Questions — Edexcel M2 (623 questions)

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Edexcel M2 Q8
15 marks Standard +0.8
A golf ball is hit with initial velocity \(u\) ms\(^{-1}\) at an angle of \(45°\) above the horizontal. The ball passes over a building which is \(15\) m tall at a distance of \(30\) m horizontally from the point where the ball was hit.
  1. Find the smallest possible value of \(u\). [7 marks]
When \(u\) has this minimum value,
  1. show that the ball does not rise higher than the top of the building. [4 marks]
  2. Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
  3. Briefly describe two modelling assumptions that you have made. [2 marks]
Edexcel M2 Q1
7 marks Moderate -0.3
The acceleration of a particle \(P\) is \((8t - 18)\) ms\(^{-2}\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving. At time \(t = 3\), \(P\) has speed \(2\) ms\(^{-1}\). Find
  1. the velocity of \(P\) at time \(t\), [4 marks]
  2. the values of \(t\) when \(P\) is instantaneously at rest. [3 marks]
Edexcel M2 Q2
7 marks Moderate -0.3
A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed 12 ms\(^{-1}\) through a pipe of radius 4 cm. Find
  1. the potential and kinetic energy given to the water each second, [5 marks]
  2. the rate, in kW, at which the pump is working. [2 marks]
[1 m\(^3\) of water has a mass of 1000 kg.]
Edexcel M2 Q3
7 marks Moderate -0.3
A particle \(P\) of mass 3 kg has position vector \(\mathbf{r} = (2t^2 - 4t)\mathbf{i} + (1 - t^2)\mathbf{j}\) m at time \(t\) seconds.
  1. Find the velocity vector of \(P\) when \(t = 3\). [3 marks]
  2. Find the magnitude of the force acting on \(P\), showing that this force is constant. [4 marks]
Edexcel M2 Q4
9 marks Standard +0.3
\includegraphics{figure_4} The diagram shows a uniform lamina \(ABCDE\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm.
  1. Find the distance of the centre of mass of the lamina from \(ED\). [4 marks]
The lamina has mass \(m\). A particle \(P\) is attached to the lamina at \(B\). The lamina is then suspended freely from \(A\) and hangs in equilibrium with \(AD\) vertical.
  1. Find, in terms of \(m\), the mass of \(P\). [5 marks]
Edexcel M2 Q5
14 marks Standard +0.3
A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.
  1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
  2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
  3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]
When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,
  1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
  2. Show that tension in the tow-bar remains unchanged. [3 marks]
Edexcel M2 Q6
15 marks Standard +0.3
Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.
  1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
  2. Find the time for which the ball is in flight. [4 marks]
  3. Find the horizontal distance travelled by the ball. [3 marks]
  4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
  5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]
Edexcel M2 Q7
16 marks Standard +0.8
Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3m\) and \(4m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.
  1. Show that \(B\) begins to move with speed \(\frac{3}{7}u(1 + e)\). [6 marks]
  2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). [3 marks]
Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(kmu\) Ns, where \(k > 2.25\). \(A\) then collides again with \(B\).
  1. Show that the speed of \(A\) after this second impact is independent of \(k\). [7 marks]
Edexcel M2 Q1
5 marks Moderate -0.8
A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]
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]