Questions — OCR MEI (4301 questions)

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OCR MEI M1 2005 January Q3
6 marks Moderate -0.8
3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.
  1. Find the values of \(x , y\) and \(z\).
  2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).
OCR MEI M1 2005 January Q4
8 marks Standard +0.3
4 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
  2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.
OCR MEI M1 2005 January Q5
7 marks Moderate -0.8
5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-3_424_472_1599_774} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on the box.
  2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
  3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.
OCR MEI M1 2005 January Q7
17 marks Standard +0.3
7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-4_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)
OCR MEI M1 2006 January Q1
6 marks Easy -1.3
1 A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_455_874_484_593} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Calculate the acceleration of the particle in the interval \(0 < t < 6\).
  2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
  3. When \(t = 0\) the particle is at A . Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\).
OCR MEI M1 2006 January Q2
5 marks Moderate -0.8
2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at \(20 ^ { \circ }\) to the vertical. The system is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_588_451_1749_806} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. What information in the question tells you that the tension is the same throughout the string AB ?
  2. What is the tension in the string AB ?
  3. Calculate the tension in the strings BC and BD .
OCR MEI M1 2006 January Q3
7 marks Moderate -0.8
3 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
  2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } )\) N. Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
  3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).
OCR MEI M1 2006 January Q4
5 marks Moderate -0.8
4 A car and its trailer travel along a straight, horizontal road. The coupling between them is light and horizontal. The car has mass 900 kg and resistance to motion 100 N , the trailer has mass 700 kg and resistance to motion 300 N , as shown in Fig. 4. The car and trailer have an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-3_400_753_1037_657} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the driving force of the car.
  2. Calculate the force in the coupling.
OCR MEI M1 2006 January Q5
6 marks Moderate -0.3
5 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.
  1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
  2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(t = 1\).
  3. Find an expression for the velocity of the particle at time \(t\).
OCR MEI M1 2006 January Q6
7 marks Moderate -0.3
6 A car is driven with constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), along a straight road. Its speed when it passes a road sign is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down two equations connecting \(a\) and \(u\). Hence find the values of \(a\) and \(u\).
  2. What distance does the car travel in the 5 seconds after passing the road sign? Section B (36 marks)
OCR MEI M1 2006 January Q7
16 marks Moderate -0.3
7 Clive and Ken are trying to move a box of mass 50 kg on a rough, horizontal floor. As shown in Fig. 7, Clive always pushes horizontally and Ken always pulls at an angle of \(30 ^ { \circ }\) to the horizontal. Each of them applies forces to the box in the same vertical plane as described below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-4_360_745_995_660} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Initially, the box is in equilibrium with Clive pushing with a force of 60 N and Ken not pulling at all.
  1. What is the resistance to motion of the box? Ken now adds a pull of 70 N to Clive's push of 60 N . The box remains in equilibrium.
  2. What now is the resistance to motion of the box?
  3. Calculate the normal reaction of the floor on the box. The frictional resistance to sliding of the box is 125 N .
    Clive now pushes with a force of 160 N but Ken does not pull at all.
  4. Calculate the acceleration of the box. Clive stops pushing when the box has a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. How far does the box then slide before coming to rest? Ken and Clive now try again. Ken pulls with a force of \(Q \mathrm {~N}\) and Clive pushes with a force of 160 N . The frictional resistance to sliding of the box is now 115 N and the acceleration of the box is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  6. Calculate the value of \(Q\).
OCR MEI M1 2006 January Q8
20 marks Standard +0.3
8 A girl throws a small stone with initial speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-5_658_757_482_648} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
  2. (A) Find the time it takes for the stone to reach its highest point.
    (B) Calculate the distance of O from the base of the wall.
    (C) Show that the stone passes over the wall with 2.5 m clearance.
  3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
  4. Calculate \(d\).
OCR MEI M1 2007 January Q1
4 marks Moderate -0.8
1 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-2_668_1360_461_354} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).
OCR MEI M1 2007 January Q2
5 marks Moderate -0.8
2 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by $$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$ Find the values of \(x\) for which the velocity of the particle is zero.
OCR MEI M1 2007 January Q3
7 marks Moderate -0.8
3 A box of mass 5 kg is at rest on a rough horizontal floor.
  1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-2_293_472_2131_794} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Draw a diagram showing all the forces acting on the box.
  3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.
OCR MEI M1 2007 January Q4
7 marks Moderate -0.8
4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-3_191_346_328_858} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
  2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
  3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.
OCR MEI M1 2007 January Q5
6 marks Moderate -0.3
5 A block of mass 4 kg slides on a horizontal plane against a constant resistance of 14.8 N . A light, inextensible string is attached to the block and, after passing over a smooth pulley, is attached to a freely hanging sphere of mass 2 kg . The part of the string between the block and the pulley is horizontal. This situation is shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-3_250_671_1466_696} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} The tension in the string is \(T \mathrm {~N}\) and the acceleration of the block and of the sphere is \(a \mathrm {~ms} ^ { - 2 }\).
