CAIE M1 (Mechanics 1) 2024 November

2 A block of mass 20 kg is held at rest at the top of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The block is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane. There is a resistance force acting on the block. As the block moves 2 m down the plane from its point of projection, the work done against this resistance force is 50 J . Find the speed of the block when it has moved 2 m down the plane.
\includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-04_2716_38_109_2012}

3 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and his bicycle is 90 kg . The power exerted by the cyclist is 250 W . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the value of the constant resistance to motion acting on the cyclist.
   The cyclist comes to the bottom of a hill inclined at \(2 ^ { \circ }\) to the horizontal.
2. Given that the power and resistance to motion are unchanged, find the steady speed which the cyclist could maintain when riding up the hill.

4
\includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_389_1134_258_468} The diagram shows two particles, \(A\) and \(B\), of masses 0.2 kg and 0.1 kg respectively. The particles are suspended below a horizontal ceiling by two strings, \(A P\) and \(B Q\), attached to fixed points \(P\) and \(Q\) on the ceiling. The particles are connected by a horizontal string, \(A B\). Angle \(A P Q = 45 ^ { \circ }\) and \(B Q P = \theta ^ { \circ }\). Each string is light and inextensible. The particles are in equilibrium.

1. Find the value of the tension in the string \(A B\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_2715_44_110_2006}
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-07_2721_34_101_20}
2. Find the value of \(\theta\) and the tension in the string \(B Q\).

5 Two particles, \(P\) and \(Q\), of masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are held at rest in the same vertical line. The heights of \(P\) and \(Q\) above horizontal ground are 1 m and 2 m respectively. \(P\) is projected vertically upwards with speed \(2 \mathrm {~ms} ^ { - 1 }\). At the same instant, \(Q\) is released from rest.

1. Find the speed of each particle immediately before they collide.
2. It is given that immediately after the collision the downward speed of \(Q\) is \(3.5 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(P\) at the instant that it reaches the ground.

6 A particle, \(P\), travels in a straight line, starting from a point \(O\) with velocity \(6 \mathrm {~ms} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = - 1.5 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 1 ,
a = 1.5 t ^ { \frac { 1 } { 2 } } - 3 t ^ { - \frac { 1 } { 2 } } & \text { for } t > 1 . \end{array}$$

1. Find the velocity of \(P\) at \(t = 1\).
2. Given that there is no change in the velocity of \(P\) when \(t = 1\), find an expression for the velocity of \(P\) for \(t > 1\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-11_2725_35_99_20}
3. Given that the velocity of \(P\) is positive for \(t \leqslant 4\), find the total distance travelled between \(t = 0\) and \(t = 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_723_762_248_653} Two particles, \(A\) and \(B\), of masses 0.2 kg and 0.3 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 bottom of a rough plane inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = 0.6\). Particle \(A\) lies on the plane, and particle \(B\) hangs vertically below the pulley, 0.25 m above horizontal ground. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between \(A\) and the plane is 1.125 . Particle \(A\) is released from rest.
4. Find the tension in the string and the magnitude of the acceleration of the particles.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_2716_38_109_2012}
5. When \(B\) reaches the ground, it comes to rest. Find the total distance that \(A\) travels down the plane from when it is released until it comes to rest. You may assume that \(A\) does not reach the pulley.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-14_2715_31_106_2016}