Questions — OCR M1 (141 questions)

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OCR M1 2011 January Q8
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    7(ii) (b(ii) (b)
    \section*{OCR} RECOGNISING ACHIEVEMENT
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    OCR M1 2012 January Q1
    1 Particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are moving in the same direction along the same straight line on a smooth horizontal surface. \(P\) is moving with speed \(2.2 \mathrm {~ms} ^ { - 1 }\) and \(Q\) is moving with speed \(0.8 \mathrm {~ms} ^ { - 1 }\) immediately before they collide. In the collision, the speed of \(P\) is reduced by \(50 \%\) and its direction of motion is unchanged.
    1. Calculate the speed of \(Q\) immediately after the collision.
    2. Find the distance \(P Q\) at the instant 3 seconds after the collision.
    OCR M1 2012 January Q2
    2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone of weight 180 N , which moves 36 m in a straight line and comes to rest 24 s after the instant of projection. The only horizontal force acting on the stone after its projection is a constant frictional force between the stone and the ice.
    1. Calculate the deceleration of the stone.
    2. Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction between the stone and the ice.
    OCR M1 2012 January Q3
    3 A car is travelling along a straight horizontal road with velocity \(32.5 \mathrm {~ms} ^ { - 1 }\). The driver applies the brakes and the car decelerates at \(( 8 - 0.6 t ) \mathrm { ms } ^ { - 2 }\), where \(t \mathrm {~s}\) is the time which has elapsed since the brakes were first applied.
    1. Show that, while the car is decelerating, its velocity is \(\left( 32.5 - 8 t + 0.3 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    2. Find the time taken to bring the car to rest.
    3. Show that the distance travelled while the car is decelerating is 75 m .
    OCR M1 2012 January Q4
    4
    \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-2_325_481_1699_792} Three horizontal forces of magnitudes \(8 \mathrm {~N} , 15 \mathrm {~N}\) and 20 N act at a point. The 8 N and 15 N forces are at right angles. The 20 N force makes an angle of \(150 ^ { \circ }\) with the 8 N force and an angle of \(120 ^ { \circ }\) with the 15 N force (see diagram).
    1. Calculate the components of the resultant force in the directions of the 8 N and 15 N forces.
    2. Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N force. The directions in which the three horizontal forces act can be altered.
    3. State the greatest and least possible magnitudes of the resultant force.
    OCR M1 2012 January Q5
    5
    \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-3_394_789_251_639} The diagram shows the ( \(t , v\) ) graph of an athlete running in a straight line on a horizontal track in a 100 m race. He starts from rest and has constant acceleration until he reaches a speed of \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = T\). He maintains this constant speed until he decelerates at a constant rate of \(1.75 \mathrm {~ms} ^ { - 2 }\) for the final 4 s of the race. He completes the race in 10 s .
    1. Calculate \(T\). The athlete races against a robot which has a displacement from the starting line of \(\left( 3 t ^ { 2 } - 0.2 t ^ { 3 } \right) \mathrm { m }\), at time \(t \mathrm {~s}\) after the start of the race.
    2. Show that the speed of the robot is \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = 5\).
    3. Find the value of \(t\) for which the decelerations of the robot and the athlete are equal.
    4. Verify that the athlete and the robot reach the finish line simultaneously.
    OCR M1 2012 January Q6
    6 A particle \(P\) of mass 0.3 kg is projected upwards along a line of greatest slope from the foot of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The initial speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction is 0.15 . The particle \(P\) comes to instantaneous rest before it reaches the top of the plane.
    1. Calculate the distance \(P\) moves up the plane.
    2. Find the time taken by \(P\) to return from its highest position on the plane to the foot of the plane.
    3. Calculate the change in the momentum of \(P\) between the instant that \(P\) leaves the foot of the plane and the instant that \(P\) returns to the foot of the plane.
