Questions M2 (1537 questions)

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Edexcel M2 Q7
14 marks Standard +0.3
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
17 marks Standard +0.3
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
4 marks Moderate -0.8
  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 \(4 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    1. Find the speed of \(P\) when \(t = 2\).
    2. 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
    1. the magnitude of the force exerted by the cylinder on the plank at \(P\),
    2. the distance \(A P\). \section*{MECHANICS 2 (A) TEST PAPER 10 Page 2}
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)
OCR M2 2007 June Q1
3 marks Easy -1.2
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
4 marks Moderate -0.8
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
8 marks Standard +0.3
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
8 marks Moderate -0.5
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
8 marks Moderate -0.3
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
9 marks Standard +0.3
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
16 marks Standard +0.8
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
16 marks Standard +0.3
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
3 marks Easy -1.2
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
4 marks Moderate -0.8
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
9 marks Standard +0.3
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
10 marks Moderate -0.5
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
12 marks Standard +0.3
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
13 marks Standard +0.3
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.
OCR M2 2008 June Q8
13 marks Standard +0.3
8
  1. Fig. 1 A uniform lamina \(A B C D\) is in the form of a right-angled trapezium. \(A B = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(A D = 17 \mathrm {~cm}\) (see Fig. 1). Taking \(x\) - and \(y\)-axes along \(A D\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-5_481_1079_991_575} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina is smoothly pivoted at \(A\) and it rests in a vertical plane in equilibrium against a fixed smooth block of height 7 cm . The mass of the lamina is 3 kg . \(A D\) makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the magnitude of the force which the block exerts on the lamina.
OCR M2 2009 June Q1
5 marks Moderate -0.3
1 A boy on a sledge slides down a straight track of length 180 m which descends a vertical distance of 40 m . The combined mass of the boy and the sledge is 75 kg . The initial speed is \(3 \mathrm {~ms} ^ { - 1 }\) and the final speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude, \(R \mathrm {~N}\), of the resistance to motion is constant. By considering the change in energy, calculate \(R\).
OCR M2 2009 June Q2
8 marks Standard +0.3
2 A car of mass 1100 kg has maximum power of 44000 W . The resistive forces have constant magnitude 1400 N .
  1. Calculate the maximum steady speed of the car on the level. The car is moving on a hill of constant inclination \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).
  2. Calculate the maximum steady speed of the car when ascending the hill.
  3. Calculate the acceleration of the car when it is descending the hill at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) working at half the maximum power.
OCR M2 2009 June Q3
10 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-2_497_951_1123_598} A uniform beam \(A B\) has weight 70 N and length 2.8 m . The beam is freely hinged to a wall at \(A\) and is supported in a horizontal position by a strut \(C D\) of length 1.3 m . One end of the strut is attached to the beam at \(C , 0.5 \mathrm {~m}\) from \(A\), and the other end is attached to the wall at \(D\), vertically below \(A\). The strut exerts a force on the beam in the direction \(D C\). The beam carries a load of weight 50 N at its end \(B\) (see diagram).
  1. Calculate the magnitude of the force exerted by the strut on the beam.
  2. Calculate the magnitude of the force acting on the beam at \(A\).
OCR M2 2009 June Q4
11 marks Moderate -0.3
4 A light inextensible string of length 0.6 m has one end fixed to a point \(A\) on a smooth horizontal plane. The other end of the string is attached to a particle \(B\), of mass 0.4 kg , which rotates about \(A\) with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) on the surface of the plane.
  1. Calculate the tension in the string. A particle \(P\) of mass 0.1 kg is attached to the mid-point of the string. The line \(A P B\) is straight and rotation continues at \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Calculate the tension in the section of the string \(A P\).
  3. Calculate the total kinetic energy of the system.
OCR M2 2009 June Q6
13 marks Standard +0.3
6 Two uniform spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.4 kg and the mass of \(B\) is 0.2 kg . The spheres \(A\) and \(B\) are travelling in the same direction in a straight line on a smooth horizontal surface, \(A\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(B\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < 5\). A collides directly with \(B\) and the impulse between them has magnitude 0.9 Ns . Immediately after the collision, the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate \(v\). \(B\) subsequently collides directly with a stationary sphere \(C\) of mass 0.1 kg and the same radius as \(A\) and \(B\). The coefficient of restitution between \(B\) and \(C\) is 0.6 .
  2. Determine whether there will be a further collision between \(A\) and \(B\).
OCR M2 2009 June Q7
14 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-4_440_657_906_744} A ball is projected with an initial speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) below the horizontal from a point on the top of a vertical wall. The point of projection is 8 m above horizontal ground. The ball hits a vertical fence which is at a horizontal distance of 9 m from the wall (see diagram).
  1. Calculate the height above the ground of the point where the ball hits the fence.
  2. Calculate the direction of motion of the ball immediately before it hits the fence.
  3. It is given that \(30 \%\) of the kinetic energy of the ball is lost when it hits the fence. Calculate the speed of the ball immediately after it hits the fence.