Edexcel M1 (Mechanics 1) 2003 November

1. A small ball is projected vertically upwards from a point \(A\). The greatest height reached by the ball is 40 m above \(A\). Calculate
   1. the speed of projection,
   2. the time between the instant that the ball is projected and the instant it returns to \(A\).
   3. A railway truck \(S\) of mass 2000 kg is travelling due east along a straight horizontal track with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The truck \(S\) collides with a truck \(T\) which is travelling due west along the same track as \(S\) with constant speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse of \(T\) on \(S\) is 28800 Ns.
   4. Calculate the speed of \(S\) immediately after the collision.
   5. State the direction of motion of \(S\) immediately after the collision.
   Given that, immediately after the collision, the speed of \(T\) is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and that \(T\) and \(S\) are moving in opposite directions,
2. calculate the mass of \(T\).
   (4)

3. \begin{figure}[h] \end{figure} A heavy suitcase \(S\) of mass 50 kg is moving along a horizontal floor under the action of a force of magnitude \(P\) newtons. The force acts at \(30 ^ { \circ }\) to the floor, as shown in Fig. 1, and \(S\) moves in a straight line at constant speed. The suitcase is modelled as a particle and the floor as a rough horizontal plane. The coefficient of friction between \(S\) and the floor is \(\frac { 3 } { 5 }\). Calculate the value of \(P\).

4. A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s , reaching a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 120 s . Finally it moves with constant deceleration, coming to rest at a point \(F\).

1. In the space below, sketch a speed-time graph to illustrate the motion of the car. The distance between \(S\) and \(F\) is 4 km .
2. Calculate the total time the car takes to travel from \(S\) to \(F\).
   (3) A motorcycle starts at \(S , 10 \mathrm {~s}\) after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate
3. the time the motorcycle takes to travel from \(S\) to \(P\),
4. the speed of the motorcycle at \(P\).
   (2)

5. A particle \(P\) of mass 3 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. At \(t = 0\), \(P\) has velocity \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(t = 4 \mathrm {~s}\), the velocity of \(P\) is \(( - 5 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the acceleration of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. the magnitude of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , P\) is at the point \(A\) with position vector ( \(6 \mathbf { i } - 29 \mathbf { j }\) ) m relative to a fixed origin \(O\). At this instant the force \(\mathbf { F }\) newtons is removed and \(P\) then moves with constant velocity. Three seconds after the force has been removed, \(P\) is at the point \(B\).
3. Calculate the distance of \(B\) from \(O\).
   (6) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{ebbf20a5-63f8-4c80-a8c4-477e1ea1eee7-4_298_1221_358_411}
   \end{figure} A non-uniform rod \(A B\) has length 5 m and weight 200 N . The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Fig. 2. The centre of mass of \(A B\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N .
4. Show that \(50 x - W = 100\). The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N .
5. Obtain another equation connecting \(W\) and \(x\).
6. Calculate the value of \(x\) and the value of \(W\). \section*{7.} \section*{Figure 3}
   \includegraphics[max width=\textwidth, alt={}]{ebbf20a5-63f8-4c80-a8c4-477e1ea1eee7-5_688_1477_379_328}
   Figure 3 shows two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac { 1 } { 5 } g\).
7. Write down an equation of motion for \(B\).
8. Find the tension in the string.
9. Prove that \(m = \frac { 16 } { 35 }\).
10. State where in the calculations you have used the information that \(P\) is a light smooth pulley. On release, \(B\) is at a height of one metre above the ground and \(A P = 1.4 \mathrm {~m}\). The particle \(B\) strikes the ground and does not rebound.
11. Calculate the speed of \(B\) as it reaches the ground.
12. Show that \(A\) comes to rest as it reaches \(P\). \section*{END}