Questions — Edexcel M2 (551 questions)

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Edexcel M2 Q1
  1. A snooker ball \(A\) is moving on a horizontal table with velocity \(( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
It collides with another ball \(B\), whose mass is twice the mass of \(A\).
After the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Find the velocity of \(B\) before the collision.
Edexcel M2 Q2
2. Charlotte, whose mass is 55 kg , is running up a straight hill inclined at \(6 ^ { \circ }\) to the horizontal. She passes two points \(P\) and \(Q , 80\) metres apart, with speeds \(2 \cdot 5 \mathrm {~ms} ^ { - 1 }\) and \(1 \cdot 5 \mathrm {~ms} ^ { - 1 }\) respectively.
Calculate, in J to the nearest whole number, the total work done by Charlotte as she runs from \(P\) to \(Q\).
Edexcel M2 Q3
3. A particle \(P\) moves in a horizontal plane such that, at time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - t ^ { \frac { 1 } { 2 } } \mathbf { j }\). When \(t = 0 , P\) is at the point with position vector \(- 10 \mathbf { i } + \mathbf { j }\) relative to a fixed origin \(O\).
  1. Find the position vector \(\mathbf { r }\) of \(P\) at time \(t\) seconds.
  2. Find the distance \(O P\) when \(t = 4\).
Edexcel M2 Q4
4. A small stone, of mass 600 grams, is released from rest a height of 2 metres above ground level and falls under gravity. The time it takes to reach the ground is \(T\) seconds. The stone is then again released from rest at the surface of a tank containing a 2 metre depth of liquid and reaches the bottom after \(2 T\) seconds. It may be assumed that the resisting force acting on the stone is constant.
  1. Find the magnitude of the resisting force exerted on the stone by the liquid.
  2. Find the speed with which the stone hits the bottom of the tank.
Edexcel M2 Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{9e1d8a2f-0c35-4398-98ff-083ec76653ec-1_367_529_2122_383} A sign-board consists of a rectangular sheet of metal, of mass \(M\), which is 3 metres wide and 1 metre high, attached to two thin metal supports, each of mass \(m\) and length 2 metres. The board stands on horizontal ground.
  1. Calculate the height above the ground of the centre of mass of the sign-board, in terms of \(M\) and \(m\). Given now that the centre of mass of the sign-board is \(2 \cdot 2\) metres above the ground, (b) find the ratio \(M : m\), in its simplest form. \section*{MECHANICS 2 (A) TEST PAPER 9 Page 2}
Edexcel M2 Q6
  1. A ball is hit with initial speed \(u \mathrm {~ms} ^ { - 1 }\), at an angle \(\theta\) above the horizontal, from a point at a height of \(h \mathrm {~m}\) above horizontal ground. The ball, which is modelled as a particle moving freely under gravity, hits the ground at a horizontal distance \(d \mathrm {~m}\) from the point of projection.
    1. Prove that \(\frac { g d ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \theta - d \tan \theta - h = 0\).
    Given further that \(u = 14 , h = 7\) and \(d = 14\), and assuming the result \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\),
  2. find the value of \(\theta\).
Edexcel M2 Q7
7. A cyclist is pedalling along a horizontal cycle track at a constant speed of \(5 \mathrm {~ms} ^ { - 1 }\). The air resistance opposing her motion has magnitude 42 N . The combined mass of the cyclist and her machine is 84 kg .
  1. Find the rate, in W , at which the cyclist is working. The cyclist now starts to ascend a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\), at a constant speed.
    She continues to work at the same rate as before, against the same air resistance.
  2. Find the constant speed at which she ascends the hill. In fact the air resistance is not constant, and a revised model takes account of this by assuming that the air resistance is proportional to the cyclist's speed.
  3. Use this model to find an improved estimate of the speed at which she ascends the hill, if her rate of working still remains constant.
