CAIE M1 (Mechanics 1) 2024 June

1 A car starts from rest and accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) for 10 s . It then travels at a constant speed for 30 s . The car then uniformly decelerates to rest over a period of 20 s .

1. Sketch a velocity-time graph for the motion of the car.
   \includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-03_762_1081_447_493}
2. Find the total distance travelled by the car.

2
\includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-04_558_606_276_715} Two forces of magnitudes 20 N and \(F \mathrm {~N}\) act at a point \(P\) in the directions shown in the diagram.

1. Given that the resultant force has no component in the \(y\)-direction, calculate the value of \(F\).
2. Given instead that \(F = 10\), find the magnitude and direction of the resultant force.

3 A train of mass 180000 kg ascends a straight hill of length 1.5 km , inclined at an angle of \(1.5 ^ { \circ }\) to the horizontal. As it ascends the hill, the total work done to overcome the resistance to motion is 12000 kJ and the speed of the train decreases from \(45 \mathrm {~ms} ^ { - 1 }\) to \(40 \mathrm {~ms} ^ { - 1 }\). Find the work done by the engine of the train as it ascends the hill, giving your answer in kJ .

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

5 A straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.

1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.

6 A particle moves in a straight line, starting from a point \(O\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v \mathrm {~ms} ^ { - 1 }\). It is given that \(\mathbf { v } = \mathrm { kt } ^ { \frac { 1 } { 2 } } - 2 \mathrm { t } - 8\), where \(k\) is a positive constant. The maximum velocity of the particle is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(k = 10\).
2. 1. Verify that \(v = 0\) when \(t = 1\) and \(t = 16\).
   2. Find the distance travelled by the particle in the first 16 s .

7 A particle \(P\) of mass 0.2 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~ms} ^ { - 1 }\).

1. Show that the speed of \(P\) when it reaches 20 m above the ground is \(15 \mathrm {~ms} ^ { - 1 }\).
   When \(P\) reaches 20 m above the ground it collides with a second particle \(Q\) of mass 0.1 kg which is moving downwards at \(20 \mathrm {~ms} ^ { - 1 } . P\) is brought to instantaneous rest in the collision.
2. Find the velocity of \(Q\) immediately after the collision.
   When \(P\) reaches the ground it rebounds back directly upwards with half of the speed that it had immediately before hitting the ground.
3. Find the height above the ground at which \(P\) and \(Q\) next collide.
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