OCR M1 (Mechanics 1) 2009 June

1
\includegraphics[max width=\textwidth, alt={}, center]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-2_462_305_274_918} Two perpendicular forces have magnitudes \(x \mathrm {~N}\) and \(3 x \mathrm {~N}\) (see diagram). Their resultant has magnitude 6 N .

1. Calculate \(x\).
2. Find the angle the resultant makes with the smaller force.

2 The driver of a car accelerating uniformly from rest sees an obstruction. She brakes immediately bringing the car to rest with constant deceleration at a distance of 6 m from its starting point. The car travels in a straight line and is in motion for 3 seconds.

1. Sketch the \(( t , v )\) graph for the car's motion.
2. Calculate the maximum speed of the car during its motion.
3. Hence, given that the acceleration of the car is \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its deceleration.

3
\includegraphics[max width=\textwidth, alt={}, center]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-2_350_1025_1704_559} The diagram shows a small block \(B\), of mass 3 kg , and a particle \(P\), of mass 0.8 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. \(B\) is held at rest on a horizontal surface, and \(P\) lies on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) towards the pulley.

1. By considering the motion of \(P\), show that the tension in the string is 3.76 N .
2. Calculate the coefficient of friction between \(B\) and the horizontal surface.

4 An object is projected vertically upwards with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate

1. the speed of the object when it is 2.1 m above the point of projection,
2. the greatest height above the point of projection reached by the object,
3. the time after projection when the object is travelling downwards with speed \(5.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-3_227_897_635_664} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A particle \(P\) of mass 0.5 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface towards a stationary particle \(Q\) of mass \(m \mathrm {~kg}\) (see Fig. 1). After the particles collide, \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in its original direction of motion, and \(Q\) has speed \(1 \mathrm {~ms} ^ { - 1 }\) more than \(P\). Show that \(v ( m + 0.5 ) = - m + 3\).
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-3_229_901_1265_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(Q\) and \(P\) are now projected towards each other with speeds \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see Fig. 2). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with its direction of motion unchanged and \(P\) has speed \(1 \mathrm {~ms} ^ { - 1 }\) more than \(Q\). Find another relationship between \(m\) and \(v\) in the form \(v ( m + 0.5 ) = a m + b\), where \(a\) and \(b\) are constants.
5. By solving these two simultaneous equations show that \(m = 0.9\), and hence find \(v\).

6 A block \(B\) of weight 10 N is projected down a line of greatest slope of a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. \(B\) travels down the plane at constant speed.

1. (a) Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane.
   (b) Hence show that the coefficient of friction is 0.364 , correct to 3 significant figures.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-4_289_711_598_758}
   \(B\) is in limiting equilibrium when acted on by a force of \(T \mathrm {~N}\) directed towards the plane at an angle of \(45 ^ { \circ }\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\).

7
\includegraphics[max width=\textwidth, alt={}, center]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-4_531_1481_1194_331} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leqslant t \leqslant 3 , S\) has velocity \(\left( 6 t - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). For \(3 < t \leqslant 22 , S\) runs at a constant speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For \(t > 22 , S\) decelerates at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) (see diagram).

1. Express the acceleration of \(S\) during the first 3 seconds in terms of \(t\).
2. Show that \(S\) runs 18 m in the first 3 seconds of motion.
3. Calculate the time \(S\) takes to run 100 m .
4. Calculate the time \(S\) takes to run 200 m . OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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