    OCR M1 2012 January Q7
    7
    \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-4_369_508_246_781} Particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.05 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth pulley. \(Q\) is attached to a particle \(R\) of mass 0.45 kg by a light inextensible string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. \(P\) is in contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the string \(Q R\) is 2.52 N during the descent of \(R\).
    1. (a) Find the acceleration of \(R\) during its descent.
      (b) By considering the motion of \(Q\), calculate the tension in the string \(P Q\) during the descent of \(R\).
    2. Find the value of \(m\).
      \(R\) strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, \(P\) does not reach the pulley and \(Q\) does not reach the surface.
    3. Calculate the greatest height of \(P\) above the surface.
    OCR M1 2013 January Q1
    1 Three horizontal forces, acting at a single point, have magnitudes \(12 \mathrm {~N} , 14 \mathrm {~N}\) and 5 N and act along bearings \(000 ^ { \circ } , 090 ^ { \circ }\) and \(270 ^ { \circ }\) respectively. Find the magnitude and bearing of their resultant.
    OCR M1 2013 January Q2
    2 A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point on the line is \(\left( t ^ { 4 } - 2 t ^ { 3 } + 5 \right) \mathrm { m }\), where \(t\) is the time in seconds. Show that, when \(t = 1.5\),
    1. \(P\) is at instantaneous rest,
    2. the acceleration of \(P\) is \(9 \mathrm {~ms} ^ { - 2 }\).
    OCR M1 2013 January Q3
    3
    \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-2_300_501_799_790} A particle \(P\) of mass 0.25 kg moves upwards with constant speed \(u \mathrm {~ms} ^ { - 1 }\) along a line of greatest slope on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The pulling force acting on \(P\) has magnitude \(T \mathrm {~N}\) and acts at an angle of \(20 ^ { \circ }\) to the line of greatest slope (see diagram). Calculate
    1. the value of \(T\),
    2. the magnitude of the contact force exerted on \(P\) by the plane. The pulling force \(T \mathrm {~N}\) acting on \(P\) is suddenly removed, and \(P\) comes to instantaneous rest 0.4 s later.
    3. Calculate \(u\).
    OCR M1 2013 January Q4
    4 The acceleration of a particle \(P\) moving in a straight line is \(\left( t ^ { 2 } - 9 t + 18 \right) \mathrm { ms } ^ { - 2 }\), where \(t\) is the time in seconds.
    1. Find the values of \(t\) for which the acceleration is zero.
    2. It is given that when \(t = 3\) the velocity of \(P\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(t = 0\).
    3. Show that the direction of motion of \(P\) changes before \(t = 1\).
    OCR M1 2013 January Q5
    5
    \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-3_462_405_258_845} A small smooth pulley is suspended from a fixed point by a light chain. A light inextensible string passes over the pulley. Particles \(P\) and \(Q\), of masses 0.3 kg and \(m \mathrm {~kg}\) respectively, are attached to the opposite ends of the string. The particles are released from rest at a height of 0.2 m above horizontal ground with the string taut; the portions of the string not in contact with the pulley are vertical (see diagram). \(P\) strikes the ground with speed \(1.4 \mathrm {~ms} ^ { - 1 }\). Subsequently \(P\) remains on the ground, and \(Q\) does not reach the pulley.
    1. Calculate the acceleration of \(P\) while it is in motion and the corresponding tension in the string.
    2. Find the value of \(m\).
    3. Calculate the greatest height of \(Q\) above the ground.
    4. It is given that the mass of the pulley is 0.5 kg . State the magnitude of the tension in the chain which supports the pulley
      (a) when \(P\) is in motion,
      (b) when \(P\) is at rest on the ground and \(Q\) is moving upwards.
    OCR M1 2013 January Q6
    6 Particle \(P\) of mass 0.3 kg and particle \(Q\) of mass 0.2 kg are 3.6 m apart on a smooth horizontal surface. \(P\) and \(Q\) are simultaneously projected directly towards each other along a straight line. Before the particles collide \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Given that the particles coalesce in the collision, calculate their common speed after they collide.
