Questions (30179 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR M3 2008 January Q3
9 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of \(A\) has components \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(B\), and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres. \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\).
  1. Find, in terms of \(e\), the component of the velocity of \(A\) along the line of centres immediately after the collision.
  2. Given that the speeds of \(A\) and \(B\) are the same immediately after the collision, and that \(3 e ^ { 2 } = 1\), find \(v\).
OCR M3 2008 January Q4
10 marks Standard +0.8
4 A particle of mass \(m \mathrm {~kg}\) is released from rest at a fixed point \(O\) and falls vertically. The particle is subject to an upward resisting force of magnitude \(0.49 m v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle when it has fallen a distance of \(x \mathrm {~m}\) from \(O\).
  1. Write down a differential equation for the motion of the particle, and show that the equation can be written as \(\left( \frac { 20 } { 20 - v } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.49\).
  2. Hence find an expression for \(x\) in terms of \(v\).
OCR M3 2008 January Q5
11 marks Standard +0.3
5 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 0.75 mgN . The other end of the string is attached to a fixed point \(O\) of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is released from rest at \(O\) and moves down the plane.
  1. Show that the maximum speed of \(P\) is reached when the extension of the string is 0.8 m .
  2. Find the maximum speed of \(P\).
  3. Find the maximum displacement of \(P\) from \(O\).
OCR M3 2008 January Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-4_497_524_276_804} A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point \(O\). With the string taut the particle is travelling in a circular path in a vertical plane. The angle between the string and the downward vertical is \(\theta ^ { \circ }\) (see diagram). When \(\theta = 0\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. At the instant when the string is horizontal, find the speed of \(P\) and the tension in the string.
  2. At the instant when the string becomes slack, find the value of \(\theta\).
OCR M3 2008 January Q7
15 marks Challenging +1.2
7 A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity \(4 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(A\). The particle is released from rest at a point 4.8 m vertically below \(A\). At time \(t \mathrm {~s}\) after \(P\) 's release \(P\) is ( \(4 + x ) \mathrm { m }\) below \(A\).
  1. Show that \(4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x\). \(P\) 's motion is simple harmonic.
  2. Write down the amplitude of \(P\) 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle \(Q\) 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 \(B\). The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time \(t \mathrm {~s}\) after \(Q\) 's release \(B Q\) makes an angle of \(\theta\) radians with the downward vertical.
  3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta\). The period of the simple harmonic motion to which \(Q\) 's motion approximates is the same as the period of \(P\) 's motion.
  4. Given that \(\theta = 0.08\) when \(t = 0\), find the speed of \(Q\) when \(t = 0.25\).
OCR M3 2009 January Q1
8 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-2_385_741_269_701} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse of magnitude 2.6 N s applied to \(P\) deflects its direction of motion through an angle \(\theta\), and reduces its speed to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). By considering an impulse-momentum triangle, or otherwise,
  1. show that \(\cos \theta = 0.6\),
  2. find the angle that the impulse makes with the original direction of motion of \(P\).
OCR M3 2009 January Q2
8 marks Standard +0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{14403602-94a6-4441-a673-65f9b98180e5-2_501_752_1133_356} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{14403602-94a6-4441-a673-65f9b98180e5-2_519_558_1183_1231} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Two uniform rods \(A B\) and \(B C\), of weights 70 N and 110 N respectively, are freely jointed at \(B\). The rods are in equilibrium in a vertical plane with \(A\) and \(C\) at the same horizontal level and \(A C = 2 \mathrm {~m}\). The \(\operatorname { rod } A B\) is freely jointed to a fixed point at \(A\) and the rod \(B C\) is freely jointed to a fixed point at \(C\). The horizontal distance between \(B\) and \(A\) is 4 m and \(B\) is 4 m below \(A C\); angle \(B A C\) is obtuse (see Fig. 1). The force exerted on the \(\operatorname { rod } A B\) at \(B\), by the \(\operatorname { rod } B C\), has horizontal and vertical components as shown in Fig. 2.
  1. By taking moments about \(A\) for the \(\operatorname { rod } A B\) find the value of \(X - Y\).
  2. By taking moments about \(C\) for the rod \(B C\) show that \(2 X - 3 Y + 165 = 0\).
  3. Find the magnitude of the force acting between \(A B\) and \(B C\) at \(B\).
OCR M3 2009 January Q3
9 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_387_181_274_982} \(A\) and \(B\) are fixed points with \(B\) at a distance of 1.8 m vertically below \(A\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to \(A\), and one end of an identical elastic string is attached to \(B\). A particle \(P\) of weight 12 N is attached to the other ends of the strings (see diagram).
  1. Verify that \(P\) is in equilibrium when it is at a distance of 1.05 m vertically below \(A\). \(P\) is released from rest at the point 1.2 m vertically below \(A\) and begins to move.
  2. Show that, when \(P\) is \(x \mathrm {~m}\) below its equilibrium position, the tensions in \(P A\) and \(P B\) are \(( 18 + 40 x ) \mathrm { N }\) and \(( 6 - 40 x ) \mathrm { N }\) respectively.
