CAIE M1 (Mechanics 1) 2024 June

1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.5 kg respectively are at rest on a smooth horizontal plane. Particle \(P\) is projected with a speed \(6 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(1 \mathrm {~ms} ^ { - 1 }\). Find the two possible speeds of \(Q\) after the collision.
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-02_2716_35_143_2012}

2
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).

3 A car travels along a straight road with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\), where \(a > 0\). The car passes through points \(A , B\) and \(C\) in that order. The speed of the car at \(A\) is \(u \mathrm {~ms} ^ { - 1 }\) in the direction \(A B\). The distance \(B C\) is twice the distance \(A B\). The car takes 8 seconds to travel from \(A\) to \(B\) and 10 seconds to travel from \(B\) to \(C\).

1. Find \(u\) in terms of \(a\).
2. Find the speed of the car at \(C\) in terms of \(a\).

4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.

1. Find the value of \(k\).
2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
   On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
2. 1. Find the acceleration of the van and the tension in the tow-bar.
   2. Find the speed of the van when it has travelled a distance of 375 m up the hill.

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the power output of the cyclist.
2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged.
   \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.

7
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track \(A B C D\) which lies in a vertical plane. The section \(A B\) is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is smooth. The section \(B C\) is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is rough. The lengths \(A B , B C\) and \(C D\) are each 2 m . A particle is released from rest at \(A\). The coefficient of friction between the particle and both \(B C\) and \(C D\) is \(\mu\). There is no change in the speed of the particle when it passes through either of the points \(B\) or \(C\).

1. It is given that \(\mu = 0.1\). Find the distance which the particle has moved up the section \(C D\) when its speed is \(1 \mathrm {~ms} ^ { - 1 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
2. It is given instead that with a different value of \(\mu\) the particle travels 1 m up the track from \(C\) before it comes instantaneously to rest. Find the value of \(\mu\) and the speed of the particle at the instant that it passes \(C\) for the second time.
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