Questions — OCR (4907 questions)

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OCR M1 2009 June Q5
11 marks Standard +0.3
  1. \includegraphics{figure_5_1} A particle \(P\) of mass \(0.5\) kg is projected with speed \(6\) m s\(^{-1}\) on a smooth horizontal surface towards a stationary particle \(Q\) of mass \(m\) kg (see Fig. 1). After the particles collide, \(P\) has speed \(v\) m s\(^{-1}\) in the original direction of motion, and \(Q\) has speed \(1\) m s\(^{-1}\) more than \(P\). Show that \(v(m + 0.5) = -m + 3\). [3]
  2. \includegraphics{figure_5_2} \(Q\) and \(P\) are now projected towards each other with speeds \(4\) m s\(^{-1}\) and \(2\) m s\(^{-1}\) respectively (see Fig. 2). Immediately after the collision the speed of \(Q\) is \(v\) m s\(^{-1}\) with its direction of motion unchanged and \(P\) has speed \(1\) m s\(^{-1}\) more than \(Q\). Find another relationship between \(m\) and \(v\) in the form \(v(m + 0.5) = am + b\), where \(a\) and \(b\) are constants. [4]
  3. By solving these two simultaneous equations show that \(m = 0.9\), and hence find \(v\). [4]
OCR M1 2009 June Q6
11 marks Standard +0.3
A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.
    1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
    2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
  2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]
OCR M1 2009 June Q7
17 marks Moderate -0.3
\includegraphics{figure_7} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leq t \leq 3\), \(S\) has velocity \((6t - t^2)\) m s\(^{-1}\). For \(3 < t \leq 22\), \(S\) runs at a constant speed of \(9\) m s\(^{-1}\). For \(t > 22\), \(S\) decelerates at \(0.6\) m s\(^{-2}\) (see diagram).
  1. Express the acceleration of \(S\) during the first \(3\) seconds in terms of \(t\). [2]
  2. Show that \(S\) runs \(18\) m in the first \(3\) seconds of motion. [5]
  3. Calculate the time \(S\) takes to run \(100\) m. [3]
  4. Calculate the time \(S\) takes to run \(200\) m. [7]
OCR M1 2016 June Q1
7 marks Moderate -0.8
A stone is released from rest on a bridge and falls vertically into a lake. The stone has velocity \(14\text{ m s}^{-1}\) 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. [4]
The lake is \(15\text{ m}\) deep and the stone has velocity \(20\text{ m s}^{-1}\) immediately before it reaches the bed of the lake.
  1. 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. [3]
OCR M1 2016 June Q2
8 marks Moderate -0.3
A particle \(P\) is projected down a line of greatest slope on a smooth inclined plane. \(P\) has velocity \(5\text{ m s}^{-1}\) at the instant when it has been in motion for \(1.6\text{ s}\) and travelled a distance of \(6.4\text{ m}\). Calculate
  1. the initial speed and the acceleration of \(P\), [5]
  2. the inclination of the plane to the vertical. [3]
OCR M1 2016 June Q3
7 marks Moderate -0.3
Two forces each of magnitude \(4\text{ N}\) have a resultant of magnitude \(6\text{ N}\).
  1. Calculate the angle between the two \(4\text{ N}\) forces. [4]
The two given forces of magnitude \(4\text{ N}\) act on a particle of mass \(m\text{ kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude \(3\text{ N}\) on the particle.
  1. Find \(m\), and give the acute angle between the surface and one of the \(4\text{ N}\) forces. [3]
OCR M1 2016 June Q4
11 marks Standard +0.3
\includegraphics{figure_4} Four particles \(A\), \(B\), \(C\) and \(D\) are on the same straight line on a smooth horizontal table. \(A\) has speed \(6\text{ m s}^{-1}\) and is at rest towards \(B\). The speed of \(B\) is \(2\text{ m s}^{-1}\) and \(B\) is moving towards \(A\). The particle \(C\) is moving with speed \(5\text{ m s}^{-1}\) away from \(B\) and towards \(D\), which is stationary (see diagram). The first collision is between \(A\) and \(B\) which have masses \(0.8\text{ kg}\) and \(0.2\text{ kg}\) respectively.
  1. After the particles collide \(A\) has speed \(4\text{ m s}^{-1}\) in its original direction of motion. Calculate the speed of \(B\) after the collision. [4]
The second collision is between \(C\) and \(D\) which have masses \(0.3\text{ kg}\) and \(0.1\text{ kg}\) respectively.
  1. The particles coalesce when they collide. Find the speed of the combined particle after this collision. [3]
The third collision is between \(B\) and the combined particle, after which no further collisions occur.
  1. Calculate the greatest possible speed of the combined particle after the third collision. [4]
OCR M1 2016 June Q5
12 marks Standard +0.3
Three forces act on a particle. The first force has magnitude \(P\text{ N}\) and acts horizontally due east. The second force has magnitude \(5\text{ N}\) and acts horizontally due west. The third force has magnitude \(2P\text{ N}\) and acts vertically upwards. The resultant of these three forces has magnitude \(25\text{ N}\).
  1. Calculate \(P\) and the angle between the resultant and the vertical. [7]
The particle has mass \(3\text{ kg}\) and rests on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.15\).
  1. Find the acceleration of the particle, and state the direction in which it moves. [5]
OCR M1 2016 June Q6
14 marks Standard +0.3
\includegraphics{figure_6} 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 \(30°\) to the horizontal. \(P\) has mass \(0.2\text{ kg}\) and is held at rest on the plane. \(Q\) has mass \(0.2\text{ kg}\) 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.
  1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released. [6]
\(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of \(0.8\text{ m}\) before it comes to rest. \(P\) does not reach the pulley.
  1. Find the speed of the particles immediately before \(Q\) strikes the floor. [5]
  2. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. [3]
OCR M1 2016 June Q7
13 marks Standard +0.3
\includegraphics{figure_7} 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 \(\text{m s}^{-1}\) and \(\text{s}\) 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 \(0.18t\text{ m s}^{-2}\) until the two particles collide when \(t = 16\). The initial velocity of \(B\) is \(U\text{ m s}^{-1}\) 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 \(9\text{ m s}^{-1}\); 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. [4]
For \(0 \leq t \leq 5\) the distance of \(B\) from \(S\) is \((Ut + 0.08t^2)\text{ m}\).
  1. Calculate \(U\) and verify that when \(t = 5\), \(B\) is \(25\text{ m}\) from \(S\). [4]
  2. Calculate the velocity of \(B\) when \(t = 16\). [5]
OCR M1 Specimen Q1
4 marks Easy -1.2
\includegraphics{figure_1} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude \(E\) newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N, and the acceleration of the truck is \(0.2 \text{ m s}^{-2}\). Find the value of \(E\). [4]
OCR M1 Specimen Q2
7 marks Moderate -0.3
\includegraphics{figure_2} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30°\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R\) N and acts at an angle \(\theta°\) to the force of magnitude 8 N, as shown in Fig. 2. Find \(R\) and \(\theta\). [7]
OCR M1 Specimen Q3
10 marks Moderate -0.8
A particle is projected vertically upwards, from the ground, with a speed of \(28 \text{ m s}^{-1}\). Ignoring air resistance, find
  1. the maximum height reached by the particle, [2]
  2. the speed of the particle when it is 30 m above the ground, [3]
  3. the time taken for the particle to fall from its highest point to a height of 30 m, [3]
  4. the length of time for which the particle is more than 30 m above the ground. [2]
OCR M1 Specimen Q4
12 marks Moderate -0.8
\includegraphics{figure_3} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s. The woman runs at a constant speed throughout. Fig. 1 shows the \((t, v)\) graph for the woman.
  1. Find the total distance run by the woman. [3]
  2. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), [3]
\includegraphics{figure_4} At time \(t = 0\), a child also starts to move, from \(A\), along \(AB\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s. Fig. 2 shows the \((t, v)\) graph for the child; it consists of two straight line segments.
  1. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s. [3]
  2. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant. [3]
OCR M1 Specimen Q5
13 marks Moderate -0.3
A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(-\frac{1}{10}t \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \(V \text{ m s}^{-1}\).
  1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\). [4]
  2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\). [2]
  3. Find the displacement of \(P\) from \(O\) when \(t = 10\). [4]
  4. Find the speed with which the particle returns to \(O\). [3]
OCR M1 Specimen Q6
13 marks Standard +0.3
\includegraphics{figure_5} Three uniform spheres \(A\), \(B\) and \(C\) have masses 0.3 kg, 0.4 kg and \(m\) kg respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \text{ m s}^{-1}\) towards \(B\) (see diagram). Air resistance may be ignored.
  1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \text{ m s}^{-1}\). Find the speed with which \(B\) starts to move. [4]
  2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \text{ m s}^{-1}\) and \(0.4 \text{ m s}^{-1}\) respectively. Find the value of \(m\). [4]
  3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\). [1]
  4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers. [4]
OCR M1 Specimen Q7
13 marks Standard +0.3
A sledge of mass 25 kg is on a plane inclined at \(30°\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2.
  1. \includegraphics{figure_6} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \text{ m s}^{-1}\) after being pulled for 10 s. Ignoring air resistance, find the tension in the cable. [6]
  2. \includegraphics{figure_7} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane. [7]
OCR M2 Q1
5 marks Standard +0.3
A uniform solid cone has vertical height 20 cm and base radius \(r\) cm. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is \(24°\) (see diagram). \includegraphics{figure_1}
  1. Find \(r\), correct to 1 decimal place. [4]
A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of \(24°\).
  1. State, with justification, whether this cone will topple. [1]
OCR M2 Q2
6 marks Moderate -0.5
A particle is projected horizontally with a speed of 6 m s\(^{-1}\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground. [6]
OCR M2 Q3
8 marks Standard +0.3
\includegraphics{figure_3} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(PQ = 0.8\) m. A small smooth bead \(B\), of mass 0.01 kg, is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius 0.6 m. \(QB\) rotates with constant angular speed \(\omega\) rad s\(^{-1}\) (see diagram).
  1. Show that the tension in the string is 0.1225 N. [3]
  2. Find \(\omega\). [3]
  3. Calculate the kinetic energy of the bead. [2]
OCR M2 Q4
9 marks Standard +0.3
\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).
  1. Find the coefficient of restitution between \(A\) and \(B\). [4]
  2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]
Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.
  1. Show that there will be another collision. [3]
OCR M2 Q5
10 marks Standard +0.3
\includegraphics{figure_5} A uniform rod \(AB\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).
  1. Show that the tension in the string is 4.5 N. [4]
  2. Find the magnitude and direction of the force acting on the rod at \(A\). [6]
OCR M2 Q6
10 marks Standard +0.3
A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5°\) to the horizontal. At a certain point \(P\) on the hill the car's speed is 20 m s\(^{-1}\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is 15 m s\(^{-1}\).
  1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. [4]
Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
  1. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW, correct to 3 significant figures. [3]
  2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]
OCR M2 Q7
11 marks Standard +0.8
\includegraphics{figure_7} A barrier is modelled as a uniform rectangular plank of wood, \(ABCD\), rigidly joined to a uniform square metal plate, \(DEFG\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m. The metal plate has mass 80 kg and side 0.5 m. The plank and plate are joined in such a way that \(CDE\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(CH\) is 0.25 m.
  1. Calculate the contact force at \(H\) when the barrier is in the closed position. [3]
In the open position, the centre of mass of the barrier is vertically above \(D\).
  1. Calculate the angle between \(AB\) and the horizontal when the barrier is in the open position. [8]
OCR M2 Q8
13 marks Standard +0.3
A particle is projected with speed 49 m s\(^{-1}\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac{x^2(1 + \tan^2 \theta)}{490}.$$ [4]
\includegraphics{figure_8} The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta_1\) and \(\theta_2\), and the corresponding points where the particle returns to the plane are \(A_1\) and \(A_2\) respectively (see diagram).
  1. Find \(\theta_1\) and \(\theta_2\). [4]
  2. Calculate the distance between \(A_1\) and \(A_2\). [5]