Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 2010 June Q7
7 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-4_1026_1339_516_404} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\)
    (A) the greatest displacement of P above its position when \(t = 0\),
    (B) the greatest distance of P from its position when \(t = 0\),
    (C) the time interval in which P is moving downwards,
    (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
  2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
  3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
  4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).
OCR MEI M1 2010 June Q8
8 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_319_1358_511_392} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure}
  1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
  2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_502_935_1411_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
  3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
  4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
  5. Find the velocity of the tub when it has moved a further 1.65 m .
    4
  		6. □ box P □
    \multirow[t]{10}{*}{4
  7. }
    8. 4
OCR MEI M1 2011 June Q1
1 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.
OCR MEI M1 2011 June Q2
2 A particle travels with constant acceleration along a straight line. A and B are points on this line 8 m apart. The motion of the particle is as follows.
  • Initially it is at A.
  • After 32 s it is at B .
  • When it is at B its speed is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is moving away from A .
In either order, calculate the acceleration and the initial velocity of the particle, making the directions clear.
OCR MEI M1 2011 June Q3
3 Force \(\mathbf { F }\) is \(\left( \begin{array} { r } - 2
3
- 4 \end{array} \right) \mathrm { N }\), force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6
y
z \end{array} \right) \mathrm { N }\) and force \(\mathbf { H }\) is \(\left( \begin{array} { r } 3
- 5
- 1 \end{array} \right) \mathrm { N }\).
  1. Given that \(\mathbf { F }\) and \(\mathbf { G }\) act in parallel lines, find \(y\) and \(z\). Forces \(\mathbf { F }\) and \(\mathbf { H }\) are the only forces acting on an object of mass 5 kg .
  2. Calculate the acceleration of the object. Calculate also the magnitude of this acceleration.
OCR MEI M1 2011 June Q4
4 Fig. 4 shows a block of mass 15 kg on a smooth plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-2_280_718_1781_715} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Show all the forces acting on the block.
  2. Calculate \(P\).
OCR MEI M1 2011 June Q5
5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(35 ^ { \circ }\) to the horizontal. Assuming that air resistance on the object may be neglected and that the speed of sound in air is \(343 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate how long after projection the noise is heard at O .
OCR MEI M1 2011 June Q6
6 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors east and north respectively. Position vectors are with respect to an origin O . Time \(t\) is in seconds. A skater has a constant acceleration of \(- 2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At \(t = 0\), his velocity is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his position vector is \(3 \mathbf { j } \mathrm {~m}\).
  1. Find expressions in terms of \(t\) for the velocity and the position vector of the skater at time \(t\).
  2. Calculate as a bearing the direction of motion of the skater when \(t = 2.5\).
OCR MEI M1 2011 June Q7
7 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q . Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-4_931_1429_520_351} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Use model A to calculate the following.
  1. The acceleration of the ring when \(t = 0.5\).
  2. The displacement of the ring from Q when
    (A) \(t = 2\),
    (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
  3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B .
  4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
  5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .
OCR MEI M1 2011 June Q8
8 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_255_1097_397_523} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure}
  1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
    The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_305_1191_1050_701} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure}
  2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
  3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
  4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_355_1294_2156_429} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
    \end{figure}
  5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the trolleys, stating whether this rod is in tension or thrust (compression).
OCR MEI M1 2012 June Q1
1 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{076ad371-b029-4d57-aa0f-8a78ed03ccf3-2_1080_1596_376_239} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.
OCR MEI M1 2012 June Q2
2 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .
  1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
  2. Find the position of the particle when \(t = 2\).
OCR MEI M1 2012 June Q3
3 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j } .$$
  1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
  2. Interpret your answer to part (i) in the cases
    (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
    (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.
OCR MEI M1 2012 June Q4
4 Fig. 4 illustrates points \(\mathrm { A } , \mathrm { B }\) and C on a straight race track. The distance AB is 300 m and AC is 500 m .
A car is travelling along the track with uniform acceleration. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{076ad371-b029-4d57-aa0f-8a78ed03ccf3-3_65_1324_897_372} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Initially the car is at A and travelling in the direction AB with speed \(5 \mathrm {~ms} ^ { - 1 }\). After 20 s it is at C .
  1. Find the acceleration of the car.
  2. Find the speed of the car at B and how long it takes to travel from A to B .
OCR MEI M1 2012 June Q5
5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{076ad371-b029-4d57-aa0f-8a78ed03ccf3-3_394_579_1644_744} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show all the forces acting on the block.
  2. Show that the frictional force acting on the block is 25 N .
  3. Calculate the normal reaction of the floor on the block.
  4. Calculate the magnitude of the total force the floor is exerting on the block.
OCR MEI M1 2012 June Q6
6 A football is kicked with speed \(31 \mathrm {~ms} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) to the horizontal. It travels towards the goal which is 50 m away. The height of the crossbar of the goal is 2.44 m .
  1. Does the ball go over the top of the crossbar? Justify your answer.
  2. State one assumption that you made in answering part (i).
OCR MEI M1 2012 June Q7
7 A train consists of a locomotive pulling 17 identical trucks.
The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).
  1. Show that \(R = 1500\).
  2. Find the tensions in the couplings between
    (A) the last two trucks,
    (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
  3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
    The driver of the train reduces the driving force to zero and the resistance forces remain the same as before.
    The train then travels at a constant speed down the slope.
  4. Find the value of \(\beta\).
OCR MEI M1 2012 June Q8
8 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).
  1. Find the distance AB .
  2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
  3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
  4. What can you say about the result of the race?
  5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
  6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.
OCR MEI M1 2013 June Q1
1 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-2_400_568_434_751} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.
OCR MEI M1 2013 June Q2
2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-2_273_1109_1297_479} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.
  1. Find its initial speed.
  2. Find the ball's flight time and range, \(R \mathrm {~m}\).
  3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
    (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.
OCR MEI M1 2013 June Q3
3 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1
0
0 \end{array} \right) , \left( \begin{array} { l } 0
1
0 \end{array} \right)\) and \(\left( \begin{array} { l } 0
0
1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1
- 1
5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1
- 4
2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2
5
0 \end{array} \right) \mathrm { N }\).
  1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
  2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.
OCR MEI M1 2013 June Q4
4 The directions of the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are east and north.
The velocity of a particle, \(\mathrm { vm } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) is given by $$\mathbf { v } = \left( 16 - t ^ { 2 } \right) \mathbf { i } + ( 31 - 8 t ) \mathbf { j } .$$ Find the time at which the particle is travelling on a bearing of \(045 ^ { \circ }\) and the speed of the particle at this time.
OCR MEI M1 2013 June Q5
5 Fig. 5 shows blocks of mass 4 kg and 6 kg on a smooth horizontal table. They are connected by a light, inextensible string. As shown, a horizontal force \(F \mathrm {~N}\) acts on the 4 kg block and a horizontal force of 30 N acts on the 6 kg block. The magnitude of the acceleration of the system is \(2 \mathrm {~ms} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-3_106_1107_1708_479} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the two possible values of \(F\).
  2. Find the tension in the string in each case.
OCR MEI M1 2013 June Q6
6 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 24 - 18 t + 3 t ^ { 2 } .$$
  1. Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
  2. Find an expression for \(x\) at time \(t \mathrm {~s}\). Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\). Section B (36 marks)
    \(7 \quad\) Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
    \end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.
OCR MEI M1 2013 June Q8
8 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_122_849_456_609} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).
  1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The resistance to motion remains 100 N .
  2. Find the speed of the sledge
    (A) 1.6 seconds after the rope breaks,
    (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_259_853_1457_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
  3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
  4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.