CAIE M1 (Mechanics 1) 2022 June

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

2 A particle \(P\) is projected vertically upwards from horizontal ground. \(P\) reaches a maximum height of 45 m . After reaching the ground, \(P\) comes to rest without rebounding.

1. Find the speed at which \(P\) was projected.
2. Find the total time for which the speed of \(P\) is at least \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is \(s\) metres at time \(t\) seconds after leaving a fixed point \(O\). The particle starts from rest and passes through points \(P , Q\) and \(R\), at times \(t = 5 , t = 10\) and \(t = 15\) respectively, and returns to \(O\) at time \(t = 20\). The distances \(O P , O Q\) and \(O R\) are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from \(t = 0\) to \(t = 20\). The graph consists of two curved segments \(A B\) and \(C D\) and two straight line segments \(B C\) and \(D E\).

1. Find the speed of the particle between \(t = 5\) and \(t = 10\).
2. Find the acceleration of the particle between \(t = 0\) and \(t = 5\), given that it is constant.
3. Find the average speed of the particle during its motion.
   \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings \(A C\) and \(B C\), of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle \(A C B = 90 ^ { \circ }\). There is a horizontal force of magnitude \(F \mathrm {~N}\) acting on the block. The block is in equilibrium.

1. In the case where \(F = 20\), find the tensions in each of the strings.
2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.

5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .

1. Find the power developed by the cyclist.
   The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
2. Find the change in kinetic energy.
3. Use an energy method to find \(d\).
   \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.

7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(b = 9\) and find the value of \(c\).
2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.