CAIE M1 (Mechanics 1) 2018 November

3 A particle of mass 1.2 kg moves in a straight line \(A B\). It is projected with speed \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) towards \(B\) and experiences a resistance force. The work done against this resistance force in moving from \(A\) to \(B\) is 25 J .

1. Given that \(A B\) is horizontal, find the speed of the particle at \(B\).
2. It is given instead that \(A B\) is inclined at \(30 ^ { \circ }\) below the horizontal and that the speed of the particle at \(B\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance force remains the same. Find the distance \(A B\).

4 A runner sets off from a point \(P\) at time \(t = 0\), where \(t\) is in seconds. The runner starts from rest and accelerates at \(1.2 \mathrm {~ms} ^ { - 2 }\) for 5 s . For the next 12 s the runner moves at constant speed before decelerating uniformly over a period of 3 s , coming to rest at \(Q\). A cyclist sets off from \(P\) at time \(t = 10\) and accelerates uniformly for 10 s , before immediately decelerating uniformly to rest at \(Q\) at time \(t = 30\).

1. Sketch the velocity-time graph for the runner and show that the distance \(P Q\) is 96 m .
   \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-06_821_1451_708_388}
2. Find the magnitude of the acceleration of the cyclist.

5
\includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .

1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\).
   \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.

6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.

1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
   When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.

7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).

1. Find the maximum velocity of the particle in this time period.
2. Find the total distance travelled until the maximum velocity is reached.
   The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
3. Find the velocity of the particle when \(t = 25\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.