Questions — OCR M3 (130 questions)

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OCR M3 2009 June Q1
1 A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The speed of the sphere immediately after it bounces is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and hence find the coefficient of restitution between the surface and the sphere.
  2. State the direction of the impulse on the sphere and find its magnitude.
OCR M3 2009 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_460_725_731_708} Two uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is 1.5 m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is 3 m (see diagram). The weight of \(A B\) is 80 N and the frictional force acting on \(A B\) at \(A\) is 14 N .
  1. Write down the horizontal component of the force acting on \(A B\) at \(B\) and show that the vertical component of this force is 33 N upwards.
  2. Given that the force acting on \(B C\) at \(C\) has magnitude 50 N , find the weight of \(B C\).
    \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_421_949_1793_598} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~ms} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
  3. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
  4. Find the coefficient of restitution between the spheres.
OCR M3 2009 June Q4
4 A motor-cycle, whose mass including the rider is 120 kg , is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \mathrm {~ms} ^ { - 1 }\) and when it has travelled a distance of \(x \mathrm {~m}\) beyond \(A\) its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine develops a constant power of 8 kW and resistances are modelled by a force of \(0.25 v ^ { 2 } \mathrm {~N}\) opposing the motion.
  1. Show that \(\frac { 480 v ^ { 2 } } { v ^ { 3 } - 32000 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
  2. Find the speed of the motor-cycle when it has travelled 500 m beyond \(A\).
OCR M3 2009 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-3_591_668_776_737} Each of two identical strings has natural length 1.5 m and modulus of elasticity 18 N . One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are 3 m apart and at the same horizontal level. \(M\) is the mid-point of \(A B\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of each of the strings. \(P\) is held at rest at the point 0.8 m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is 2 m below \(M\) (see diagram).
  1. Find the maximum tension in each of the strings during \(P\) 's motion.
  2. By considering energy,
    (a) show that the value of \(m\) is 0.42 , correct to 2 significant figures,
    (b) find the speed of \(P\) at \(M\).
OCR M3 2009 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_368_131_274_1005} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(O P\), at time \(t \mathrm {~s}\), is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is 0.05 .
  1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { L } \sin \theta\).
  2. Hence show that the motion of \(P\) is approximately simple harmonic.
  3. Given that the period of the approximate simple harmonic motion is \(\frac { 4 } { 7 } \pi \mathrm {~s}\), find the value of \(L\).
  4. Find the value of \(\theta\) when \(t = 0.7 \mathrm {~s}\), and the value of \(t\) when \(\theta\) next takes this value.
  5. Find the speed of \(P\) when \(t = 0.7 \mathrm {~s}\).
    \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_422_501_1500_822} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(O P\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).
  6. Find \(v ^ { 2 }\) in terms of \(u , a , g\) and \(\theta\) and show that \(R = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )\).
  7. Given that \(P\) just reaches the highest point of the circle, find \(u ^ { 2 }\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v ^ { 2 }\) is \(a g\).
  8. Given instead that \(P\) oscillates between \(\theta = \pm \frac { 1 } { 6 } \pi\) radians, find \(u ^ { 2 }\) in terms of \(a\) and \(g\).
OCR M3 2010 June Q1
1 A small ball of mass 0.8 kg is moving with speed \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 4 Ns . The speed of the ball immediately afterwards is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\).
OCR M3 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-2_691_767_529_689} Two uniform rods \(A B\) and \(B C\) are of equal length and each has weight 100 N . The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A B\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(A B\) (see diagram).
  1. By taking moments about \(A\) for \(A B\), find the vertical component of the force on \(A B\) at \(B\). Hence find the vertical component of the contact force on \(B C\) at \(C\).
  2. Calculate the magnitude of the frictional force on \(B C\) at \(C\) and state its direction. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_452_345_264_900} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after hitting the wall the components have magnitudes \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively (see Fig. 1).
OCR M3 2010 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-4_234_1003_1007_571} Particles \(P _ { 1 }\) and \(P _ { 2 }\) are each moving with simple harmonic motion along the same straight line. \(P _ { 1 }\) 's motion has centre \(C _ { 1 }\), period \(2 \pi \mathrm {~s}\) and amplitude \(3 \mathrm {~m} ; P _ { 2 }\) 's motion has centre \(C _ { 2 }\), period \(\frac { 4 } { 3 } \pi \mathrm {~s}\) and amplitude 4 m . The points \(C _ { 1 }\) and \(C _ { 2 }\) are 6.5 m apart. The displacements of \(P _ { 1 }\) and \(P _ { 2 }\) from their centres of oscillation at time \(t \mathrm {~s}\) are denoted by \(x _ { 1 } \mathrm {~m}\) and \(x _ { 2 } \mathrm {~m}\) respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x _ { 1 } = 3\) and \(x _ { 2 } = 4\).
