CAIE M1 (Mechanics 1) 2022 November

2 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is \(2 \mathrm {~ms} ^ { - 1 }\). Find the loss of kinetic energy of the system due to the collision.

3 A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg .

1. The car is moving along a straight level road at a constant speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW . Find this speed.
3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N . The resistance to the motion of the car remains 1400 N . Find the acceleration of the system and the tension in the cable.
   \includegraphics[max width=\textwidth, alt={}, center]{167f782c-3047-41f9-90a8-32ccdc19216d-06_378_631_255_757} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of \(18 ^ { \circ }\) to the horizontal. The block is pulled up the plane by a light string that makes an angle of \(26 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\) (see diagram). The coefficient of friction between the block and plane is 0.65 .
4. The acceleration of the block is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find \(T\).
5. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion.

5 A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(- 20 \mathrm {~ms} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after leaving \(O\) is given by \(a = 12 - 2 t\).

1. Sketch a velocity-time graph for \(0 \leqslant t \leqslant 12\), indicating the times when \(P\) is at rest.
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 12\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-10_410_723_260_717} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure} Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.
3. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane.
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-11_412_899_276_589} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.