Questions — OCR M1 (171 questions)

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OCR M1 2013 January Q5
14 marks Standard +0.8
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
    1. when \(P\) is in motion,
    2. when \(P\) is at rest on the ground and \(Q\) is moving upwards.
OCR M1 2013 January Q6
15 marks Moderate -0.8
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.
    1. State which particle is in motion after the collision and find the speed of this particle.
    2. Find the time taken after the collision for the moving particle to return to its initial position.
    3. 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 .
      1. 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 \(B\).
      2. 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 2011 June Q1
4 marks Easy -1.8
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
10 marks Standard +0.3
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
10 marks Moderate -0.3
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
10 marks Moderate -0.8
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
10 marks Standard +0.3
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
OCR M1 2011 June Q6
11 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_348_1109_1345_516} A small smooth ring \(R\) of weight 7 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\) at the same horizontal level. A horizontal force of magnitude 5 N is applied to \(R\). The string is taut. In the equilibrium position the angle \(A R B\) is a right angle, and the portion of the string attached to \(B\) makes an angle \(\theta\) with the horizontal (see diagram).
  1. Explain why the tension \(T \mathrm {~N}\) is the same in each part of the string.
  2. By resolving horizontally and vertically for the forces acting on \(R\), form two simultaneous equations in \(T \cos \theta\) and \(T \sin \theta\).
  3. Hence find \(T\) and \(\theta\).
OCR M1 2011 June Q7
17 marks Standard +0.3
7 A particle \(P\) is projected from a fixed point \(O\) on a straight line. The displacement \(x\) m of \(P\) from \(O\) at time \(t \mathrm {~s}\) after projection is given by \(x = 0.1 t ^ { 3 } - 0.3 t ^ { 2 } + 0.2 t\).
  1. Express the velocity and acceleration of \(P\) in terms of \(t\).
  2. Show that when the acceleration of \(P\) is zero, \(P\) is at \(O\).
  3. Find the values of \(t\) when \(P\) is stationary. At the instant when \(P\) first leaves \(O\), a particle \(Q\) is projected from \(O\). \(Q\) moves on the same straight line as \(P\) and at time \(t \mathrm {~s}\) after projection the velocity of \(Q\) is given by \(\left( 0.2 t ^ { 2 } - 0.4 \right) \mathrm { ms } ^ { - 1 } . P\) and \(Q\) collide first when \(t = T\).
  4. Show that \(T\) satisfies the equation \(t ^ { 2 } - 9 t + 18 = 0\), and hence find \(T\).
OCR M1 2012 June Q1
6 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_305_295_264_868} Two perpendicular forces of magnitudes \(F \mathrm {~N}\) and 8 N act at a point \(O\) (see diagram). Their resultant has magnitude 17 N .
  1. Calculate \(F\) and find the angle which the resultant makes with the 8 N force. A third force of magnitude \(E \mathrm {~N}\), acting in the same plane as the two original forces, is now applied at the point \(O\). The three forces of magnitudes \(E N , F N\) and \(8 N\) are in equilibrium.
  2. State the value of \(E\) and the angle between the directions of the \(E \mathrm {~N}\) and 8 N forces.
OCR M1 2012 June Q2
8 marks Moderate -0.8
2 A particle is projected vertically upwards with speed \(7 \mathrm {~ms} ^ { - 1 }\) from a point on the ground.
  1. Find the speed of the particle and its distance above the ground 0.4 s after projection.
  2. Find the total distance travelled by the particle in the first 0.9 s after projection.
OCR M1 2012 June Q3
7 marks Standard +0.3
3 \(\mathrm { v } \left( \mathrm { ms } ^ { - 1 } \right)\) \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_449_1121_1500_440}
not to scale The diagram shows the \(( t , v )\) graphs for two athletes, \(A\) and \(B\), who run in the same direction in the same straight line while they exchange the baton in a relay race. \(A\) runs with constant velocity \(10 \mathrm {~ms} ^ { - 1 }\) until he decelerates at \(5 \mathrm {~ms} ^ { - 2 }\) and subsequently comes to rest. \(B\) has constant acceleration from rest until reaching his constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The baton is exchanged 2 s after \(B\) starts running, when both athletes have speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) is 1 m ahead of \(A\).