  3. Show that \(P\) moves with simple harmonic motion of period 0.777 s , correct to 3 significant figures.
  4. Find the speed with which \(P\) passes through the equilibrium position. \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-3_540_655_1564_744} One end of a light inextensible string of length 0.5 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. With the string taut and horizontal, \(P\) is projected with a velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downward. \(P\) begins to move in a vertical circle with centre \(O\). While the string remains taut the angular displacement of \(O P\) is \(\theta\) radians from its initial position, and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).
OCR M3 2009 January Q5
10 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-4_369_953_269_596} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 3 kg and 4 kg respectively. They are moving on a horizontal surface, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. The directions of motion of \(A\) and \(B\) make angles \(\alpha\) and \(\beta\) respectively with the line of centres of the spheres, where \(\sin \alpha = \cos \beta = 0.6\) (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the angle that the velocity of \(A\) makes, immediately after impact, with the line of centres of the spheres.
[0pt] [10]
OCR M3 2009 January Q6
12 marks Standard +0.8
6 A stone of mass 0.125 kg falls freely under gravity, from rest, until it has travelled a distance of 10 m . The stone then continues to fall in a medium which exerts an upward resisting force of \(0.025 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the stone \(t \mathrm {~s}\) after the instant that it enters the resisting medium.
  1. Show by integration that \(v = 49 - 35 \mathrm { e } ^ { - 0.2 t }\).
  2. Find how far the stone travels during the first 3 seconds in the medium.
OCR M3 2009 January Q7
15 marks Standard +0.3
7 A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to a fixed point \(O\). The particle is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\), the particle is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy show that \(v ^ { 2 } = 39.2 + 19.6 x - 12.5 x ^ { 2 }\).
  2. Hence find
    (a) the maximum extension of the string,
    (b) the maximum speed of the particle,
    (c) the maximum magnitude of the acceleration of the particle. \footnotetext{OCR
    RECOGNISING ACHIEVEMENT
OCR M3 2010 January Q1
7 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_323_639_255_753} A particle \(P\) of mass 0.4 kg is moving horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\), in a direction which makes an angle \(( 180 - \theta ) ^ { \circ }\) with the direction of motion of \(P\). Immediately after the impulse acts \(P\) moves horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is turned through an angle of \(60 ^ { \circ }\) by the impulse (see diagram). Find \(I\) and \(\theta\).
OCR M3 2010 January Q2
7 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_421_759_936_694} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and \(B\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres (see diagram). The coefficient of restitution is 0.6 . The direction of motion of \(B\) after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Find the value of \(v\).
OCR M3 2010 January Q3
9 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-3_812_773_260_685} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\), have weights \(2 W\) and \(W\) respectively. The rods are freely jointed to each other at \(B\), and \(B C\) is freely jointed to a fixed point at \(C\). The rods are held in equilibrium in a vertical plane by a light string attached to \(A\) and perpendicular to \(A B\). The rods \(A B\) and \(B C\) make angles \(45 ^ { \circ }\) and \(\alpha\), respectively, with the horizontal. The tension in the string is \(T\) (see diagram).
  1. By taking moments about \(B\) for \(A B\), show that \(W = \sqrt { 2 } T\).
  2. Find the value of \(\tan \alpha\).
OCR M3 2010 January Q4
11 marks Challenging +1.2
4 A particle \(P\) of mass 0.2 kg travels in a straight line on a horizontal surface. It passes through a point \(O\) on the surface with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude \(0.2 \left( v + v ^ { 2 } \right) \mathrm { N }\) acts on \(P\) in the direction opposite to its motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) when it is at a distance \(x \mathrm {~m}\) from \(O\).
  1. Show that \(\frac { 1 } { 1 + v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
  2. By solving the differential equation in part (i) show that \(\frac { - \mathrm { e } ^ { x } } { 3 - \mathrm { e } ^ { x } } \frac { \mathrm {~d} x } { \mathrm {~d} t } = - 1\), where \(t\) s is the time taken for \(P\) to travel \(x \mathrm {~m}\) from \(O\).
  3. Hence find the value of \(t\) when \(x = 1\).
OCR M3 2010 January Q5
11 marks Standard +0.8
5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).
  1. Verify that the distance \(A B\) is 4 m .
  2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).
OCR M3 2010 January Q7
14 marks Challenging +1.2
7 A particle \(P\) of mass 0.5 kg is attached to one end of each of two identical light elastic strings of natural length 1.6 m and modulus of elasticity 19.6 N . The other ends of the strings are attached to fixed points \(A\) and \(B\) on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4.8 m and \(A\) is higher than \(B\).
  1. Find the distance \(A P\) for which \(P\) is in equilibrium on the line \(A B\). \(P\) is released from rest at a point on \(A B\) where both strings are taut. The strings remain taut during the subsequent motion of \(P\) and \(t\) seconds after release the distance \(A P\) is \(( 2.5 + x ) \mathrm { m }\).
