Questions — OCR MEI M1 (276 questions)

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OCR MEI M1 Q4
6 marks Moderate -0.8
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
18 marks Standard +0.3
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 Q2
8 marks Standard +0.3
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
5 marks Moderate -0.5
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
16 marks Moderate -0.3
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
8 marks Moderate -0.8
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
17 marks Moderate -0.3
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
8 marks Moderate -0.8
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
8 marks Moderate -0.8
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
7 marks Moderate -0.3
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
5 marks Moderate -0.3
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
7 marks Moderate -0.3
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
17 marks Moderate -0.3
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
8 marks Moderate -0.3
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
7 marks Standard +0.3
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 .
OCR MEI M1 Q4
7 marks Standard +0.3
4 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 \(u \mathrm {~ms} { } ^ { 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~ms} { } ^ { 1 }\).
  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?
OCR MEI M1 Q1
6 marks Moderate -0.3
1 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 Q2
6 marks Standard +0.3
2 The directions of the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are east and north.
The velocity of a particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), 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.
[0pt] [6]
OCR MEI M1 Q3
7 marks Standard +0.3
3 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 Q4
8 marks Moderate -0.3
4 The three forces \(\left. \begin{array} { r } - 1 \\ 14 \\ - 8 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 3 \\ - 9 \\ 10 \end{array} \right) \mathrm { N }\) and \(\mathbf { F } \mathrm { N }\) act on a body of mass 4 kg in deep space and give it an acceleration of \(\left. \quad \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\). At one instant the velocity of the body is \(\left. \begin{array} { r } - 3 \\ 3 \\ 6 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Calculate the velocity and also the speed of the body 3 seconds later.
OCR MEI M1 Q5
8 marks Moderate -0.3
5 The position vector of a toy boat of mass 1.5 kg is modelled as \(\mathbf { r } = ( 2 + t ) \mathbf { i } + \left( 3 t - t ^ { 2 } \right) \mathbf { j }\) where lengths are in metres, \(t\) is the time in seconds, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal, perpendicular unit vectors and the origin is O .
  1. Find the velocity of the boat when \(t = 4\).
  2. Find the acceleration of the boat and the horizontal force acting on the boat.
  3. Find the cartesian equation of the path of the boat referred to \(x\) - and \(y\)-axes in the directions of \(\mathbf { i }\) and \(\mathbf { j }\), respectively, with origin O . You are not required to simplify your answer.
OCR MEI M1 Q6
5 marks Moderate -0.8
6 An object of mass 5 kg has a constant acceleration of \(\binom { - 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) for \(0 \leqslant t \leqslant 4\), where \(t\) is the time in seconds.
  1. Calculate the force acting on the object. When \(t = 0\), the object has position vector \(\binom { - 2 } { 3 } \mathrm {~m}\) and velocity \(\binom { 4 } { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the position vector of the object when \(t = 4\).
OCR MEI M1 Q7
5 marks Moderate -0.8
7 An object of mass 5 kg has a constant acceleration of \(\binom { - 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) for \(0 \leqslant t \leqslant 4\), where \(t\) is the time in seconds.
  1. Calculate the force acting on the object. When \(t = 0\), the object has position vector \(\binom { - 2 } { 3 } \mathrm {~m}\) and velocity \(\binom { 4 } { 5 } \mathrm {~ms} ^ { - 1 }\).
  2. Find the position vector of the object when \(t = 4\).
OCR MEI M1 Q1
7 marks Moderate -0.5
1 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 Q2
6 marks Moderate -0.5
2 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 \(t\).