CAIE M1 (Mechanics 1) 2020 November

1 A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and collides directly with \(B\). In the collision the two particles coalesce.

1. Find the speed of the combined particle after the collision.
2. Find the loss of kinetic energy of the system due to the collision.

2 A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N .

1. Find, in kW , the rate at which the engine of the car is working when it is travelling at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at 15 kW .

3
\includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-04_378_969_258_587} Coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N} , 10 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\).

4 A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t \mathrm {~s}\) after leaving \(O\) it has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest.
\includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-06_540_606_258_767} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor.

6 A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin ^ { - 1 } 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill and \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.

1. Use an energy method to find the constant driving force as the car and trailer travel up the hill.
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   After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged.
2. Find the acceleration of the system and the tension in the tow-bar.
   \includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-10_440_738_262_699} Three points \(A , B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal, with \(A B = 1 \mathrm {~m}\) and \(B C = 1 \mathrm {~m}\), as shown in the diagram. A particle of mass 0.2 kg is released from rest at \(A\) and slides down the plane. The part of the plane from \(A\) to \(B\) is smooth. The part of the plane from \(B\) to \(C\) is rough, with coefficient of friction \(\mu\) between the plane and the particle.
3. Given that \(\mu = \frac { 1 } { 2 } \sqrt { 3 }\), find the speed of the particle at \(C\).
4. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.