Questions — Edexcel (10514 questions)

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Edexcel M2 Q6
11 marks Moderate -0.3
  1. Two smooth spheres \(A\) and \(B\) have equal radii and masses 0.4 kg and 0.8 kg respectively. They are moving in opposite directions along the same straight line, with speeds \(3 \mathrm {~ms} ^ { - 1 }\) and 2 \(\mathrm { ms } ^ { - 1 }\) respectively, and collide directly. The coefficient of restitution between \(A\) and \(B\) is 0.8 .
    1. Calculate the speeds of \(A\) and \(B\) after the impact, stating in each case whether the direction of motion has been reversed.
    2. Find the kinetic energy, in J, lost in the impact.
    3. A point of light, \(P\), is moving along a straight line in such a way that, \(t\) seconds after passing through a fixed point \(O\) on the line, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = \frac { 1 } { 2 } t ^ { 2 } - 4 t + 10\). Calculate
    1. the velocity of \(P 6\) seconds after it passes \(O\),
    2. the magnitude of the acceleration of \(P\) when \(t = 1\),
    3. the minimum speed of \(P\),
    4. the distance travelled by \(P\) in the fourth second after it passes \(O\).
    5. A bullet is fired out of a window at a height of 5.2 m above horizontal ground. The initial velocity of the bullet is \(392 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the vertical, where \(\sin \alpha = \frac { 1 } { 20 }\), as shown.
      Find \includegraphics[max width=\textwidth, alt={}, center]{63133ab4-9381-4777-a575-1207219948b7-2_335_490_1343_1419}
    1. the range of times after firing during which the bullet is 15 m or more above ground level,
    2. the greatest height above the ground reached by the bullet,
    3. the horizontal distance travelled by the bullet before it reaches its highest point.
    Certain modelling assumptions have been made about the bullet.
  2. State these assumptions and suggest a way in which the model could be refined.
  3. State, with a reason, whether you think this refinement would make a significant difference to the answers.
    (2 marks)
Edexcel M2 Q1
6 marks Moderate -0.3
  1. Two identical particles are approaching each other along a straight horizontal track. Just before they collide, they are moving with speeds \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between the particles is \(\frac { 1 } { 2 }\).
Find the speeds of the particles immediately after the impact.
Edexcel M2 Q2
8 marks Standard +0.3
2. A particle \(P\) of mass 3 kg moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin, \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - 3 t \right) \mathbf { i } + \frac { 1 } { 6 } t ^ { 3 } \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
  1. Find the velocity of \(P\) when \(t = 0\).
  2. Find the kinetic energy lost by \(P\) in the interval \(0 \leq t \leq 2\).
Edexcel M2 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-2_424_360_1155_648} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform ladder of mass 15 kg and length 8 m which rests against a smooth vertical wall at \(A\) with its lower end on rough horizontal ground at \(B\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\) and the ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 2\). A man of mass 75 kg ascends the ladder until he reaches a point \(P\). The ladder is then on the point of slipping.
  1. Write down suitable models for
    1. the ladder,
    2. the man.
  2. Find the distance \(A P\).
Edexcel M2 Q4
11 marks Standard +0.3
4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),
  1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
  2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)
Edexcel M2 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-3_591_609_785_623} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plane lamina \(A B C D E G\) in the shape of a letter ' \(L\) ' consisting of a rectangle \(A B F G\) joined to another rectangle \(C D E F\). The sides \(A B\) and \(D E\) are both 8 cm long and the sides \(E G\) and \(G A\) are of length 24 cm and 32 cm respectively.
  1. Show that the centre of mass of the lamina lies on the line \(B F\).
  2. Find the distance of the centre of mass from the line \(A B\). The uniform lamina in Figure 2 is a model of the letter ' \(L\) ' in a sign above a shop. The letter is normally suspended from a wall at \(A\) and \(B\) so that \(A B\) is horizontal but the fixing at \(B\) has broken and the letter hangs in equilibrium from the point \(A\).
  3. Find, in degrees to one decimal place, the acute angle \(A G\) makes with the vertical.
Edexcel M2 Q6
13 marks Standard +0.3
6. The engine of a car of mass 1200 kg is working at a constant rate of 90 kW as the car moves along a straight horizontal road. The resistive forces opposing the motion of the car are constant and of magnitude 1800 N .
  1. Find the acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find, in kJ, the kinetic energy of the car when it is travelling at maximum speed. The car ascends a hill which is straight and makes an angle \(\alpha\) with the horizontal. The power output of the engine and the non-gravitational forces opposing the motion remain the same. Given that the car can attain a maximum speed of \(25 \mathrm {~ms} ^ { - 1 }\),
  3. find, in degrees correct to one decimal place, the value of \(\alpha\).
    (5 marks)
Edexcel M2 Q7
15 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-4_236_942_1101_479} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows the path of a golf ball which is hit from the point \(O\) with speed \(49 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the ball is in a vertical plane containing \(O\) and the hole at which the ball is aimed. The hole is 170 m from \(O\) and on the same horizontal level as \(O\).