  2. Use Newton's second law to obtain an equation of the form \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = k x\). State the property of the constant \(k\) for which the equation indicates that \(P\) 's motion is simple harmonic, and find the period of this motion.
  3. Given that \(x = 0.5\) when \(t = 0\), find the values of \(x\) for which the speed of \(P\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). {www.ocr.org.uk}) after the live examination series.
    If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1 GE.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR M3 2011 January Q1
6 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_476_583_258_781} A ball of mass 0.5 kg is moving with speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line when it is struck by a bat. The impulse exerted by the bat has magnitude 15 N s and the ball is deflected through an angle of \(90 ^ { \circ }\) (see diagram). Find
  1. the direction of the impulse,
  2. the speed of the ball immediately after it is struck.
OCR M3 2011 January Q2
6 marks Standard +0.3
2 A particle of mass 0.4 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.5 m . The particle is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). The particle moves in a complete circle. Find the tension in the string when
  1. the string is horizontal,
  2. the particle is vertically above \(O\).
OCR M3 2011 January Q3
9 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_586_1435_1537_354} A uniform \(\operatorname { rod } P Q\) has weight 72 N . A non-uniform \(\operatorname { rod } Q R\) has weight 54 N and its centre of mass is at \(C\), where \(Q C = 2 C R\). The rods are freely jointed to each other at \(Q\). The rod \(P Q\) is freely jointed to a fixed point of a vertical wall at \(P\) and the rod \(Q R\) rests on horizontal ground at \(R\). The rod \(P Q\) is 2.8 m long and is horizontal. The point \(R\) is 1.44 m below the level of \(P Q\) and 4 m from the wall (see diagram).
  1. Find the vertical component of the force exerted by the wall on \(P Q\).
  2. Hence show that the normal component of the force exerted by the ground on \(Q R\) is 90 N .
  3. Given that the friction at \(R\) is limiting, find the coefficient of friction between the rod \(Q R\) and the ground.
OCR M3 2011 January Q4
11 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-3_497_1157_255_493} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg and \(B\) has mass 0.3 kg . Immediately before the collision \(A\) is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\), and \(B\) is moving with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.7. Find
  1. the speed of \(B\) immediately after the collision,
  2. the angle turned through by the direction of motion of \(A\) as a result of the collision.
OCR M3 2011 January Q5
13 marks Standard +0.3
5 A particle \(P\) of mass 0.05 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.5 m and modulus of elasticity 2.45 N .
  1. Show that the equilibrium position of \(P\) is 0.6 m below \(O\). \(P\) is held at rest at a point 0.675 m vertically below \(O\) and then released. At time \(t \mathrm {~s}\) after \(P\) is released, its downward displacement from the equilibrium position is \(x \mathrm {~m}\).
  2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 98 x\).
  3. Find the value of \(x\) and the magnitude and direction of the velocity of \(P\) when \(t = 0.2\).
OCR M3 2011 January Q6
12 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-4_638_473_260_836} A particle \(P\), of mass 3.5 kg , is in equilibrium suspended from the top \(A\) of a smooth slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 40 } { 49 }\), by an elastic rope of natural length 4 m and modulus of elasticity 112 N (see diagram). Another particle \(Q\), of mass 0.5 kg , is released from rest at \(A\) and slides freely downwards until it reaches \(P\) and becomes attached to it.
  1. Find the value of \(V ^ { 2 }\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) immediately before it becomes attached to \(P\), and show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(\frac { 1 } { 2 } \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The combined particles slide downwards for a distance of \(X \mathrm {~m}\), before coming instantaneously to rest at \(B\).
  2. Show that \(28 X ^ { 2 } - 8 X - 5 = 0\).
OCR M3 2011 January Q7
15 marks Challenging +1.2
7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) and falls vertically. Air resistance of magnitude \(\frac { v ^ { 2 } } { 2000 } \mathrm {~N}\) acts upwards on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) when it has fallen a distance of \(x \mathrm {~m}\).
  1. Show that \(\left( \frac { 400 v } { 3920 - v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
  2. Find \(v ^ { 2 }\) in terms of \(x\) and hence show that \(v ^ { 2 } < 3920\) for all values of \(x\).
  3. Find the work done against the air resistance while \(P\) is falling, from \(O\), to the point where its downward acceleration is \(5.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). 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.
    If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1 GE.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
OCR M3 2012 January Q1
8 marks Standard +0.3
1 A particle \(P\) of mass 0.05 kg is moving on a smooth horizontal surface with speed \(2 \mathrm {~ms} ^ { - 1 }\), when it is struck by a horizontal blow in a direction perpendicular to its direction of motion. The magnitude of the impulse of the blow is \(I\) Ns. The speed of \(P\) after the blow is \(2.5 \mathrm {~ms} ^ { - 1 }\).
  1. Find the value of \(I\). Immediately before the blow \(P\) is moving parallel to a smooth vertical wall. After the blow \(P\) hits the wall and rebounds from the wall with speed \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the coefficient of restitution between \(P\) and the wall.