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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 2010 January Q1
5 marks Easy -1.2
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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
8 marks Moderate -0.8
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
8 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
    8 marks Standard +0.3
    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
    7 marks Moderate -0.3
    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
    5 marks Moderate -0.8
    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
    7 marks Moderate -0.3
    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
    8 marks Moderate -0.3
    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
    8 marks Moderate -0.8
    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 $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$