CAIE M1 (Mechanics 1) 2024 November

1 An athlete has mass \(m \mathrm {~kg}\) .The athlete runs along a horizontal road against a constant resistance force of magnitude 24 N .The total work done by the athlete in increasing his speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(6 \mathrm {~ms} ^ { - 1 }\) while running a distance of 50 metres is 1541 J . Find the value of \(m\) .
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-04_464_1116_247_478} Coplanar forces of magnitudes \(16 \mathrm {~N} , 12 \mathrm {~N} , 24 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium.

3 A car of mass 1600 kg travels up a slope inclined at an angle of \(\sin ^ { - 1 } 0.08\) to the horizontal. There is a constant resistance of magnitude 240 N acting on the car.

1. It is given that the car travels at a constant speed of \(32 \mathrm {~ms} ^ { - 1 }\). Find the power of the engine of the car.
2. Find the acceleration of the car when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at 95\% of the power found in (a).

4 Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8 \mathrm {~ms} ^ { - 1 }\). After \(A\) and \(B\) collide, \(A\) moves with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the greater of the two possible total losses of kinetic energy due to the collision.
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-06_2722_43_107_2004}
\includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-07_197_1142_254_460} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string.The string is at an angle of \(30 ^ { \circ }\) above the plane and has tension \(T \mathrm {~N}\)(see diagram).The coefficient of friction between the particle and the plane is 0.5 .

1. Given that the particle is on the point of moving,find the value of \(T\) .
2. Given instead that the particle is accelerating at \(0.2 \mathrm {~ms} ^ { - 2 }\) ,find the value of \(T\) .

6 A particle moves in a straight line. It starts from rest, at time \(t = 0\), and accelerates at \(0.6 t \mathrm {~ms} ^ { - 2 }\) for 4 s , reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle then travels at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 11 s , and finally slows down, with constant deceleration, stopping after a further 5 s .

1. Show that \(V = 4.8\).
2. Sketch a velocity-time graph for the motion.
   \includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-08_2722_40_107_2010}
3. Find an expression, in terms of \(t\), for the velocity of the particle for \(15 \leqslant t \leqslant 20\).
4. Find the total distance travelled by the particle.
   \includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-10_592_608_251_731} Two particles, \(A\) and \(B\), of masses 3 kg and 5 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held with the string taut and its straight parts vertical. Particle \(A\) is 1 m above a horizontal plane, and particle \(B\) is 2 m above the plane (see diagram). The particles are released from rest. In the subsequent motion, \(A\) does not reach the pulley, and after \(B\) reaches the plane it remains in contact with the plane.
5. Find the tension in the string and the time taken for \(B\) to reach the plane.
   \includegraphics[max width=\textwidth, alt={}, center]{3a6ecf05-127f-4ddf-959e-233f6bae9171-10_2718_42_107_2007}
6. Find the time for which \(A\) is at least 3.25 m above the plane.
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