Questions — OCR M1 (171 questions)

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OCR M1 2015 June Q4
9 marks Moderate -0.3
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
11 marks Moderate -0.3
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
14 marks Standard +0.3
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
15 marks Standard +0.3
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 2014 June Q1
7 marks Moderate -0.3
1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.
  1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
  2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
  3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.
OCR M1 2014 June Q2
7 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes \(2.5 \mathrm {~N} , F \mathrm {~N}\) and 2.4 N act on the particle on bearings \(\theta ^ { \circ } , 180 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram). The particle is in equilibrium.
  1. Find \(\theta\) and \(F\). The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
  2. Find the magnitude and direction of the acceleration of the particle.
OCR M1 2014 June Q3
8 marks Moderate -0.8
3 A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\) seconds after it passes through a fixed point \(A\) is given by \(\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }\). Find
  1. the velocity of \(P\) when it passes through \(A\),
  2. the displacement of \(P\) from \(A\) when \(t = 1.5\),
  3. the velocity of \(P\) when it has acceleration \(6 \mathrm {~ms} ^ { - 2 }\).
OCR M1 2014 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles \(P\) and \(Q\) are moving towards each other with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) along the same straight line on a smooth horizontal surface (see diagram). \(P\) has mass 0.2 kg and \(Q\) has mass 0.3 kg . The two particles collide.
  1. Show that \(Q\) must change its direction of motion in the collision.
  2. Given that \(P\) and \(Q\) move with equal speed after the collision, calculate both possible values for their speed after they collide.
OCR M1 2014 June Q5
12 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle \(P\) can move in a straight line on a horizontal surface. At time \(t\) seconds the displacement of \(P\) from a fixed point \(A\) on the line is \(x \mathrm {~m}\). The diagram shows the \(( t , x )\) graph for \(P\). In the interval \(0 \leqslant t \leqslant 10\), either the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\), or \(P\) is at rest.
  1. Show by calculation that \(T = 1.75\).
  2. State the velocity of \(P\) when
    1. \(t = 2\),
    2. \(t = 8\),
    3. \(t = 9\).
    4. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 10\). For \(t > 10\), the displacement of \(P\) from \(A\) is given by \(x = 20 t - t ^ { 2 } - 96\).
    5. Calculate the value of \(t\), where \(t > 10\), for which the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
OCR M1 2014 June Q6
14 marks Moderate -0.3
6 A particle \(P\) of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on \(P\), and \(P\) is in limiting equilibrium.
  1. Calculate the coefficient of friction between \(P\) and the surface.
  2. Find the magnitude and direction of the contact force exerted by the surface on \(P\).
  3. \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-4_190_579_580_598} The initial 3 N force continues to act on \(P\) in its original direction. An additional force of magnitude \(T \mathrm {~N}\), acting in the same vertical plane as the 3 N force, is now applied to \(P\) at an angle of \(\theta ^ { \circ }\) above the horizontal (see diagram). \(P\) is again in limiting equilibrium.
    1. Given that \(\theta = 0\), find \(T\).
    2. Given instead that \(\theta = 30\), calculate \(T\).
OCR M1 2014 June Q7
16 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(M\) is the mid-point of \(A B\). Two particles \(P\) and \(Q\), joined by a taut light inextensible string, are placed on the plane at \(A\) and \(M\) respectively. The particles are simultaneously projected with speed \(0.6 \mathrm {~ms} ^ { - 1 }\) down the line of greatest slope (see diagram). The particles move down the plane with acceleration \(0.9 \mathrm {~ms} ^ { - 2 }\). At the instant 2 s after projection, \(P\) is at \(M\) and \(Q\) is at \(B\). The particle \(Q\) subsequently remains at rest at \(B\).
  1. Find the distance \(A B\). The plane is rough between \(A\) and \(M\), but smooth between \(M\) and \(B\).
  2. Calculate the speed of \(P\) when it reaches \(B\). \(P\) has mass 0.4 kg and \(Q\) has mass 0.3 kg .
  3. By considering the motion of \(Q\), calculate the tension in the string while both particles are moving down the plane.
  4. Calculate the coefficient of friction between \(P\) and the plane between \(A\) and \(M\). \section*{END OF QUESTION PAPER}
OCR M1 2010 January Q5
11 marks Moderate -0.3
  1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
  2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
  3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).
OCR M1 2015 June Q3
8 marks Standard +0.3
  1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
  2. Find \(T\).
  3. Calculate the distance \(A\) and \(B\) each travel.
OCR M1 Q1
7 marks Moderate -0.3
\includegraphics{figure_1} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m\) kg is threaded on the string and is pulled by a force of magnitude \(1.6\) N acting upwards at \(45°\) to the horizontal. The section \(AR\) of the string makes an angle of \(30°\) with the downward vertical and the section \(BR\) is horizontal (see diagram). The ring is in equilibrium with the string taut.
