Questions M1 (1912 questions)

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Edexcel M1 2013 January Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-12_337_1084_228_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows two particles \(A\) and \(B\), of mass \(2 m\) and \(4 m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,
  1. give a reason why the magnitudes of the accelerations of the two particles are the same,
  2. write down an equation of motion for each particle,
  3. find the acceleration of each particle. Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),
  4. find the distance \(X Y\) in terms of \(h\).
OCR M1 2010 January Q1
Moderate -0.8
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
Moderate -0.8
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
Standard +0.3
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
Standard +0.3
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
Standard +0.3
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
Standard +0.3
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.
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OCR M1 2011 January Q1
Moderate -0.8
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
Moderate -0.5
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
Moderate -0.8
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
Standard +0.3
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
Standard +0.8
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
Moderate -0.3
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
Standard +0.3
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.
OCR M1 2011 January Q8
Moderate -0.8
8
6
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    7(ii) (b(ii) (b)
    \section*{OCR} RECOGNISING ACHIEVEMENT
    •
    OCR M1 2012 January Q1
    Moderate -0.8
    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
    Moderate -0.3
    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
    Standard +0.3
    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
    Moderate -0.3
    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
    Moderate -0.3
    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
    Standard +0.3
    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
    Standard +0.3
    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
    Moderate -0.8
    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
    Moderate -0.8
    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
    Standard +0.3
    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 \(u\).