Questions M1 (1912 questions)

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OCR M1 2013 January Q4
Standard +0.3
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
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
    (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
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.
    (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
Moderate -0.8
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
Moderate -0.8
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
Standard +0.3
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
Moderate -0.3
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
Moderate -0.3
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
Moderate -0.3
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.