Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 Q2
2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-002_478_397_1211_872} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. State what information in the question tells you that
    (A) the tension is the same throughout the string,
    (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
  2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
  3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).
OCR MEI M1 Q5
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]{9a79f274-1a3f-4d11-9775-313d82075035-003_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. 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]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  4. 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 \(t\).
  5. 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 \(X = 30\).
  6. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  7. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)
OCR MEI M1 Q7
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]{9a79f274-1a3f-4d11-9775-313d82075035-004_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 Q1
1 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north. At time \(t\) hours the position vectors of two hikers, Ashok and Kumar, are given by: $$\begin{array} { l l } \text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t ,
\text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } . \end{array}$$
  1. Prove that the two hikers meet and give the coordinates of the point where this happens.
  2. Compare the speeds of the two hikers.
OCR MEI M1 Q2
2 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10
& a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$
  1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
  2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
  3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
  4. Find the value of \(t\) when the box hits the ground.
  5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
    (A) The box should be dropped from a height of 500 m instead of 1000 m .
    (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 ,
    a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$
OCR MEI M1 Q3
3 In this question the origin is a point on the ground. 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.
\includegraphics[max width=\textwidth, alt={}, center]{cb72a1c4-f769-4348-ad7f-66c3c96e1732-3_401_686_368_721} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.
  • During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
  • During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
  • Alesha's mass, including her equipment, is 100 kg .
  • Initially, her position vector is \(\left( \begin{array} { r } - 75
    90
    750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5
    0
    - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
    1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.
At a certain time during the dive, forces of \(\left( \begin{array} { r } 0
0
- 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0
0
880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50
- 20
0 \end{array} \right) \mathrm { N }\) are acting on Alesha.
  • Suggest how these forces could arise.
  • Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.
  • Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.
  • Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.
    •
    OCR MEI M1 Q4
    4 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 \(t\) s. Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\).
    OCR MEI M1 Q5
    5 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 Q1
    1 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$ where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).
    1. Write down the displacement of the particle when \(t = 0\).
    2. Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
    3. Find an expression in terms of \(t\) for the acceleration of the particle.
    4. Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
    5. Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
    6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
    7. Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).
    OCR MEI M1 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, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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 Q4
    4 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 .
    2. Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).
    OCR MEI M1 Q5
    2 marks
    5 The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a particle moving along a straight line is given by $$v = 3 t ^ { 2 } - 12 t + 14$$ where \(t\) is the time in seconds.
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\).
    3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times.
      [0pt] [2]
    OCR MEI M1 Q2
    2 \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34e4ce80-21b0-48f5-865c-de4dd837f7c5-1_98_836_1073_718} \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 Q3
    3 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 Q4
    4 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]{34e4ce80-21b0-48f5-865c-de4dd837f7c5-3_1027_1333_372_435} \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 Q5
    5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34e4ce80-21b0-48f5-865c-de4dd837f7c5-4_581_1085_453_567} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
    1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
    2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
    3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).
    OCR MEI M1 Q6
    6 Fig. 7 is a sketch of part of the velocity-time graph for the motion of an insect walking in a straight line. Its velocity, \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\), at time \(t\) seconds for the time interval \(- 3 \leqslant t \leqslant 5\) is given by $$v = t ^ { 2 } - 2 t - 8 .$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34e4ce80-21b0-48f5-865c-de4dd837f7c5-5_624_886_549_631} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
    1. Write down the velocity of the insect when \(t = 0\).
    2. Show that the insect is instantaneously at rest when \(t = - 2\) and when \(t = 4\).
    3. Determine the velocity of the insect when its acceleration is zero. Write down the coordinates of the point A shown in Fig. 7.
    4. Calculate the distance travelled by the insect from \(t = 1\) to \(t = 4\).
    5. Write down the distance travelled by the insect in the time interval \(- 2 \leqslant t \leqslant 4\).
    6. How far does the insect walk in the time interval \(1 \leqslant t \leqslant 5\) ?
    OCR MEI M1 Q1
    1 Fig. 4 illustrates a straight horizontal road. \(A\) and \(B\) are points on the road which are 215 metres apart and \(M\) is the mid-point of AB . When a car passes A its speed is \(12 \mathrm {~ms} ^ { - 1 }\) in the direction AB . It then accelerates uniformly and when it reaches \(B\) its speed is \(31 \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-1_140_1160_455_488} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
    1. Find the car's acceleration.
    2. Find how long it takes the car to travel from A to B .
    3. Find how long it takes the car to travel from A to M .
    4. Explain briefly, in terms of the speed of the car, why the time taken to travel from A to M is more than half the time taken to travel from A to B .
    OCR MEI M1 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]{b9e41fac-9f4b-4165-af03-67ebdcb326de-1_285_1117_1450_497} \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 Q3
    3 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 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]{b9e41fac-9f4b-4165-af03-67ebdcb326de-2_90_1335_982_331} \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 20s 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 Q5
    5 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 \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 \(t = 2\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.
    3. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
    4. How much time does it take for P to catch up with Q and how far does P travel in this time?
    OCR MEI M1 Q1
    1 A car of mass 1000 kg is travelling along a straight, level road. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-1_150_868_316_602} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure}
    1. Calculate the acceleration of the car when a resultant force of 2000 N acts on it in the direction of its motion. How long does it take the car to increase its speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(12.5 \mathrm {~ms} ^ { - 1 }\) ? The car has an acceleration of \(1.4 \mathrm {~ms} ^ { - 2 }\) when there is a driving force of 2000 N .
    2. Show that the resistance to motion of the car is 600 N . A trailer is now atached to the car, as shown in Fig. 6.2. The car still has a driving force of 2000 N and resistance to motion of 600 N . The trailer has a mass of 800 kg . The tow-bar connecting the car and the trailer is light and horizontal. The car and trailer are accelerating at \(0.7 \mathrm {~ms} ^ { 2 }\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-1_165_883_1279_554} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
      \end{figure}
    3. Show that the resistance to the motion of the trailer is 140 N .
    4. Calculate the force in the tow bar. The driving force is now removed and a braking force of 610 N is applied to the car. All the resistances to motion remain as before. The trailer has no brakes.
    5. Calculate the new acceleration. Calculate also the force in the tow-bar, stating whether it is a tension or a thrust (compression).
    OCR MEI M1 Q2
    2 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]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-2_289_1132_571_507} \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 Q3
    3 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]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-2_115_1169_1719_499} \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 \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 .