Edexcel M2 Q8
8. Two ships \(A\) and \(B\), of masses \(m\) and km respectively, are moving towards each other in heavy fog along the same straight line, both with speed \(u\). The ships collide and immediately after the collision they drift away from each other, both their directions of motion having been reversed. The speed of \(A\) after the impact is \(\frac { 1 } { 5 } u\) and the speed of \(B\) after the impact is \(v\).
  1. Show that \(v = u \left( \frac { 6 } { 5 k } - 1 \right)\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
  2. Show that \(v = u \left( 2 e - \frac { 1 } { 5 } \right)\).
  3. Use your answers to parts (a) and (b) to find the rational numbers \(p\) and \(q\) such that \(p \leq k < q\).
    (9 marks)
Edexcel M2 Q1
  1. Particles of mass \(2 m , 3 m\) and \(5 m\) are placed at the points in the \(x - y\) plane with coordinates \(( - 1,5 ) , ( 0,6 )\) and \(( 3 , - 2 )\) respectively.
    Find the coordinates of the centre of mass of this system of particles.
  2. A lorry of mass 3800 kg is pulling a trailer of mass 1200 kg along a straight horizontal road. At a particular moment, the lorry and trailer are moving at a speed of \(10 \mathrm {~ms} ^ { - 1 }\) and accelerating at \(0.8 \mathrm {~ms} ^ { - 2 }\). The resistances to the motion of the lorry and the trailer are constant and of magnitude 1600 N and 600 N respectively.
    Find the rate, in kW , at which the engine of the lorry is working.
  3. A bullet of mass 0.05 kg is fired with speed \(u \mathrm {~ms} ^ { - 1 }\) from a gun, which recoils at a speed of \(0.008 u \mathrm {~ms} ^ { - 1 }\) in the opposite direction to that in which the bullet is fired.
    1. Find the mass of the gun.
    2. Find, in terms of \(u\), the kinetic energy given to the bullet and to the gun at the instant of firing.
    3. If the total kinetic energy created in firing the gun is 5100 J , find the value of \(u\).
    4. The acceleration of a particle \(P\) at time \(t \mathrm {~s}\) is \(\mathbf { a } \mathrm { ms } ^ { - 2 }\), where \(\mathbf { a } = 4 \mathrm { e } ^ { t } \mathbf { i } - \mathrm { e } ^ { t } \mathbf { j }\). When \(t = 0 , P\) has velocity \(4 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    5. Find the speed of \(P\) when \(t = 2\).
    6. Find the time at which the direction of motion of \(P\) is parallel to the vector \(5 \mathbf { i } - \mathbf { j }\).
    \includegraphics[max width=\textwidth, alt={}]{63133ab4-9381-4777-a575-1207219948b7-1_323_383_1992_429}
    A uniform plank \(A B\), of mass 3 kg and length 2 m , rests in equilibrium with the point \(P\) in contact with a smooth cylinder. The end \(B\) rests on a rough horizontal surface and the coefficient of friction between the plank and the surface is \(\frac { 1 } { 3 } . A B\) makes an angle of \(60 ^ { \circ }\) with the horizontal.
    If the plank is in limiting equilibrium in this position, find
  4. the magnitude of the force exerted by the cylinder on the plank at \(P\),
  5. the distance \(A P\). \section*{MECHANICS 2 (A) TEST PAPER 10 Page 2}
Edexcel M2 Q6
  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
    4. the velocity of \(P 6\) seconds after it passes \(O\),
    5. the magnitude of the acceleration of \(P\) when \(t = 1\),
    6. the minimum speed of \(P\),
    7. the distance travelled by \(P\) in the fourth second after it passes \(O\).
    8. 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}
    9. the range of times after firing during which the bullet is 15 m or more above ground level,
    10. the greatest height above the ground reached by the bullet,
    11. 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
  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
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
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
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
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
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
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
  1. 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
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
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
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
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
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
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
  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 \(\mathbf { F }\).