  1. Suggest a suitable model for the motion of the golf ball. Find, correct to 3 significant figures,
  2. the distance beyond the hole at which the ball hits the ground,
  3. the magnitude and direction of the velocity of the ball when it is directly above the hole.
Edexcel M2 Q1
7 marks Moderate -0.3
A bullet of mass 25 g is fired directly at a fixed wooden block of thickness 4 cm and passes through it. When the bullet hits the block, it is travelling horizontally at \(200 \mathrm {~ms} ^ { - 1 }\). The block exerts a constant resistive force of 8000 N on the bullet.
  1. Find the work done by the block on the bullet. By using the Work-Energy principle,
  2. show that the bullet emerges from the block with speed \(120 \mathrm {~ms} ^ { - 1 }\).
Edexcel M2 Q2
7 marks Standard +0.3
2. A car is travelling along a straight horizontal road against resistances to motion which are constant and total 2000 N . When the engine of the car is working at a rate of \(H\) kilowatts, the maximum speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(H\). The car driver wishes to overtake another vehicle so she increases the rate of working of the engine by \(20 \%\) and this results in an initial acceleration of \(0.32 \mathrm {~ms} ^ { - 2 }\). Assuming that the resistances to motion remain constant,
  2. find the mass of the car.
    (4 marks)
Edexcel M2 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{086ace58-0aa9-4f36-95c3-5698d14f511e-2_369_684_1356_555} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) placed with edge \(B C\) along the line of greatest slope of a plane inclined at an angle \(\theta\) to the horizontal. The lengths \(A C\) and \(B C\) are 15 cm and 9 cm respectively and \(\angle A B C\) is a right angle.
  1. Find the distance of the centre of mass of the lamina from
    1. \(\quad A B\),
    2. \(B C\). Assuming that the plane is rough enough to prevent the lamina from slipping,
  2. find in degrees, correct to 1 decimal place, the maximum value of \(\theta\) for which the lamina remains in equilibrium.
    (4 marks)
Edexcel M2 Q4
10 marks Moderate -0.3
4. The velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds is given by \(\mathbf { v } = 3 t \mathbf { i } - t ^ { 2 } \mathbf { j }\).
  1. Find the magnitude of the acceleration of \(P\) when \(t = 2\). When \(t = 0\), the displacement of \(P\) from a fixed origin \(O\) is \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
  2. Show that the displacement of \(P\) from \(O\) when \(t = 6\) is given by \(k ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(k\) is an integer which you should find.
    (6 marks)
Edexcel M2 Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{086ace58-0aa9-4f36-95c3-5698d14f511e-3_417_851_778_614} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform rod \(A B\) of length \(2 a\) and mass 8 kg is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium inclined at an angle of \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(F\) newtons acting horizontally at \(B\) which is below the level of \(A\) as shown in Figure 2.
  1. Find, correct to 3 significant figures, the value of \(F\).
  2. Show that the magnitude of the reaction at the hinge is 133 N , correct to 3 significant figures, and find to the nearest degree the acute angle which the reaction makes with the vertical.
Edexcel M2 Q6
15 marks Standard +0.3
6. A particle \(P\) is projected from a point \(A\) on horizontal ground with speed \(u\) at an angle of elevation \(\alpha\) and moves freely under gravity. \(P\) hits the ground at the point \(B\).
  1. Show that \(A B = \frac { u ^ { 2 } } { g } \sin 2 \alpha\). An archer fires an arrow with an initial speed of \(45 \mathrm {~ms} ^ { - 1 }\) at a target which is level with the point of projection and at a distance of 80 m . Given that the arrow hits the target,
  2. find in degrees, correct to 1 decimal place, the two possible angles of projection.
  3. Write down, with a reason, which of the two possible angles of projection would give the shortest time of flight.
    (2 marks)
  4. Show that the minimum time of flight is 1.8 seconds, correct to 1 decimal place.
    (2 marks)
Edexcel M2 Q7
16 marks Standard +0.8
7. A smooth sphere \(A\) of mass \(4 m\) is moving on a smooth horizontal plane with speed \(u\). It collides directly with a stationary smooth sphere \(B\) of mass \(5 m\) and with the same radius as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).