  1. Give a reason why the tension in the part \(AR\) of the string is the same as that in the part \(BR\). [1]
  2. Show that the tension in the string is \(0.754\) N, correct to 3 significant figures. [3]
  3. Find the value of \(m\). [3]
OCR M1 Q2
7 marks Standard +0.3
\includegraphics{figure_2} Particles \(A\) and \(B\), of masses \(0.2\) kg and \(0.3\) kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is \(0.4\) N and the air resistance acting on \(B\) is \(0.25\) N (see diagram). The downward acceleration of each of the particles is \(a\) m s\(^{-2}\) and the tension in the string is \(T\) N.
  1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\). [4]
  2. Find the values of \(a\) and \(T\). [3]
OCR M1 Q3
8 marks Standard +0.3
Two small spheres \(P\) and \(Q\) have masses \(0.1\) kg and \(0.2\) kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).
  1. Find the value of \(u\). [4]
After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.
  1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]
OCR M1 Q4
9 marks Standard +0.3
A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find
  1. the value of \(u\) and the acceleration of the particle, [7]
  2. the value of \(\alpha\) in degrees. [2]
OCR M1 Q5
12 marks Standard +0.8
\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.
  1. Find the coefficient of friction between the wire and the ring \(A\). [7]
  2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
  3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]
OCR M1 Q6
12 marks Standard +0.3
A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) N in a direction at an angle \(\alpha\) to the upward vertical.
  1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20°\) find
    1. the value of \(P\), [3]
    2. the magnitude of the resultant, [2]
    3. the magnitude of the acceleration of the particle. [2]
  2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]
OCR M1 Q7
17 marks Standard +0.3
\includegraphics{figure_7} A car \(P\) starts from rest and travels along a straight road for \(600\) s. The \((t, v)\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find
  1. the distance travelled by \(P\), [3]
  2. the deceleration of \(P\) during the interval \(500 < t < 600\). [2]
Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for \(600\) s. At time \(t\) s after starting the velocity of \(Q\) is \((600t^2 - t^3) \times 10^{-6}\) m s\(^{-1}\).
  1. Find an expression in terms of \(t\) for the acceleration of \(Q\). [2]
  2. Find how much less \(Q\)'s deceleration is than \(P\)'s when \(t = 550\). [2]
  3. Show that \(Q\) has its maximum velocity when \(t = 400\). [2]
  4. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\). [6]
OCR M1 Q1
7 marks Standard +0.2
\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).
  1. Show that \(a = -1.4\). [4]
Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).
  1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]
OCR M1 Q2
7 marks Moderate -0.3
\includegraphics{figure_2} An object of mass \(0.08\) kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration \(1.25\) m s\(^{-2}\). The string makes a constant angle \(\alpha\) with the downward vertical and the tension in the string is \(T\) N (see diagram).
  1. By applying Newton's second law horizontally to the object, find the value of \(T \sin \alpha\). [2]
  2. Find the value of \(T\). [5]
OCR M1 Q3
11 marks Moderate -0.3
A motorcyclist starts from rest at a point \(O\) and travels in a straight line. His velocity after \(t\) seconds is \(v\) m s\(^{-1}\), for \(0 \leq t \leq T\), where \(v = 7.2t - 0.45t^2\). The motorcyclist's acceleration is zero when \(t = T\).
  1. Find the value of \(T\). [4]
  2. Show that \(v = 28.8\) when \(t = T\). [1]
For \(t \geq T\) the motorcyclist travels in the same direction as before, but with constant speed \(28.8\) m s\(^{-1}\).
  1. Find the displacement of the motorcyclist from \(O\) when \(t = 31\). [6]
OCR M1 Q4
11 marks Moderate -0.3
\includegraphics{figure_4} A block of mass \(2\) kg is at rest on a rough horizontal plane, acted on by a force of magnitude \(12\) N at an angle of \(15°\) upwards from the horizontal (see diagram).
  1. Find the frictional component of the contact force exerted on the block by the plane. [2]
  2. Show that the normal component of the contact force exerted on the block by the plane has magnitude \(16.5\) N, correct to 3 significant figures. [2]
It is given that the block is on the point of sliding.
  1. Find the coefficient of friction between the block and the plane. [2]
The force of magnitude \(12\) N is now replaced by a horizontal force of magnitude \(20\) N. The block starts to move.
  1. Find the acceleration of the block. [5]
OCR M1 Q5
11 marks Standard +0.3
A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).
  1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
  2. Find the deceleration of the car for \(0 < t < 8\). [2]
  3. Find the value of \(T\). [4]