CAIE M1 (Mechanics 1) 2020 November

1 A particle \(P\) is projected vertically upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .

1. Find \(v\).
2. Find the greatest height of \(P\) above the ground.

2 A box of mass 5 kg is pulled at a constant speed a distance of 15 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.

1. Find the work done against friction.
2. Find the change in gravitational potential energy of the box.
3. Find the work done by the pulling force.

3 A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal table. The string makes an angle of \(24 ^ { \circ }\) above the horizontal and the tension in the string is 30 N .

1. Draw a diagram showing all the forces acting on the block.
2. Find the coefficient of friction between the block and the table.

4 Two small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and \(m \mathrm {~kg}\) respectively, lie on a smooth horizontal plane. Initially, sphere \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collision \(A\) moves with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the two possible values of the loss of kinetic energy due to the collision.

5 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t ^ { 2 } - 20 t + 21\).

1. Find the values of \(t\) for which \(P\) is at instantaneous rest.
2. Find the initial acceleration of \(P\).
3. Find the minimum velocity of \(P\).
4. Find the distance travelled by \(P\) during the time when its velocity is negative.

6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

1. The car and caravan are travelling along a straight horizontal road.
   1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
   2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).

7
\includegraphics[max width=\textwidth, alt={}, center]{ac4bb5a0-c7c0-4e1d-9e76-64f92ae28066-10_214_1461_255_342} As shown in the diagram, particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle \(A\) is on plane \(P\), which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. Particle \(B\) is on plane \(Q\), which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.

1. It is given that plane \(P\) is smooth, plane \(Q\) is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle \(B\) and plane \(Q\).
2. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m . [You should assume that this occurs before \(A\) reaches the pulley or \(B\) reaches the bottom of plane \(Q\).] [6]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.