  1. Find the value of \(t\) at which \(A\) starts to decelerate.
  2. Calculate the distance between \(A\) and \(B\) at the instant when \(B\) starts to run.
OCR M1 2012 June Q4
10 marks Moderate -0.8
4 A block \(B\) of weight 28 N is pulled at constant speed across a rough horizontal surface by a force of magnitude 14 N inclined at \(30 ^ { \circ }\) above the horizontal.
  1. Show that the coefficient of friction between the block and the surface is 0.577 , correct to 3 significant figures. The 14 N force is suddenly removed, and the block decelerates, coming to rest after travelling a further 3.2 m .
  2. Calculate the speed of the block at the instant the 14 N force was removed.
OCR M1 2012 June Q5
13 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.
  1. Calculate the tension in the string while \(Q\) is in motion.
  2. Calculate the momentum lost by \(Q\) when it reaches the surface.
  3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}
OCR M1 2012 June Q6
13 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_328_698_255_657} A particle \(P\) lies on a slope inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is attached to one end of a taut light inextensible string which passes through a small smooth ring \(Q\) of mass \(m \mathrm {~kg}\). The portion \(P Q\) of the string is horizontal and the other portion of the string is inclined at \(40 ^ { \circ }\) to the vertical. A horizontal force of magnitude \(H \mathrm {~N}\), acting away from \(P\), is applied to \(Q\) (see diagram). The tension in the string is 6.4 N , and the string is in the vertical plane containing the line of greatest slope on which \(P\) lies. Both \(P\) and \(Q\) are in equilibrium.
  1. Calculate \(m\).
  2. Calculate \(H\).
  3. Given that the weight of \(P\) is 32 N , and that \(P\) is in limiting equilibrium, show that the coefficient of friction between \(P\) and the slope is 0.879 , correct to 3 significant figures. \(Q\) and the string are now removed.
  4. Determine whether \(P\) remains in equilibrium.
OCR M1 2012 June Q7
15 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
  2. (a) State the value of \(V _ { 0 }\).
    (b) Calculate the distance \(C\) moves before it strikes \(R\).
    (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
  3. Find the speed of \(C\) immediately after it strikes \(R\).
OCR M1 2013 June Q1
6 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_135_917_274_575} Three particles \(P , Q\) and \(R\) have masses \(0.1 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and 0.6 kg respectively. The particles travel along the same straight line on a smooth horizontal table and have velocities \(1.5 \mathrm {~ms} ^ { - 1 } , 1.1 \mathrm {~ms} ^ { - 1 }\) and \(0.8 \mathrm {~ms} ^ { - 1 }\) respectively (see diagram). \(P\) collides with \(Q\) and then \(Q\) collides with \(R\). In the second collision \(Q\) and \(R\) coalesce and subsequently move with a velocity of \(1 \mathrm {~ms} ^ { - 1 }\).
  1. Find the speed of \(Q\) immediately before the second collision.
  2. Calculate the change in momentum of \(P\) in the first collision.
OCR M1 2013 June Q2
8 marks Moderate -0.8
2 A particle \(P\) is projected vertically upwards and reaches its greatest height 0.5 s after the instant of projection. Calculate
  1. the speed of projection of \(P\),
  2. the greatest height of \(P\) above the point of projection. It is given that the point of projection is 0.539 m above the ground.
  3. Find the speed of \(P\) immediately before it strikes the ground.
OCR M1 2013 June Q3
9 marks Moderate -0.8
3 Two forces of magnitudes 8 N and 12 N act at a point \(O\).
  1. Given that the two forces are perpendicular to each other, find
    1. the angle between the resultant and the 12 N force,
    2. the magnitude of the resultant.
    3. It is given instead that the resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts in a direction perpendicular to the 8 N force (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_248_388_1877_826}
      (a) Calculate the angle between the resultant and the 12 N force.
      (b) Find \(R\).
