Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 2008 January Q6
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
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.
OCR MEI M1 2010 January Q1
1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{eafaf02f-bcd4-4368-a282-61ef1ad074da-2_766_1065_500_539} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
  2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
  3. State the direction of motion of the ring when
    (A) \(t = 1\),
    (B) \(t = 2.75\),
    (C) \(t = 3.25\).
OCR MEI M1 2010 January Q2
2 A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\binom { - 1 } { 2 } \mathrm {~m}\) with velocity \(\binom { 2 } { - 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\); after 4 seconds the particle has velocity \(\binom { 12 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the acceleration of the particle.
  2. Calculate the position of the particle at the end of the 4 seconds.
  3. Calculate the force acting on the particle.
OCR MEI M1 2010 January Q3
3 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
  2. Find \(\mathbf { G }\) and \(\mathbf { H }\).
OCR MEI M1 2010 January Q4
4 A box of mass 2.5 kg is on a smooth horizontal table, as shown in Fig. 4. A light string AB is attached to the table at A and the box at B . AB is at an angle of \(50 ^ { \circ }\) to the vertical. Another light string is attached to the box at C ; this string is inclined at \(15 ^ { \circ }\) above the horizontal and the tension in it is 20 N . The box is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{eafaf02f-bcd4-4368-a282-61ef1ad074da-3_403_1063_1085_539} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the horizontal component of the force exerted on the box by the string at C .
  2. Calculate the tension in the string AB .
  3. Calculate the normal reaction of the table on the box. The string at C is replaced by one inclined at \(15 ^ { \circ }\) below the horizontal with the same tension of 20 N .
  4. Explain why this has no effect on the tension in string AB .
OCR MEI M1 2011 January Q1
1 An object C is moving along a vertical straight line. Fig. 1 shows the velocity-time graph for part of its motion. Initially C is moving upwards at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after 10 s it is moving downwards at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-2_878_933_479_607} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} C then moves as follows.
  • In the interval \(10 \leqslant t \leqslant 15\), the velocity of C is constant at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
  • In the interval \(15 \leqslant t \leqslant 20\), the velocity of C increases uniformly so that C has zero velocity at \(t = 20\).
    1. Complete the velocity-time graph for the motion of C in the time interval \(0 \leqslant t \leqslant 20\).
    2. Calculate the acceleration of C in the time interval \(0 < t < 10\).
    3. Calculate the displacement of C from \(t = 0\) to \(t = 20\).
OCR MEI M1 2011 January Q2
2 Fig. 2 shows two forces acting at A. The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-3_264_922_479_609} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
  2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.
OCR MEI M1 2011 January Q3
3 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start
  • P is travelling directly towards Q with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • Q is instantaneously at rest and has an acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directly towards P .
    \(P\) continues with the same velocity and \(Q\) continues with the same acceleration. The cars collide \(T\) seconds after the start.
    1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start.
At the start, \(P\) is 90 m from \(Q\).
  • Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).
    •
    OCR MEI M1 2011 January Q4
    4 At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf { r } = \binom { 8 t } { 10 t ^ { 2 } - 2 t ^ { 3 } } ,$$ where \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are perpendicular unit vectors east and north respectively and distances are in metres.
    1. When \(t = 1\), the particle is at P . Find the bearing of P from O .
    2. Find the velocity of the particle at time \(t\) and show that it is never zero.
    3. Determine the time(s), if any, when the acceleration of the particle is zero.
    OCR MEI M1 2011 January Q5
    5 Fig. 5 shows two boxes, A of mass 12 kg and B of mass 6 kg , sliding in a straight line on a rough horizontal plane. The boxes are connected by a light rigid rod which is parallel to the line of motion. The only forces acting on the boxes in the line of motion are those due to the rod and a constant force of \(F \mathrm {~N}\) on each box. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-4_246_1006_479_568} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} The boxes have an initial speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and come to rest after sliding a distance of 0.375 m .
    1. Calculate the deceleration of the boxes and the value of \(F\).
    2. Calculate the magnitude of the force in the rod and state, with a reason, whether it is a tension or a thrust (compression).
    OCR MEI M1 2012 January Q1
    1 Fig. 1 shows two blocks of masses 3 kg and 5 kg connected by a light string which passes over a smooth, fixed pulley. Initially the blocks are held at rest but then they are released. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0330185f-d79d-4a78-9fa2-29ec345c2856-2_490_303_520_881} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Find the acceleration of the blocks when they start to move, and the tension in the string.
    OCR MEI M1 2012 January Q2
    2 Fig. 2 shows a small object, P , of weight 20 N , suspended by two light strings. The strings are tied to points A and B on a sloping ceiling which is at an angle of \(60 ^ { \circ }\) to the upward vertical. The string AP is at \(60 ^ { \circ }\) to the downward vertical and the string BP makes an angle of \(30 ^ { \circ }\) with the ceiling. The object is in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0330185f-d79d-4a78-9fa2-29ec345c2856-2_430_670_1546_699} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Show that \(\angle \mathrm { APB } = 90 ^ { \circ }\).
