CAIE M1 (Mechanics 1) 2021 November

1
\includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-02_611_1351_260_397} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of six straight line segments. The car accelerates from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s , and then travels at this speed for a further 20 s . The car then decelerates to a speed of \(6 \mathrm {~ms} ^ { - 1 }\) over a period of 5 s . This speed is maintained for a further \(( T - 30 ) \mathrm { s }\). The car then accelerates again to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(( 50 - T ) \mathrm { s }\), before decelerating to rest over a period of 10 s .

1. Given that during the two stages of the motion when the car is accelerating, the accelerations are equal, find the value of \(T\).
2. Find the total distance travelled by the car during the motion.

2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.

1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
   The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
2. Find the least possible value of the braking force which will cause the rope to become slack.

3
\includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-04_416_792_260_674} The diagram shows a semi-circular track \(A B C\) of radius 1.8 m which is fixed in a vertical plane. The points \(A\) and \(C\) are at the same horizontal level and the point \(B\) is at the bottom of the track. The section \(A B\) is smooth and the section \(B C\) is rough. A small block is released from rest at \(A\).

1. Show that the speed of the block at \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of \(B\). The work done against the resistance force during the motion of the block from \(B\) to this point is 4.5 J .
2. Find the mass of the block.

4 A cyclist starts from rest at a point \(A\) and travels along a straight road \(A B\), coming to rest at \(B\). The displacement of the cyclist from \(A\) at time \(t \mathrm {~s}\) after the start is \(s \mathrm {~m}\), where $$s = 0.004 \left( 75 t ^ { 2 } - t ^ { 3 } \right)$$

1. Show that the distance \(A B\) is 250 m .
2. Find the maximum velocity of the cyclist.

5 A railway engine of mass 75000 kg is moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.01\). The engine is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine is working at 960 kW . There is a constant force resisting the motion of the engine.

1. Find the resistance force.
   The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the power of the engine is now reduced to 900 kW . The resistance to motion is no longer constant, but in the next 60 s the work done against the resistance force is 46500 kJ .
2. Find the speed of the engine at the end of the 60 s .

6
\includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.

1. Given that \(X = 100\), find the tension in the lower string.
2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]

7
\includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-10_501_416_262_861} Particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively. The particles are initially held at rest 6.4 m apart on the same line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\) (see diagram). Particle \(P\) is released from rest and slides down the line of greatest slope. Simultaneously, particle \(Q\) is projected up the same line of greatest slope at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between each particle and the plane is 0.6 .

1. Show that the acceleration of \(Q\) up the plane is \(- 11.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the time for which the particles are in motion before they collide.
3. The particles coalesce on impact. Find the speed of the combined particle immediately after the impact.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.