  1. State expressions for \(x _ { 1 }\) and \(x _ { 2 }\) in terms of \(t\), which are valid until the particles collide. The particles collide when \(t = 5.99\), correct to 3 significant figures.
  2. Find the distance travelled by \(P _ { 2 }\) before the collision takes place.
  3. Find the velocities of \(P _ { 1 }\) and \(P _ { 2 }\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions.
OCR M3 2010 June Q6
6 A bungee jumper of weight \(W \mathrm {~N}\) is joined to a fixed point \(O\) by a light elastic rope of natural length 20 m and modulus of elasticity 32000 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.
  1. Given that the jumper just reaches a point 25 m below \(O\), find the value of \(W\).
  2. Find the maximum speed reached by the jumper.
  3. Find the maximum value of the deceleration of the jumper during the downward motion.
OCR M3 2010 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-5_447_693_255_726} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length 0.7 m . A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(O P\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\).
  2. Given that the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(O Q\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v ^ { 2 }\) in terms of \(\theta\), and show that the tension in the string \(O Q\) is \(14.7 m ( 1 + 2 \cos \theta ) \mathrm { N }\), where \(m \mathrm {~kg}\) is the mass of \(Q\).
  3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(O Q\) becomes slack.
  4. Show that \(V ^ { 2 } = 0.8575\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) when it reaches its greatest height (after the string \(O Q\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position.
OCR M3 2011 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-2_355_572_260_788} A particle \(P\) of mass 0.3 kg is moving in a straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(\cos \theta = 0.8\) and find the value of \(I\).
OCR M3 2011 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-2_403_999_982_575} Two uniform rods \(A B\) and \(A C\), of lengths 3 m and 4 m respectively, have weights 300 N and 400 N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is 2.4 m above the surface (see diagram).
  1. Show that the force exerted by the surface on \(A B\) is 374 N and find the force exerted by the surface on \(A C\).
  2. Find the tension in the string.
  3. Find the horizontal and vertical components of the force exerted on \(A B\) at \(A\) and state their directions.
OCR M3 2011 June Q3
3 A particle \(P\) of mass 0.25 kg is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface and moves in a straight line on the surface. The only horizontal force acting on \(P\) has magnitude \(0.2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after it is projected from \(O\). This force is directed towards \(O\).
  1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) passes through a point \(X\) with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the average speed of \(P\) for its motion between \(O\) and \(X\).
OCR M3 2011 June Q4
4 One end of a light inextensible string of length 2 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(O P\) makes an angle of 0.15 radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(O P\) makes an angle of \(\theta\) radians with the downward vertical.
  1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. Using the simple harmonic approximation,
  2. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = - 0.1\),
  3. find the angular speed of \(O P\) and the linear speed of \(P\) when \(t = 0.5\).
    \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-3_606_1006_973_568} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).
  4. Given that \(\sin \beta = 0.96\) and \(\frac { v _ { B } } { u _ { B } } = 1.2\), find the value of \(\sin \gamma\).
  5. Given also that, before the collision, the component of \(A\) 's velocity parallel to the line of centres is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the values of \(u _ { B }\) and \(v _ { B }\).
  6. Find the coefficient of restitution between the spheres.
  7. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5 m \mathrm {~J}\), where \(m \mathrm {~kg}\) is the mass of \(A\), find the value of \(v _ { A }\).
OCR M3 2011 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-4_559_525_258_808} A particle \(P\) of weight 6 N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 0.8 m . The string has natural length \(\frac { 1 } { 10 } \pi \mathrm {~m}\) and modulus of elasticity \(9 \mathrm {~N} . P\) is released from rest at a point \(X\) on the sphere where \(O X\) makes an angle of \(\frac { 1 } { 4 } \pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(O P\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac { 1 } { 6 } \pi , P\) passes through the point \(Y\).
  1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2 \cdot 4 ( \sqrt { 3 } - \sqrt { 2 } ) \mathrm { J }\) and the elastic potential energy in the string decreases by \(0.4 \pi \mathrm {~J}\).
  2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac { 1 } { 6 } \pi\), and hence find the maximum speed of \(P\).
OCR M3 2011 June Q7
7 One end of a light inextensible string of length 0.8 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.8 m vertically below \(O\) with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string makes an angle \(\theta\) with the downward vertical.
  1. While the string remains taut, show that \(v ^ { 2 } = 15.68 ( 1 + \cos \theta )\), and find the tension in the string in terms of \(\theta\).
  2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\).