    2. Draw a labelled triangle of forces to represent the three forces acting on P .
    3. Hence, or otherwise, find the tensions in the two strings.
    OCR MEI M1 2012 January Q3
    3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
    Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 ,
    a = 0 & \text { for } t > 4 . \end{array}$$
    1. Show that Nina's speed, \(v \mathrm {~ms} ^ { - 1 }\), is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 ,
      v = 8 & \text { for } t > 4 . \end{array}$$
    2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
    3. Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).
    OCR MEI M1 2012 January Q4
    4 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$ where distances are in metres and the origin is a point on the level ground.
    1. Write down
      (A) the height from which P is projected,
      (B) the value of \(g\) in this model.
    2. Find the displacement of P from \(t = 3\) to \(t = 5\).
    3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$
    OCR MEI M1 2012 January Q5
    5 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$
    1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
    2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
    3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.
    OCR MEI M1 2012 January Q6
    6 Robin is driving a car of mass 800 kg along a straight horizontal road at a speed of \(40 \mathrm {~ms} ^ { - 1 }\).
    Robin applies the brakes and the car decelerates uniformly; it comes to rest after travelling a distance of 125 m .
    1. Show that the resistance force on the car when the brakes are applied is 5120 N .
    2. Find the time the car takes to come to rest. For the rest of this question, assume that when Robin applies the brakes there is a constant resistance force of 5120 N on the car. The car returns to its speed of \(40 \mathrm {~ms} ^ { - 1 }\) and the road remains straight and horizontal.
      Robin sees a red light 155 m ahead, takes a short time to react and then applies the brakes.
      The car comes to rest before it reaches the red light.
    3. Show that Robin's reaction time is less than 0.75 s . The 'stopping distance' is the total distance travelled while a driver reacts and then applies the brakes to bring the car to rest. For the rest of this question, assume that Robin is still driving the car described above and has a reaction time of 0.675 s . (This is the figure used in calculating the stopping distances given in the Highway Code.)
    4. Calculate the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) on a horizontal road. The car then travels down a hill which has a slope of \(5 ^ { \circ }\) to the horizontal.
    5. Find the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) down this hill.
    6. By what percentage is the stopping distance increased by the fact that the car is going down the hill? Give your answer to the nearest \(1 \%\).
    OCR MEI M1 2013 January Q1
    1 Fig. 1 shows a block of mass 3 kg on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
    A force \(P \mathrm {~N}\) is applied to the block parallel to the plane in the upwards direction.
    The plane is rough so that a frictional force of 10 N opposes the motion.
    The block is moving at constant speed up the plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-3_214_622_657_724} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure}
    1. Mark and label all the forces acting on the block.
    2. Calculate the magnitude of the normal reaction of the plane on the block.
    3. Calculate the magnitude of the force \(P\).
    OCR MEI M1 2013 January Q2
    2 In this question, the unit vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north.
    Distance is measured in metres and time, \(t\), in seconds.
    A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car.
    When \(t = 0\), the displacement of the car from the origin is \(\binom { 0 } { - 2 } \mathrm {~m}\), and the car has velocity \(\binom { 2 } { 0 } \mathrm {~ms} ^ { - 1 }\). The acceleration of the car is constant and is \(\binom { - 1 } { 1 } \mathrm {~ms} ^ { - 2 }\).
    1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\).
    2. Find the distance of the car from the child when \(t = 8\).
    OCR MEI M1 2013 January Q3
    3 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
    Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-4_291_1132_534_479} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
    1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
    2. Find the resistance force acting on the car along the line \(l\).
    3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.
    OCR MEI M1 2013 January Q4
    4 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-4_113_1173_1576_447} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).
    1. Find the initial velocity and the acceleration of the particle.
    2. Prove that the particle is never at P .
    OCR MEI M1 2013 January Q5
    5 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
    Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-5_229_953_434_557} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
    1. Find the time it takes for the stone to reach the water.
    2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.
    OCR MEI M1 2013 January Q6
    6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
    The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13f555cc-d506-48e5-a0e4-225cae4251dc-6_1025_1504_641_260} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
    1. Use Model P to estimate
      (A) the distance the runner has gone at the end of 12 seconds,
      (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
    2. Verify that Model Q gives the correct speed for \(t = 8\).
    3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
    4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
    5. Find the greatest acceleration of the runner according to each model.
    OCR MEI M1 2005 June Q1
    1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{04848aba-9e64-4265-a4a5-e9336b958a05-2_737_1274_502_461} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure}
    1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
    2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
    3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
    4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.
    OCR MEI M1 2005 June Q2
    2 A particle moves along the \(x\)-axis with velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) given by $$v = 24 t - 6 t ^ { 2 }$$ The positive direction is in the sense of \(x\) increasing.
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. Find the times, \(t _ { 1 }\) and \(t _ { 2 }\), at which the particle has zero speed.
    3. Find the distance travelled between the times \(t _ { 1 }\) and \(t _ { 2 }\).