OCR M1 (Mechanics 1) 2010 January

1 A particle \(P\) is projected vertically downwards from a fixed point \(O\) with initial speed \(4.2 \mathrm {~ms} ^ { - 1 }\), and takes 1.5 s to reach the ground. Calculate

1. the speed of \(P\) when it reaches the ground,
2. the height of \(O\) above the ground,
3. the speed of \(P\) when it is 5 m above the ground.

2 Two horizontal forces of magnitudes 12 N and 19 N act at a point. Given that the angle between the two forces is \(60 ^ { \circ }\), calculate

1. the magnitude of the resultant force,
2. the angle between the resultant and the 12 N force.

3
\includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-2_153_1009_978_570} Three particles \(P , Q\) and \(R\), are travelling in the same direction in the same straight line on a smooth horizontal surface. \(P\) has mass \(m \mathrm {~kg}\) and speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 } , Q\) has mass 0.8 kg and speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(R\) has mass 0.4 kg and speed \(2.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. A collision occurs between \(P\) and \(Q\), after which \(P\) and \(Q\) move in opposite directions, each with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
   (a) the value of \(m\),
   (b) the change in the momentum of \(P\).
2. When \(Q\) collides with \(R\) the two particles coalesce. Find their subsequent common speed.

4
\includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_494_255_258_945} Particles \(P\) and \(Q\), of masses 0.4 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the sections of the string not in contact with the pulley are vertical. \(P\) rests in limiting equilibrium on a plane inclined at \(60 ^ { \circ }\) to the horizontal (see diagram).

1. (a) Calculate the components, parallel and perpendicular to the plane, of the contact force exerted by the plane on \(P\).
   (b) Find the coefficient of friction between \(P\) and the plane.
   \(P\) is held stationary and a particle of mass 0.2 kg is attached to \(Q\). With the string taut, \(P\) is released from rest.
2. Calculate the tension in the string and the acceleration of the particles.
   \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_579_1195_1553_475} The \(( t , v )\) diagram represents the motion of two cyclists \(A\) and \(B\) who are travelling along a horizontal straight road. At time \(t = 0 , A\), who cycles with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), overtakes \(B\) who has initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From time \(t = 0 B\) cycles with constant acceleration for 20 s . When \(t = 20\) her speed is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which she subsequently maintains.

1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).

6 A swimmer \(C\) swims with velocity \(v \mathrm {~ms} ^ { - 1 }\) in a swimming pool. At time \(t \mathrm {~s}\) after starting, \(v = 0.006 t ^ { 2 } - 0.18 t + k\), where \(k\) is a constant. \(C\) swims from one end of the pool to the other in 28.4 s .

1. Find the acceleration of \(C\) in terms of \(t\).
2. Given that the minimum speed of \(C\) is \(0.65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = 2\).
3. Express the distance travelled by \(C\) in terms of \(t\), and calculate the length of the pool.

7
\includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-4_129_798_756_676} A winch drags a \(\log\) of mass 600 kg up a slope inclined at \(10 ^ { \circ }\) to the horizontal by means of an inextensible cable of negligible mass parallel to the slope (see diagram). The coefficient of friction between the \(\log\) and the slope is 0.15 , and the \(\log\) is initially at rest at the foot of the slope. The acceleration of the \(\log\) is \(0.11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the tension in the cable. The cable suddenly breaks after dragging the log a distance of 10 m .
2. (a) Show that the deceleration of the log while continuing to move up the slope is \(3.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures.
   (b) Calculate the time taken, after the cable breaks, for the log to return to its original position at the foot of the slope. {www.ocr.org.uk}) after the live examination series.
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