    2. It is given instead that one particle is at rest immediately after the collision.
      (a) State which particle is in motion after the collision and find the speed of this particle.
      (b) Find the time taken after the collision for the moving particle to return to its initial position.
      (c) On a single diagram sketch the \(( t , v )\) graphs for the two particles, with \(t = 0\) as the instant of their initial projection.
      \(7 \quad A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(45 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected from \(A\) towards \(B\) with speed \(5 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between the plane and \(P\) is 0.2 .
    3. Given that the level of \(A\) is above the level of \(B\), calculate the speed of \(P\) when it passes through the point \(B\), and the time taken to travel from \(A\) to \(B\).
    4. Given instead that the level of \(A\) is below the level of \(B\),
      (a) show that \(P\) does not reach \(B\),
      (b) calculate the difference in the momentum of \(P\) for the two occasions when it is at \(A\).
    OCR M1 2009 June Q1
    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.
    OCR M1 2009 June Q2
    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.
    OCR M1 2009 June Q3
    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.
    OCR M1 2009 June Q4
    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 \(v\).
    OCR M1 2009 June Q6
    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 \(T\).
    OCR M1 2009 June Q7
    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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    OCR M1 2011 June Q1
    1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.
    OCR M1 2011 June Q2
    2 Particles \(P\) and \(Q\), of masses 0.45 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley. The particles are released from rest with the string taut and both particles 0.36 m above a horizontal surface. \(Q\) descends with acceleration \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). When \(Q\) strikes the surface, it remains at rest.
    1. Calculate the tension in the string while both particles are in motion.
    2. Find the value of \(m\).
    3. Calculate the speed at which \(Q\) strikes the surface.
    4. Calculate the greatest height of \(P\) above the surface. (You may assume that \(P\) does not reach the pulley.)
    OCR M1 2011 June Q3
    3 A block \(B\) of mass 0.8 kg is pulled across a horizontal surface by a force of 6 N inclined at an angle of \(60 ^ { \circ }\) to the upward vertical. The coefficient of friction between the block and the surface is 0.2 . Calculate
    1. the vertical component of the force exerted on \(B\) by the surface,
    2. the acceleration of \(B\). The 6 N force is removed when \(B\) has speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. Calculate the time taken for \(B\) to decelerate from a speed of \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to rest.
    OCR M1 2011 June Q4
    4
    \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-2_657_1495_1539_324} A car travelling on a straight road accelerates from rest to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . It continues at constant speed for 11 s and then decelerates to rest in 2 s . The driver gets out of the car and walks at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 20 s back to a shop which he enters. Some time later he leaves the shop and jogs to the car at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He arrives at the vehicle 60 s after it began to accelerate from rest. The diagram, which has six straight line segments, shows the \(( t , v )\) graph for the motion of the driver.
    1. Calculate the initial acceleration and final deceleration of the car.
    2. Calculate the distance the car travels.
    3. Calculate the length of time the driver is in the shop.
    OCR M1 2011 June Q5
    5
    \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_362_1065_258_539} Three particles \(P , Q\) and \(R\) lie on a line of greatest slope of a smooth inclined plane. \(P\) has mass 0.5 kg and initially is at the foot of the plane. \(R\) has mass 0.3 kg and initially is at the top of the plane. \(Q\) has mass 0.2 kg and is between \(P\) and \(R\) (see diagram). \(P\) is projected up the line of greatest slope with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant when \(Q\) and \(R\) are released from rest. Each particle has an acceleration of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) down the plane.
    1. \(P\) and \(Q\) collide 0.4 s after being set in motion. Immediately after the collision \(Q\) moves up the plane with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed and direction of motion of \(P\) immediately after the collision.
    2. 0.6 s after its collision with \(P , Q\) collides with \(R\) and the two particles coalesce. Find the speed and direction of motion of the combined particle immediately after the collision