  1. Write down the equation of motion of the block and also the equation of motion of the sphere, each in terms of \(T\) and \(a\).
  2. Find the values of \(T\) and \(a\).
OCR MEI M1 2007 January Q6
7 marks Moderate -0.8
6 The velocity of a model boat, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \binom { - 5 } { 10 } + t \binom { 6 } { - 8 }$$ where \(t\) is the time in seconds and the vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are east and north respectively.
  1. Show that when \(t = 2.5\) the boat is travelling south-east (i.e. on a bearing of \(135 ^ { \circ }\) ). Calculate its speed at this time. The boat is at a point O when \(t = 0\).
  2. Calculate the bearing of the boat from O when \(t = 2.5\).
OCR MEI M1 2007 January Q7
18 marks Standard +0.3
7 A horizontal force of 24 N acts on a block of mass 12 kg on a horizontal plane. The block is initially at rest. This situation is first modelled assuming the plane is smooth.
  1. Write down the acceleration of the block according to this model. The situation is now modelled assuming a constant resistance to motion of 15 N .
  2. Calculate the acceleration of the block according to this new model. How much less distance does the new model predict that the block will travel in the first 4 seconds? The 24 N force is removed and the block slides down a slope at \(5 ^ { \circ }\) to the horizontal. The speed of the block at the top of the slope is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 7. The answers to parts (iii) and (iv) should be found using the assumption that the resistance to the motion of the block is still a constant 15 N . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-5_255_901_1128_575} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  3. Calculate the acceleration of the block in the direction of its motion.
  4. For how much time does the block slide down the slope before coming to rest and how far does it slide in that time? Measurements show that the block actually comes to rest in 3.5 seconds.
  5. Assuming that the error in the prediction is due only to the value of the resistance, calculate the true value of the resistance.
OCR MEI M1 2008 January Q1
6 marks Easy -1.3
1 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.
  1. Sketch a velocity-time graph for the motion.
  2. Calculate the distance travelled by the cyclist.
OCR MEI M1 2008 January Q2
7 marks Moderate -0.8
2 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).
  1. Give the acceleration of the particle as a vector.
  2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
  3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~ms} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.
OCR MEI M1 2008 January Q3
8 marks Moderate -0.8
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-2_344_716_1324_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at \(40 ^ { \circ }\) to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N .
  1. The sphere has a mass of \(m \mathrm {~kg}\). Calculate the value of \(m\).
  2. Calculate the frictional force acting on the block.
  3. Calculate the normal reaction of the plane on the block.
OCR MEI M1 2008 January Q4
7 marks Moderate -0.8
4 Force \(\mathbf { F }\) is \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right) \mathrm { N }\) and force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ 2 \\ 4 \end{array} \right) \mathrm { N }\).
  1. Find the resultant of \(\mathbf { F }\) and \(\mathbf { G }\) and calculate its magnitude.
  2. Forces \(\mathbf { F } , 2 \mathbf { G }\) and \(\mathbf { H }\) act on a particle which is in equilibrium. Find \(\mathbf { H }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-3_99_841_676_651} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} A toy car is moving along the straight line \(\mathrm { O } x\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at \(\mathrm { A } , 3 \mathrm {~m}\) from O as shown in Fig. 5. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 2 + 12 t - 3 t ^ { 2 }$$ Calculate the distance of the car from O when its acceleration is zero.
OCR MEI M1 2008 January Q6
17 marks Moderate -0.3
6 A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1.
  1. When the man is descending with an acceleration \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, how much time does it take for his speed to increase from \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards to \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards? \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_373_460_365_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} How far does he descend in this time? The man has a mass of 80 kg . All resistances to motion may be neglected.
  2. Calculate the tension in the wire when the man is being lowered
    (A) with an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards,
    (B) with an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards. Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.
  3. For safety reasons, the tension in the wire should not exceed 2500 N . What is the maximum acceleration allowed when the man is being raised? At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N .
  4. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_442_460_1589_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} tension in the wire.
OCR MEI M1 2008 January Q7
19 marks Moderate -0.3
7 A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O , as shown in Fig. 7. Air resistance is negligible. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-5_600_1029_447_557} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The initial horizontal component of the velocity of the firework is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that the firework is \(\left( 28 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) above the ground \(t\) seconds after its projection. When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the other with a quarter of this speed.
  3. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion.
  4. Find the distance between these parts of the firework
    (A) when they reach the ground,
    (B) when they are 10 m above the ground.
  5. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac { 1 } { 90 } \left( 120 x - x ^ { 2 } \right)\), referred to the coordinate axes shown in Fig. 7.