.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

1. the normal reaction of the plane on \(P\),
2. the value of \(X\). The force of magnitude \(X\) newtons is now removed.
3. Show that \(P\) remains in equilibrium on the plane.

5. Figure 4
\includegraphics[max width=\textwidth, alt={}, center]{975a5462-0ba3-448a-8a5a-fc90da640d57-08_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\),
2. the tension in the string,
3. the value of \(\mu\).
4. State how in your calculations you have used the information that the string is inextensible.

1. A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(A B C\), where \(A B = 24 \mathrm {~m}\) and \(A C = 30 \mathrm {~m}\). The stone is subject to a constant resistance to motion of magnitude 0.3 N . At \(A\) the speed of \(S\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(B\) the speed of \(S\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
   1. the deceleration of \(S\),
   2. the speed of \(S\) at \(C\).
   3. Show that the mass of \(S\) is 0.1 kg .
   At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(C A\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N .
2. Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest.

7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.