Questions M1 (1912 questions)

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OCR M1 2013 June Q3
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
    (a) the angle between the resultant and the 12 N force,
    (b) the magnitude of the resultant.
  2. 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
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
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
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
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
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
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).
OCR M1 2015 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).
  1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
  2. the magnitude of the force exerted on \(P\) by the surface,
  3. the angle between the surface and the 10 N force.
OCR M1 2015 June Q5
5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:
  1. \(\theta = 0\);
  2. \(\theta = 20\);
  3. \(\theta = 70\);
  4. \(\theta = 90\).
OCR M1 2015 June Q6
6 A particle \(P\) moves in a straight line on a horizontal surface. \(P\) passes through a fixed point \(O\) on the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. Calculate the velocity of \(P\) when \(t = 3\).
  2. Find the distance \(O P\) when \(t = 3\). A second particle \(Q\), having the same mass as \(P\), moves along the same straight line. The displacement of \(Q\) from \(O\) is \(\left( k - 2 t ^ { 3 } \right) \mathrm { m }\), where \(k\) is a constant. When \(t = 3\) the particles collide and coalesce.
  3. Find the value of \(k\).
  4. Find the common velocity of the particles immediately after their collision.
OCR M1 2015 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424}
\(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).
  1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
  2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
  3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
  4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
    (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR M1 2016 June Q1
1 A stone is released from rest on a bridge and falls vertically into a lake. The stone has velocity \(14 \mathrm {~ms} ^ { - 1 }\) when it enters the lake.
  1. Calculate the distance the stone falls before it enters the lake, and the time after its release when it enters the lake. The lake is 15 m deep and the stone has velocity \(20 \mathrm {~ms} ^ { - 1 }\) immediately before it reaches the bed of the lake.
  2. Given that there is no sudden change in the velocity of the stone when it enters the lake, find the acceleration of the stone while it is falling through the lake.
OCR M1 2016 June Q2
2 A particle \(P\) is projected down a line of greatest slope on a smooth inclined plane. \(P\) has velocity \(5 \mathrm {~ms} ^ { - 1 }\) at the instant when it has been in motion for 1.6 s and travelled a distance of 6.4 m . Calculate
  1. the initial speed and the acceleration of \(P\),
  2. the inclination of the plane to the vertical.
OCR M1 2016 June Q3
3 Two forces each of magnitude 4 N have a resultant of magnitude 6 N .
  1. Calculate the angle between the two 4 N forces. The two given forces of magnitude 4 N act on a particle of mass \(m \mathrm {~kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude 3 N on the particle.
  2. Find \(m\), and give the acute angle between the surface and one of the 4 N forces.
OCR M1 2016 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-2_144_1317_1655_372} Four particles \(A , B , C\) and \(D\) are on the same straight line on a smooth horizontal table. \(A\) has speed \(6 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(B\). The speed of \(B\) is \(2 \mathrm {~ms} ^ { - 1 }\) and \(B\) is moving towards \(A\). The particle \(C\) is moving with speed \(5 \mathrm {~ms} ^ { - 1 }\) away from \(B\) and towards \(D\), which is stationary (see diagram). The first collision is between \(A\) and \(B\) which have masses 0.8 kg and 0.2 kg respectively.
  1. After the particles collide \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) in its original direction of motion. Calculate the speed of \(B\) after the collision. The second collision is between \(C\) and \(D\) which have masses 0.3 kg and 0.1 kg respectively.
  2. The particles coalesce when they collide. Find the speed of the combined particle after this collision. The third collision is between \(B\) and the combined particle, after which no further collisions occur.
  3. Calculate the greatest possible speed of the combined particle after the third collision.
OCR M1 2016 June Q5
5 Three forces act on a particle. The first force has magnitude \(P \mathrm {~N}\) and acts horizontally due east. The second force has magnitude 5 N and acts horizontally due west. The third force has magnitude \(2 P \mathrm {~N}\) and acts vertically upwards. The resultant of these three forces has magnitude 25 N .
  1. Calculate \(P\) and the angle between the resultant and the vertical. The particle has mass 3 kg and rests on a rough horizontal table. The coefficient of friction between the particle and the table is 0.15 .
  2. Find the acceleration of the particle, and state the direction in which it moves.
    \includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-3_485_876_797_598} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) has mass 0.2 kg and is held at rest on the plane. \(Q\) has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is 0.4 . The particle \(P\) is released.
  3. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released.
    \(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of 0.8 m before it comes to rest. \(P\) does not reach the pulley.
  4. Find the speed of the particles immediately before \(Q\) strikes the floor.
  5. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion.
OCR M1 2016 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-4_652_1068_255_500} The diagram shows the ( \(t , v\) ) graphs for two particles \(A\) and \(B\) which move on the same straight line. The units of \(v\) and \(t\) are \(\mathrm { ms } ^ { - 1 }\) and s respectively. Both particles are at the point \(S\) on the line when \(t = 0\). The particle \(A\) is initially at rest, and moves with acceleration \(0.18 t \mathrm {~ms} ^ { - 2 }\) until the two particles collide when \(t = 16\). The initial velocity of \(B\) is \(U \mathrm {~ms} ^ { - 1 }\) and \(B\) has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion \(B\) has a constant velocity of \(9 \mathrm {~ms} ^ { - 1 }\); finally \(B\) moves with constant deceleration for one second before it collides with \(A\).
  1. Calculate the value of \(t\) at which the two particles have the same velocity. For \(0 \leqslant t \leqslant 5\) the distance of \(B\) from \(S\) is \(\left( U t + 0.08 t ^ { 3 } \right) \mathrm { m }\).
  2. Calculate \(U\) and verify that when \(t = 5 , B\) is 25 m from \(S\).
  3. Calculate the velocity of \(B\) when \(t = 16\). \section*{END OF QUESTION PAPER}
OCR MEI M1 2009 January Q1
1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$
  1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
  2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the total distance travelled by the particle during the 15 seconds.
OCR MEI M1 2009 January Q2
2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-2_684_1070_1064_536} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.
  1. Find the velocity of the particle when \(t = 2\).
  2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?
OCR MEI M1 2009 January Q3
3 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 2009 January Q4
4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \(\left( 9.8 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) higher than O . When descending, the stone hits a plum which is 3.675 m higher than O . Air resistance should be neglected. Calculate the horizontal distance of the plum from O .
OCR MEI M1 2009 January Q5
5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_456_476_833_833} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the upward acceleration.
  2. Find the reaction on the man of the lift floor.
OCR MEI M1 2009 January Q6
6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_318_271_1800_938} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). Section B (36 marks)
OCR MEI M1 2009 January Q7
7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-4_327_1013_482_566} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.
  1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
  2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
  3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
  4. Calculate the frictional force between the slope and the sledge.
OCR MEI M1 2009 January Q8
8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
  2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
  3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
  4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
  5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).