Questions M2 (1391 questions)

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Edexcel M2 Q8
8. An aeroplane, travelling horizontally at a speed of \(55 \mathrm {~ms} ^ { - 1 }\) 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,
  2. the horizontal distance moved by the packet during this time. The packet will split open if it hits the ground at a speed in excess of \(125 \mathrm {~ms} ^ { - 1 }\).
  3. Determine, with explanation, whether the packet will split open.
  4. 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. One of the leaflets is stuck to the front of the packet and becomes detached as it leaves the aeroplane.
  5. If the leaflet is modelled as a particle, state how long it takes to reach the ground.
  6. Comment on the model of the leaflet as a particle.
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)
OCR M2 2007 June Q1
1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of \(35 ^ { \circ }\) with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m .
OCR M2 2007 June Q2
2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(27 ^ { \circ }\).
OCR M2 2007 June Q3
3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a time \(t\) seconds.
  1. Calculate the value of \(t\).
  2. Calculate the acceleration of the rocket at the instant when its speed is \(120 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR M2 2007 June Q4
4 A ball is projected from a point \(O\) on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. At time \(t\) seconds after projection the ball is at the point \(( x , y )\) referred to horizontal and vertically upward axes through \(O\). Air resistance may be neglected.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence show that \(y = 3 x - \frac { 1 } { 10 } x ^ { 2 }\). The ball hits the sea at a point which is 25 m below the level of \(O\).
  2. Find the horizontal distance between the cliff and the point where the ball hits the sea.
OCR M2 2007 June Q5
5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find
  1. the increase in kinetic energy,
  2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
  3. Calculate the distance \(A B\).
OCR M2 2007 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-3_670_613_274_767} A particle \(P\) of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point \(A\) and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.4 m long. \(P B\) makes an angle of \(60 ^ { \circ }\) with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string \(A P\) is 5 N . Calculate
  1. the tension in the string \(P B\),
  2. the angular speed of \(P\),
  3. the kinetic energy of \(P\).
OCR M2 2007 June Q7
7 Two small spheres \(A\) and \(B\), with masses 0.3 kg and \(m \mathrm {~kg}\) respectively, lie at rest on a smooth horizontal surface. \(A\) is projected directly towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and hits \(B\). The direction of motion of \(A\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(1 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that \(m = 0.7\).
  2. Find \(e\). B continues to move at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The coefficient of restitution between \(B\) and the wall is \(f\).
  3. Find the range of values of \(f\) for which there will be a second collision between \(A\) and \(B\).
  4. Find, in terms of \(f\), the magnitude of the impulse that the wall exerts on \(B\).
  5. Given that \(f = \frac { 3 } { 4 }\), calculate the final speeds of \(A\) and \(B\), correct to 1 decimal place.
OCR M2 2007 June Q8
8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_451_481_274_833} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height \(h \mathrm {~m}\). The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point \(O\) (see Fig. 1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from \(O\).
  1. Show that \(h = 1.2\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_620_1065_1297_541} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). The tension in the string is \(T \mathrm {~N}\) and the frictional force acting on the object is \(F \mathrm {~N}\).
  2. By taking moments about \(O\), express \(F\) in terms of \(T\).
  3. Find another equation connecting \(T\) and \(F\). Hence calculate the tension and the frictional force.
OCR M2 2008 June Q1
1 A car is pulled at constant speed along a horizontal straight road by a force of 200 N inclined at \(35 ^ { \circ }\) to the horizontal. Given that the work done by the force is 5000 J , calculate the distance moved by the car.
OCR M2 2008 June Q2
2 A bullet of mass 9 grams passes horizontally through a fixed vertical board of thickness 3 cm . The speed of the bullet is reduced from \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it passes through the board. The board exerts a constant resistive force on the bullet. Calculate the magnitude of this resistive force.
OCR M2 2008 June Q3
3 The resistance to the motion of a car of mass 600 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The car ascends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 10 }\). The power exerted by the car's engine is 12000 W and the car has constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 0.6\). The power exerted by the car's engine is increased to 16000 W .
  2. Calculate the maximum speed of the car while ascending the hill. The car now travels on horizontal ground and the power remains 16000 W .
  3. Calculate the acceleration of the car at an instant when its speed is \(32 \mathrm {~ms} ^ { - 1 }\).
OCR M2 2008 June Q4
4 A golfer hits a ball from a point \(O\) on horizontal ground with a velocity of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The horizontal range of the ball is \(R\) metres and the time of flight is \(t\) seconds.
  1. Express \(t\) in terms of \(\theta\), and hence show that \(R = 125 \sin 2 \theta\). The golfer hits the ball so that it lands 110 m from \(O\).
  2. Calculate the two possible values of \(t\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_672_403_267_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A toy is constructed by attaching a small ball of mass 0.01 kg to one end of a uniform rod of length 10 cm whose other end is attached to the centre of the plane face of a uniform solid hemisphere with radius 3 cm . The rod has mass 0.02 kg , the hemisphere has mass 0.5 kg and the rod is perpendicular to the plane face of the hemisphere (see Fig. 1).
OCR M2 2008 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_794_735_264_705} A particle \(P\) of mass 0.5 kg is attached to points \(A\) and \(B\) on a fixed vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical (see diagram). The particle moves with constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.4 m .
  1. Calculate the tensions in the two strings. The particle now moves with constant angular speed \(\omega\) rad s \(^ { - 1 }\) and the string \(B P\) is on the point of becoming slack.
  2. Calculate \(\omega\).
OCR M2 2008 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).
  1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
  2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.