OCR M1 2013 June Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-3_298_540_262_735} The diagram shows the \(( t , v )\) graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\). The car passes through \(A\) with velocity \(18 \mathrm {~ms} ^ { - 1 }\), and moves with constant acceleration \(2.4 \mathrm {~ms} ^ { - 2 }\) until \(t = 5\). The car subsequently moves with constant velocity until it is 300 m from \(A\). When the car is more than 300 m from \(A\), it has constant deceleration \(6 \mathrm {~ms} ^ { - 2 }\), until it comes to rest.
  1. Find the greatest speed of the car.
  2. Calculate the value of \(t\) for the instant when the car begins to decelerate.
  3. Calculate the distance from \(A\) of the car when it is at rest.
OCR M1 2013 June Q5
10 marks Standard +0.3
5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the top of a smooth inclined plane of length \(2 d\) metres. After its projection \(P\) moves downwards along a line of greatest slope with acceleration \(4 \mathrm {~ms} ^ { - 2 }\). At the instant 3 s after projection \(P\) has moved half way down the plane. \(P\) reaches the foot of the plane 5 s after the instant of projection.
  1. Form two simultaneous equations in \(u\) and \(d\), and hence calculate the speed of projection of \(P\) and the length of the plane.
  2. Find the inclination of the plane to the horizontal.
  3. Given that the contact force exerted on \(P\) by the plane has magnitude 6 N , calculate the mass of \(P\).
OCR M1 2013 June Q6
14 marks Standard +0.3
6 A particle \(P\) moves in a straight line. At time \(t\) s after passing through a point \(O\) of the line, the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). Given that \(x = 0.06 t ^ { 3 } - 0.45 t ^ { 2 } - 0.24 t\), find
  1. the velocity and the acceleration of \(P\) when \(t = 0\),
  2. the value of \(x\) when \(P\) has its minimum velocity, and the speed of \(P\) at this instant,
  3. the positive value of \(t\) when the direction of motion of \(P\) changes.
OCR M1 2013 June Q7
15 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-4_310_579_255_721} A block \(B\) is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.6 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth pulley fixed to the top of the plane. A particle \(Q\) of mass 0.5 kg is attached to the other end of the string. The portion of the string attached to \(P\) is parallel to a line of greatest slope of the plane, the portion of the string attached to \(Q\) is vertical and the string is taut. The particles are released from rest and start to move with acceleration \(1.4 \mathrm {~ms} ^ { - 2 }\) (see diagram). It is given that \(B\) is in equilibrium while \(P\) moves on its upper surface.
  1. Find the tension in the string while \(P\) and \(B\) are in contact.
  2. Calculate the coefficient of friction between \(P\) and \(B\).
  3. Given that the weight of \(B\) is 7 N , calculate the set of possible values of the coefficient of friction between \(B\) and the plane.
OCR M1 2015 June Q1
7 marks Moderate -0.8
1 A particle \(P\) is projected vertically downwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 30 m above the ground.
  1. Calculate the speed of \(P\) when it reaches the ground.
  2. Find the distance travelled by \(P\) in the first 0.4 s of its motion.
  3. Calculate the time taken for \(P\) to travel the final 15 m of its descent.
OCR M1 2015 June Q2
8 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_138_1118_680_463} Three particles \(P , Q\) and \(R\) with masses \(0.4 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are moving along the same straight line on a smooth horizontal surface. \(P\) and \(Q\) are moving towards each other with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(R\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the same direction as \(Q\) (see diagram).
  1. Immediately after the collision between \(P\) and \(Q\) their directions of motion have been reversed, but their speeds are unchanged. Calculate \(u\). The next collision is between \(Q\) and \(R\). After the collision between \(Q\) and \(R\), particle \(Q\) is at rest and \(R\) has speed \(9 \mathrm {~ms} ^ { - 1 }\).
  2. Calculate \(m\). \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_547_1506_1521_251} Two travellers \(A\) and \(B\) make the same journey on a long straight road. Each traveller walks for part of the journey and rides a bicycle for part of the journey. They start their journeys at the same instant, and they end their journeys simultaneously after travelling for \(T\) hours. \(A\) starts the journey cycling at a steady \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) for 1 hour. \(A\) then leaves the bicycle at the side of the road, and completes the journey walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \(B\) begins the journey walking at a steady \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). When \(B\) finds the bicycle where \(A\) left it, \(B\) cycles at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) to complete the journey (see diagram).