CAIE M1 (Mechanics 1) 2020 June

2 A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There are constant resistance forces of 250 N on the car and 100 N on the trailer.

1. Find the tension in the tow-bar.
2. Find the power of the engine of the car at the instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A particle \(P\) is projected vertically upwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 2.8 m above horizontal ground.

1. Find the greatest height above the ground reached by \(P\).
2. Find the length of time for which \(P\) is at a height of more than 3.6 m above the ground.
   The diagram shows a ring of mass 0.1 kg threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is 0.8 . A force of magnitude \(T \mathrm {~N}\) acts on the ring in a direction at \(30 ^ { \circ }\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest.
3. Find the greatest value of \(T\) for which the ring remains at rest.
4. Find the acceleration of the ring when \(T = 3\).

5
\includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-08_412_339_260_897} A child of mass 35 kg is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length 4 m . Initially \(P\) is held at an angle of \(45 ^ { \circ }\) to the vertical (see diagram).

1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point.
2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X \mathrm {~J}\). The speed of \(P\) at its lowest point is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(X\).

6 A particle moves in a straight line \(A B\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(A\) is given by \(v = k \left( t ^ { 2 } - 10 t + 21 \right)\), where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is 2.85 m when \(t = 3\) and is 2.4 m when \(t = 6\).

1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\).
2. Find the displacement of the particle from \(A\) when its velocity is a minimum.

7
\includegraphics[max width=\textwidth, alt={}, center]{c175972b-e298-4c86-bbef-3b05c6aca76f-12_399_1121_262_511} A particle \(P\) of mass 0.3 kg , lying on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of 2.5 m and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass 0.2 kg lies at rest on the horizontal plane 1.5 m from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\).

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the coefficient of friction between \(P\) and the horizontal plane.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.