Questions — OCR M1 (141 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR M1 2010 June Q4
4 A particle \(P\) moving in a straight line has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after passing through a fixed point \(O\). It is given that \(v = 3.2 - 0.2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 5\). Calculate
  1. the value of \(t\) when \(P\) is at instantaneous rest,
  2. the acceleration of \(P\) when it is at instantaneous rest,
  3. the greatest distance of \(P\) from \(O\).
OCR M1 2010 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.
  1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
  2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
  3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.
OCR M1 2010 June Q6
6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.
  1. Calculate
    (a) the acceleration of \(B\) and the tension in the string,
    (b) the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
  2. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.
OCR M1 2010 June Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_305_748_260_699} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).
  1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
  2. Given that the upper block is in limiting equilibrium, find \(\mu\).
  3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.
OCR M1 2010 June Q8
8 \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
6 (ii)
\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR M1 2010 June Q10
10
7
7
  • 7
  		• (continued)
    \multirow[t]{26}{*}{7
  • }
    • \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR M1 Specimen Q1
    1
    \includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_99_812_310_635} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude \(E\) newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N , and the acceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(E\).
    OCR M1 Specimen Q2
    2 \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_166_518_824_351} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_168_755_822_1043} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30 ^ { \circ }\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at an angle \(\theta ^ { \circ }\) to the force of magnitude 8 N , as shown in Fig. 2. Find \(R\) and \(\theta\).
    OCR M1 Specimen Q3
    3 A particle is projected vertically upwards, from the ground, with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring air resistance, find
    1. the maximum height reached by the particle,
    2. the speed of the particle when it is 30 m above the ground,
    3. the time taken for the particle to fall from its highest point to a height of 30 m ,
    4. the length of time for which the particle is more than 30 m above the ground. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_569_1132_258_516} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s . The woman runs at a constant speed throughout. Fig. 1 shows the \(( t , v )\) graph for the woman.
    5. Find the total distance run by the woman.
    6. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_424_1135_1233_513} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} At time \(t = 0\), a child also starts to move, from \(A\), along \(A B\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s . Fig. 2 shows the ( \(t , v\) ) graph for the child; it consists of two straight line segments.
    7. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s .
    8. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant.
    OCR M1 Specimen Q5
    5 A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(- \frac { 1 } { 10 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\).
    2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\).
    3. Find the displacement of \(P\) from \(O\) when \(t = 10\).
    4. Find the speed with which the particle returns to \(O\).
    OCR M1 Specimen Q6
    6
    \includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres \(A , B\) and \(C\) have masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) (see diagram). Air resistance may be ignored.
    1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed with which \(B\) starts to move.
    2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
    3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\).
    4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers.
    OCR M1 Specimen Q7
    7 A sledge of mass 25 kg is on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2 .
    1. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_289_488_1493_849} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after being pulled for 10 s . Ignoring air resistance, find the tension in the cable.
    2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_291_490_2149_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane.
    OCR M1 2010 January Q1
    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.
    OCR M1 2010 January Q2
    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.
    OCR M1 2010 January Q3
    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.
    OCR M1 2010 January Q4
    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.
    OCR M1 2010 January Q6
    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.
    OCR M1 2010 January Q7
    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.
      If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
      OCR 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. }
    OCR M1 2011 January Q1
    1 Two particles \(P\) and \(Q\) are projected directly towards each other on a smooth horizontal surface. \(P\) has mass 0.5 kg and initial speed \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(Q\) has mass 0.8 kg and initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a collision between \(P\) and \(Q\), the speed of \(P\) is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the direction of its motion is reversed. Calculate
    1. the change in the momentum of \(P\),
    2. the speed of \(Q\) after the collision.
    OCR M1 2011 January Q2
    2
    \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-02_597_885_676_630} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate
    1. \(\alpha\),
    2. \(F\).
    OCR M1 2011 January Q3
    3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.
    1. Calculate the speed of the particle when it strikes the ground.
    2. Calculate the time after projection when the particle reaches the ground.
    3. Sketch on separate diagrams
      (a) the \(( t , v )\) graph,
      (b) the \(( t , x )\) graph,
      representing the motion of the particle.
    OCR M1 2011 January Q4
    4
    \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_156_1141_258_502} A block \(B\) of mass 0.8 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string inclined at \(10 ^ { \circ }\) to the horizontal. They are pulled across a horizontal surface with acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), by a horizontal force of 2 N applied to \(B\) (see diagram).
    1. Given that contact between \(B\) and the surface is smooth, calculate the tension in the string.
    2. Calculate the coefficient of friction between \(P\) and the surface.
    OCR M1 2011 January Q5
    5
    \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614}
    \(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.
    1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
      (b) Calculate \(\theta\).
      (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
    2. Calculate the speeds of the particles immediately before they collide.
    OCR M1 2011 January Q6
    6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).
    1. Find the displacement of the particle from \(O\) when \(t = 0\).
    2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
    3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    OCR M1 2011 January Q7
    7 A particle \(P\) of mass 0.6 kg is projected up a line of greatest slope of a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves with deceleration \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and comes to rest before reaching the top of the plane.
    1. Calculate the frictional force acting on \(P\), and the coefficient of friction between \(P\) and the plane.
    2. Find the magnitude of the contact force exerted on \(P\) by the plane and the angle between the contact force and the upward direction of the line of greatest slope,
      (a) when \(P\) is in motion,
      (b) when \(P\) is at rest.