CAIE M1 (Mechanics 1) 2010 November

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\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_608_831_258_657} Two particles \(P\) and \(Q\) move vertically under gravity. The graphs show the upward velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particles at time \(t \mathrm {~s}\), for \(0 \leqslant t \leqslant 4 . P\) starts with velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) starts from rest.

1. Find the value of \(V\). Given that \(Q\) reaches the horizontal ground when \(t = 4\), find
2. the speed with which \(Q\) reaches the ground,
3. the height of \(Q\) above the ground when \(t = 0\).

2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of 40 kW . The resistance to motion of the car is constant and equal to 800 N . The car passes through the point \(A\) on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).

3
\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).

4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).

5 A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope \(A B\). The distance \(A B\) is 8 m . The particle starts at \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves with constant acceleration \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of the particle at the instant it reaches \(B\).
2. Given that the work done against the frictional force as the particle moves from \(A\) to \(B\) is 7 J , find the angle of inclination of the plane. When the particle is at the point \(X\) its speed is the same as the average speed for the motion from \(A\) to \(B\).
3. Find the work done by the frictional force for the particle's motion from \(A\) to \(X\).

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\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-3_476_1305_1519_420} A smooth slide \(A B\) is fixed so that its highest point \(A\) is 3 m above horizontal ground. \(B\) is \(h \mathrm {~m}\) above the ground. A particle \(P\) of mass 0.2 kg is released from rest at a point on the slide. The particle moves down the slide and, after passing \(B\), continues moving until it hits the ground (see diagram). The speed of \(P\) at \(B\) is \(v _ { B }\) and the speed at which \(P\) hits the ground is \(v _ { G }\).

1. In the case that \(P\) is released at \(A\), it is given that the kinetic energy of \(P\) at \(B\) is 1.6 J . Find
   (a) the value of \(h\),
   (b) the kinetic energy of the particle immediately before it reaches the ground,
   (c) the ratio \(v _ { G } : v _ { B }\).
2. In the case that \(P\) is released at the point \(X\) of the slide, which is \(H \mathrm {~m}\) above the ground (see diagram), it is given that \(v _ { G } : v _ { B } = 2.55\). Find the value of \(H\) correct to 2 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-4_384_679_258_733} Particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(P\) hangs freely and \(Q\) is in contact with the table. A force of magnitude 3.2 N acts on \(Q\), upwards and away from the pulley, at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
3. The system is in limiting equilibrium with \(P\) about to move upwards. Find the coefficient of friction between \(Q\) and the table. The force of magnitude 3.2 N is now removed and \(P\) starts to move downwards.
4. Find the acceleration of the particles and the tension in the string. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }