Questions M1 (2067 questions)

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OCR M1 2007 January Q4
10 marks Moderate -0.3
\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses 0.8 kg, 0.6 kg and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \text{ m s}^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \text{ m s}^{-1}\) and \(0.5 \text{ m s}^{-1}\) respectively (see diagram).
  1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \text{ m s}^{-1}\). Find the speed of \(L\) after the collision. [4]
  2. \(M\) then collides with \(N\).
    1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
    2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]
OCR M1 2007 January Q5
12 marks Standard +0.3
A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leq t \leq 4\) the acceleration is \(1.8t \text{ m s}^{-2}\), and for \(4 \leq t \leq 7\) the particle has constant acceleration \(7.2 \text{ m s}^{-2}\).
  1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leq t \leq 4\). [3]
  2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\). [4]
  3. Find the displacement of the particle from \(A\) when \(t = 7\). [5]
OCR M1 2007 January Q6
12 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s, coming to rest 8 m above ground level.
  1. Find the greatest speed reached by the hoist during this stage. [2]
The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at \(0.02 \text{ m s}^{-2}\) for the first 40 s of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). Find
  1. the value of \(V\), [3]
  2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
  3. the deceleration of the hoist in the final part of the third stage. [3]
OCR M1 2007 January Q7
15 marks Standard +0.3
A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6.
  1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N, correct to 3 significant figures. [3]
  2. Find the acceleration of \(P\) when it is moving
    1. up the plane,
    2. down the plane.
    [4]
  3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \text{ m s}^{-1}\).
    1. Find the length of time before \(P\) reaches its highest point.
    2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
    [8]
OCR M1 2009 June Q1
6 marks Moderate -0.8
\includegraphics{figure_1} Two perpendicular forces have magnitudes \(x\) N and \(3x\) N (see diagram). Their resultant has magnitude \(6\) N.
  1. Calculate \(x\). [3]
  2. Find the angle the resultant makes with the smaller force. [3]
OCR M1 2009 June Q2
9 marks Moderate -0.8
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]
  2. Calculate the maximum speed of the car during its motion. [3]
  3. Hence, given that the acceleration of the car is \(2.4\) m s\(^{-2}\), calculate its deceleration. [4]
OCR M1 2009 June Q3
9 marks Standard +0.3
\includegraphics{figure_3} 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 \(30°\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2\) m s\(^{-2}\) towards the pulley.
  1. By considering the motion of \(P\), show that the tension in the string is \(3.76\) N. [4]
  2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]
OCR M1 2009 June Q4
9 marks Moderate -0.8
An object is projected vertically upwards with speed \(7\) m s\(^{-1}\). Calculate
  1. the speed of the object when it is \(2.1\) m above the point of projection, [3]
  2. the greatest height above the point of projection reached by the object, [3]
  3. the time after projection when the object is travelling downwards with speed \(5.7\) m s\(^{-1}\). [3]
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]