Questions M2 (1537 questions)

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Edexcel M2 Q2
7 marks Standard +0.3
\includegraphics{figure_2} A key is modelled as a lamina which consists of a circle of radius 3 cm, with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm, with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(AB\). [7 marks]
Edexcel M2 Q3
7 marks Standard +0.3
A van of mass 1600 kg is moving with constant speed down a straight road inclined at 7° to the horizontal. The non-gravitational resistance to the van's motion has a constant magnitude of 2000 N and the engine of the van is working at a rate of 1.5 kW. Find
  1. the constant speed of the van, [5 marks]
  2. the acceleration of the van if the resistance is suddenly reduced to 1900 N. [2 marks]
Edexcel M2 Q4
10 marks Standard +0.3
\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.
  1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
  2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]
Edexcel M2 Q5
13 marks Standard +0.3
Two railway trucks \(A\) and \(B\), whose masses are \(6m\) and \(5m\) respectively, are moving in the same direction along a straight track with speeds \(5u\) and \(3u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(kv\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\). Modelling the trucks as particles,
  1. show that
    1. \(v = \frac{45u}{5k + 6}\),
    2. \(v = \frac{2eu}{k - 1}\).
    [8 marks]
  2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). [5 marks]
Edexcel M2 Q6
16 marks Standard +0.3
A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after 3.3 seconds.
  1. Calculate the height above ground from which the lead was dropped. [2 marks]
According to a simple model, the ball hits the ground at the same time as the lead.
  1. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. [2 marks]
The piece of lead is now thrown again from \(P\), with speed 7 ms\(^{-1}\) at an angle of 30° to the horizontal, as shown. \includegraphics{figure_6}
  1. Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown. [4 marks]
  2. Deduce that \(y = \frac{\sqrt{3}}{3}x - \frac{2}{15}x^2\). [3 marks]
  3. Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\). [5 marks]
Edexcel M2 Q7
17 marks Standard +0.8
\includegraphics{figure_7} A uniform ladder \(AB\), of mass \(m\) kg and length \(2a\) m, rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{3}{4}\).
  1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac{\sqrt{13}}{3}mg\). [7 marks]
  2. Find, to the nearest degree, the direction of this resultant force at \(B\). [3 marks]
The peg will break when the horizontal force acting on it exceeds \(2mg\) N. A painter of mass \(6m\) kg starts to climb the ladder from \(B\).
  1. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb. [7 marks]
Edexcel M2 Q1
5 marks Moderate -0.8
A heavy ball, of mass 2 kg, rolls along a horizontal surface. It strikes a vertical wall at a speed of 4 ms\(^{-1}\) and rebounds. The coefficient of restitution between the ball and the wall is 0.4. Find the kinetic energy lost in the impact. [5 marks]
Edexcel M2 Q2
7 marks Moderate -0.8
The velocity, \(v\) ms\(^{-1}\), of a particle at time \(t\) s is given by \(v = 4t^2 - 9\).
  1. Find the acceleration of the particle when it is instantaneously at rest. [3 marks]
  2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. [4 marks]
Edexcel M2 Q3
7 marks Moderate -0.8
A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf{r} = t^2\mathbf{i} - 2t\mathbf{j}\).
  1. Find the velocity vector of \(P\) at time \(t\) seconds. [2 marks]
  2. Show that the direction of the acceleration of \(P\) is constant. [2 marks]
  3. Find the value of \(t\) when the acceleration of \(P\) has magnitude 12 ms\(^{-2}\). [3 marks]
Edexcel M2 Q4
8 marks Standard +0.8
A uniform plank of wood \(XY\), of mass 1.4 kg, rests with its upper end \(X\) against a rough vertical wall and its lower end \(Y\) on rough horizontal ground. The coefficient of friction between the plank and both the wall and the ground is \(\mu\). The plank is in limiting equilibrium at both ends and the vertical component of the force exerted on the plank by the ground has magnitude 12 N. Find the value of \(\mu\), to 2 decimal places. [8 marks]
Edexcel M2 Q5
9 marks Standard +0.3
A motor-cycle and its rider have a total mass of 460 kg. The maximum rate at which the cycle's engine can work is 25 920 W and the maximum speed of the cycle on a horizontal road is 36 ms\(^{-1}\). A variable resisting force acts on the cycle and has magnitude \(kv^2\), where \(v\) is the speed of the cycle in ms\(^{-1}\).
  1. Show that \(k = \frac{5}{8}\). [4 marks]
  2. Find the acceleration of the cycle when it is moving at 25 ms\(^{-1}\) on the horizontal road, with its engine working at full power. [5 marks]
Edexcel M2 Q6
9 marks Standard +0.3
\(PQR\) is a triangular lamina with \(PQ = 18\) cm, \(QR = 24\) cm and \(PR = 30\) cm.
  1. Verify that angle \(PQR\) is a right angle and find the distances of the centre of mass of the lamina from
    1. \(PQ\),
    2. \(QR\).
    [5 marks]
\includegraphics{figure_6} The lamina is held in a vertical plane and placed on a line of greatest slope of a rough plane inclined at an angle \(\theta\) to the horizontal, as shown.
  1. Find the largest value of \(\theta\) for which equilibrium will not be broken by toppling. [4 marks]
Edexcel M2 Q7
14 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9m\) and \(4m\) respectively, are moving towards each other along a straight line with speeds 4 ms\(^{-1}\) and 6 ms\(^{-1}\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.
  1. Show that the speed of \(B\) after the impact cannot be more than 3 ms\(^{-1}\). [5 marks]
The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that \(e < \frac{3}{10}\). [5 marks]
  2. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\). [4 marks]
Edexcel M2 Q8
16 marks Standard +0.3
An aeroplane, travelling horizontally at a speed of 55 ms\(^{-1}\) at a height of 600 metres above horizontal ground, drops a sealed packet of leaflets. Find
  1. the time taken by the packet to reach the ground, [3 marks]
  2. the horizontal distance moved by the packet during this time. [2 marks]
The packet will split open if it hits the ground at a speed in excess of 125 ms\(^{-1}\).
  1. Determine, with explanation, whether the packet will split open. [5 marks]
  2. Find the lowest speed at which the aeroplane could be travelling, at the same height of 600 m, to ensure that the packet will split open when it hits the ground. [3 marks]
One of the leaflets is stuck to the front of the packet and becomes detached as it leaves the aeroplane.
  1. If the leaflet is modelled as a particle, state how long it takes to reach the ground. [1 mark]
  2. Comment on the model of the leaflet as a particle. [2 marks]
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]
OCR M2 2013 January Q1
5 marks Easy -1.2
A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find
  1. the work done by the force, [3]
  2. the power with which the force is working. [2]
OCR M2 2013 January Q2
7 marks Standard +0.3
A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(kv^{\frac{3}{2}}\) N, where \(v\) ms\(^{-1}\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W, the car has speed 15 ms\(^{-1}\) and is accelerating at 0.4 ms\(^{-2}\).
  1. Find the value of \(k\). [4]
It is given that the greatest steady speed of the car on this road is 30 ms\(^{-1}\).
  1. Find the greatest power that the engine can produce. [3]
OCR M2 2013 January Q3
9 marks Standard +0.3
A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find
  1. the speed of \(A\) immediately after the collision, [4]
  2. the distance \(A\) moves up the plane after the collision. [2]
The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.
  1. Find the value of \(m\). [3]