  3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR M3 2012 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-3_442_636_255_715}
\(B\) is a point on a smooth plane surface inclined at an angle of \(15 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.45 kg is released from rest at the point \(A\) which is 2.5 m vertically above \(B\). The particle \(P\) rebounds from the surface at an angle of \(60 ^ { \circ }\) to the line of greatest slope through \(B\), with a speed of \(u \mathrm {~ms} ^ { - 1 }\). The impulse exerted on \(P\) by the surface has magnitude \(I\) Ns and is in a direction making an angle of \(\theta ^ { \circ }\) with the upward vertical through \(B\) (see diagram).
  1. Explain why \(\theta = 15\).
  2. Find the values of \(u\) and \(I\).
OCR M3 2012 June Q3
3 A particle \(P\) of mass \(m \mathrm {~kg}\) is released from rest and falls vertically. When \(P\) has fallen a distance of \(x \mathrm {~m}\) it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and air resistance of magnitude \(\frac { 1 } { 400 } m v ^ { 2 } \mathrm {~N}\).
  1. Find \(v ^ { 2 }\) in terms of \(x\) and show that \(v ^ { 2 }\) must be less than 3920 .
  2. Find the speed of \(P\) when it has fallen 100 m .
OCR M3 2012 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-4_524_611_255_703} A hollow cylinder is fixed with its axis horizontal. The inner surface of the cylinder is smooth and has radius 0.6 m . A particle \(P\) of mass 0.45 kg is projected horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the lowest point of a vertical cross-section of the cylinder and moves in the plane of the cross-section, which is perpendicular to the axis of the cylinder. While \(P\) remains in contact with the surface, its speed is \(v \mathrm {~ms} ^ { - 1 }\) when \(O P\) makes an angle \(\theta\) with the downward vertical at \(O\), where \(O\) is the centre of the cross-section (see diagram). The force exerted on \(P\) by the surface is \(R \mathrm {~N}\).
  1. Show that \(v ^ { 2 } = 4.24 + 11.76 \cos \theta\) and find an expression for \(R\) in terms of \(\theta\).
  2. Find the speed of \(P\) at the instant when it leaves the surface.
OCR M3 2012 June Q5
5 One end of a light elastic string, of natural length 0.78 m and modulus of elasticity 0.8 mgN , is attached to a fixed point \(O\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves down the plane without reaching the bottom. Find
  1. the maximum speed of \(P\) in the subsequent motion,
  2. the distance of \(P\) from \(O\) when it is at its lowest point.
OCR M3 2012 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the speed of \(A\) after the collision,
  2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
  3. the value of \(m\),
  4. the coefficient of restitution between \(A\) and \(B\).
    \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
  5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
  6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
  7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
  8. the distance travelled by \(P\),
  9. the speed of \(P\).
OCR M3 2013 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-2_435_665_255_699} A small object \(W\) of weight 100 N is attached to one end of each of two parallel light elastic strings. One string is of natural length 0.4 m and has modulus of elasticity 20 N ; the other string is of natural length 0.6 m and has modulus of elasticity 30 N . The upper ends of both strings are attached to a horizontal ceiling and \(W\) hangs in equilibrium at a distance \(d \mathrm {~m}\) below the ceiling (see diagram). Find \(d\).
OCR M3 2013 June Q2
2 A particle of mass 0.3 kg is projected horizontally under gravity with velocity \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point 0.4 m above a smooth horizontal plane. The particle first hits the plane at point \(A\); it bounces and hits the plane a second time at point \(B\). The distance \(A B\) is 1 m . Calculate
  1. the vertical component of the velocity of the particle when it arrives at \(A\), and the time taken for the particle to travel from \(A\) to \(B\),
  2. the coefficient of restitution between the particle and the plane,
  3. the impulse exerted by the plane on the particle at \(A\).
OCR M3 2013 June Q3
3 A particle \(P\) of mass 0.2 kg moves on a smooth horizontal plane. Initially it is projected with velocity \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) towards another fixed point \(A\). At time \(t\) s after projection, \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the direction \(O A\) being positive. A force of \(( 1.5 t - 1 ) \mathrm { N }\) acts on \(P\) in the direction parallel to \(O A\).
  1. Find an expression for \(v\) in terms of \(t\).
  2. Find the time when the velocity of \(P\) is next \(0.8 \mathrm {~ms} ^ { - 1 }\).
  3. Find the times when \(P\) subsequently passes through \(O\).
  4. Find the distance \(P\) travels in the third second of its motion.
OCR M3 2013 June Q4
4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find
  1. the velocity of \(A\) immediately after the collision,
  2. the angle turned through by the direction of motion of \(B\) as a result of the collision.