CAIE M1 (Mechanics 1) 2024 March

2 A particle is projected vertically upwards from horizontal ground. The speed of the particle 2 seconds after it is projected is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is travelling downwards.

1. Find the speed of projection of the particle.
2. Find the distance travelled by the particle between the two times at which its speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A crate of mass 600 kg is being pulled up a line of greatest slope of a rough plane at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by a rope attached to a winch. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and the rope is parallel to the plane. The winch is working at a constant rate of 8 kW . Find the coefficient of friction between the crate and the plane.
\includegraphics[max width=\textwidth, alt={}, center]{1ca74dfc-9bef-475c-a7d1-77b95c487f4b-05_483_953_269_557} Four coplanar forces act at a point. The magnitudes of the forces are \(F N , 2 F N , 3 F N\) and \(30 N\). The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).

5 A particle moves in a straight line starting from a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(O\) is given by $$\mathrm { v } = \mathrm { t } ^ { 3 } - \frac { 9 } { 2 } \mathrm { t } ^ { 2 } + 1 \text { for } 0 \leqslant t \leqslant 4$$ You may assume that the velocity of the particle is positive for \(t < \frac { 1 } { 2 }\), is zero at \(t = \frac { 1 } { 2 }\) and is negative for \(t > \frac { 1 } { 2 }\).

1. Find the distance travelled between \(t = 0\) and \(t = \frac { 1 } { 2 }\).
2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant.

6 A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F \mathrm {~N}\) acting on the trailer. The driving force of the car's engine is 3000 N .

1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar.
2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J . The speed of the car at the start of the 50 m is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car at the end of the 50 m .
   \includegraphics[max width=\textwidth, alt={}, center]{1ca74dfc-9bef-475c-a7d1-77b95c487f4b-10_680_887_269_596} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(A B C\). Particles \(P\) and \(Q\) are each of mass \(m \mathrm {~kg}\). The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(A B\) is 0.75 m and the length of \(B C\) is 3.25 m . The section \(A B\) of the plane is smooth and the section \(B C\) is rough. The coefficient of friction between each particle and the section \(B C\) is 0.25 . Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).
3. Verify that particle \(P\) reaches \(B 0.5 \mathrm {~s}\) after it is released, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find the time that it takes from the instant the two particles are released until they collide.
   The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25 .
5. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\).
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