  1. Show that after the collision the speed of \(B\) is 4 times greater than the speed of \(A\).
    (7 marks)
    Sphere \(B\) subsequently hits a smooth vertical wall at right angles. After rebounding from the wall, \(B\) collides with \(A\) again and as a result of this collision, \(B\) comes to rest. Given that the coefficient of restitution between \(B\) and the wall is \(e\),
  2. find \(e\). END
Edexcel M2 Q1
7 marks Moderate -0.5
  1. A particle \(P\) of mass 2 kg is subjected to a force \(\mathbf { F }\) such that its displacement, \(\mathbf { r }\) metres, from a fixed origin, \(O\), at time \(t\) seconds is given by
$$\mathbf { r } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + \left( 3 - 4 t ^ { 2 } \right) \mathbf { j }$$
  1. Show that the acceleration of \(P\) is constant.
  2. Find the magnitude of \(\mathbf { F }\).
Edexcel M2 Q2
8 marks Standard +0.3
2. A pump raises water from a well 12 metres below the ground and ejects the water through a pipe of diameter 10 cm at a speed of \(6 \mathrm {~ms} ^ { - 1 }\). Given that the mass of \(1 \mathrm {~m} ^ { 3 }\) of water is 1000 kg ,
  1. find, in terms of \(\pi\), the mass of water discharged by the pipe every second,
  2. find in kJ , correct to 3 significant figures, the total mechanical energy gained by the water per second.
Edexcel M2 Q3
9 marks Moderate -0.3
3. A particle moves in a straight horizontal line such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by \(v = 2 t ^ { 2 } - 9 t + 4\). Initially, the particle has displacement 9 m from a fixed point \(O\) on the line.
  1. Find the initial velocity of the particle.
  2. Show that the particle is at rest when \(t = 4\) and find the other value of \(t\) when it is at rest.
  3. Find the displacement of the particle from \(O\) when \(t = 6\).
Edexcel M2 Q4
9 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f8ece90a-5042-4db1-9855-ffe74333a899-3_407_341_201_635} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform ladder of mass \(m\) and length \(2 a\) resting against a rough vertical wall with its lower end on rough horizontal ground. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 2 }\) and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\). Given that the ladder is in limiting equilibrium when it is inclined at an angle \(\theta\) to the horizontal, show that \(\tan \theta = \frac { 5 } { 4 }\).
(9 marks)
Edexcel M2 Q5
13 marks Moderate -0.3
5. A firework company is testing its new brand of firework, the Sputnik Special. One of the company's employees lights a Sputnik Special on a large area of horizontal ground and it takes off at a small angle to the vertical. After a flight lasting 8 seconds it lands at a distance of 24 metres from the point where it was launched. The employee models the firework as a particle and ignores air resistance and any loss of mass which the Sputnik Special experiences. Using this model, find for this flight of the Sputnik Special,
  1. the horizontal and vertical components of the initial velocity,
  2. the initial speed, correct to 3 significant figures,
  3. the maximum height attained.
  4. Comment on the suitability of the modelling assumptions made by the employee.
Edexcel M2 Q6
14 marks Standard +0.3
6. Three uniform spheres \(A , B\) and \(C\) of equal radius have masses \(3 m , 2 m\) and \(2 m\) respectively. Initially, the spheres are at rest on a smooth horizontal table with their centres in a straight line and with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\),
  1. show that the speeds of \(A\) and \(B\) after the collision are \(\frac { 1 } { 3 } u\) and \(u\) respectively.
    (6 marks)
    The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that \(A\) and \(B\) collide again,
  2. show that \(e > \frac { 1 } { 3 }\).
    (8 marks)
Edexcel M2 Q7
15 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f8ece90a-5042-4db1-9855-ffe74333a899-4_542_625_959_589} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform lamina \(A B C D\) formed by removing an isosceles triangle \(B C D\) from an equilateral triangle \(A B D\) of side \(2 d\). The point \(C\) is the centroid of triangle \(A B D\).
  1. Find the area of triangle \(B C D\) in terms of \(d\).
  2. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 4 } { 9 } \sqrt { 3 } d\).
    (8 marks)
    The lamina is freely suspended from the point \(B\) and hangs at rest.
  3. Find in degrees, correct to 1 decimal place, the acute angle that the side \(A B\) makes with the vertical.
Edexcel M2 Q1
6 marks Moderate -0.3
  1. A particle \(P\) moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin \(O\) is given by
$$\mathbf { r } = \left( \frac { 3 } { 2 } t ^ { 2 } - 3 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - k t \right) \mathbf { j } ,$$ where \(k\) is a constant and \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
  1. Find an expression for the velocity of \(P\) at time \(t\).
  2. Given that \(P\) comes to rest instantaneously, find the value of \(k\).
Edexcel M2 Q2
6 marks Standard +0.3
2. Two smooth spheres \(P\) and \(Q\) of equal radius and of mass \(2 m\) and \(5 m\) respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, \(P\) and \(Q\) travel in opposite directions with speeds of \(3 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. Given that the coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\), find the speeds of \(P\) and \(Q\) before the collision.
(6 marks)
Edexcel M2 Q3
10 marks Standard +0.3
3. A car of mass 1200 kg experiences a resistance to motion, \(R\) newtons, which is proportional to its speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the power output of the car engine is 90 kW and the car is travelling along a horizontal road, its maximum speed is \(50 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(R = 36 v\). The car ascends a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 14 }\).
  2. Find, correct to 3 significant figures, the maximum speed of the car up the hill assuming that the power output of the engine is unchanged.
    (6 marks)