CAIE M1 (Mechanics 1) 2021 November

1
\includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-03_471_613_254_766} A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with it after the impact.

1. Calculate the speed with which the combined post and object moves immediately after the impact.
2. There is a constant force resisting the motion of magnitude 4800 N . Calculate the distance the post is driven into the ground.

2 A particle of mass 8 kg is suspended in equilibrium by two light inextensible strings which make angles of \(60 ^ { \circ }\) and \(45 ^ { \circ }\) above the horizontal.

1. Draw a diagram showing the forces acting on the particle.
2. Find the tensions in the strings.

3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.

1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.

4 A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.

1. The car moves along a horizontal section of the road at a constant speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the work done against the resisting force during the first 8 seconds.
   2. Calculate, in kW , the power developed by the engine of the car.
   3. Given that this power is suddenly increased by 12 kW , find the instantaneous acceleration of the car.
2. The car now travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a section of the road inclined at \(\theta ^ { \circ }\) to the horizontal, with the engine working at 64 kW . Find the value of \(\theta\).

5 A particle \(P\) moves in a straight line, starting from rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\) the acceleration of \(P\) is \(k \left( 16 - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(k\) is a positive constant, and the displacement from \(O\) is \(s \mathrm {~m}\). The velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\).

1. Show that \(s = \frac { 1 } { 64 } t ^ { 2 } \left( 96 - t ^ { 2 } \right)\).
2. Find the speed of \(P\) at the instant that it returns to \(O\).
3. Find the maximum displacement of the particle from \(O\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.

1. Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
   The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
2. Find the acceleration of the 4 kg particle and the tensions in the strings.
3. In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a 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.