CAIE M1 (Mechanics 1) 2021 November

1 A bus moves from rest with constant acceleration for 12 s . It then moves with constant speed for 30 s before decelerating uniformly to rest in a further 6 s . The total distance travelled is 585 m .

1. Find the constant speed of the bus.
2. Find the magnitude of the deceleration.

2 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses km kg and \(m \mathrm {~kg}\) respectively, where \(k > 1\), are free to move on a smooth horizontal plane. \(A\) is moving towards \(B\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(A\) and \(B\) coalesce and move with speed \(4 \mathrm {~ms} ^ { - 1 }\).

1. Find \(k\).
2. Find, in terms of \(m\), the loss of kinetic energy due to the collision.

3
\includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-04_519_616_260_762} Coplanar forces of magnitudes \(24 \mathrm {~N} , P \mathrm {~N} , 20 \mathrm {~N}\) and 36 N act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the values of \(P\) and \(\theta\).

4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A force of magnitude \(P \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to prevent the particle sliding down the plane. The coefficient of friction between the particle and the plane is 0.35 .

1. Draw a sketch showing the forces acting on the particle.
2. Find the least possible value of \(P\).

5 A car of mass 1600 kg travels at constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle of \(\sin ^ { - 1 } 0.12\) to the horizontal.

1. Find the change in potential energy of the car in 30 s .
2. Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.
3. Calculate, in kW , the power developed by the engine of the car.
4. Given that this power is suddenly decreased by \(15 \%\), find the instantaneous deceleration of the car.

6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6
v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).

1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
2. Sketch the velocity-time graph.
3. Find the total distance travelled by \(P\) during the 14 s .
   \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) does not reach the pulley.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.