Distance from velocity-time graph

3 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-3_727_899_267_625} A boy runs from a point \(A\) to a point \(C\). He pauses at \(C\) and then walks back towards \(A\) until reaching the point \(B\), where he stops. The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the boy's velocity at time \(t\) seconds after leaving \(A\). The boy runs and walks in the same straight line throughout.

1. Find the distances \(A C\) and \(A B\).
2. Sketch the graph of \(x\) against \(t\), where \(x\) metres is the boy's displacement from \(A\). Show clearly the values of \(t\) and \(x\) when the boy arrives at \(C\), when he leaves \(C\), and when he arrives at \(B\). [3]

3 \includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-2_314_1193_1366_276} \({ } ^ { P A } { } _ { P A } ^ { P B }\) \(P\) \(P\) \begin{verbatim} " \end{verbatim}

1 hour 15 minutes
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES. Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value for the acceleration due to gravity is needed, use \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.

3 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-2_520_719_1137_712} \(A\) and \(B\) are fixed points of a vertical wall with \(A\) vertically above \(B\). A particle \(P\) of mass 0.7 kg is attached to \(A\) by a light inextensible string of length \(3 \mathrm {~m} . P\) is also attached to \(B\) by a light inextensible string of length \(2.5 \mathrm {~m} . P\) is maintained in equilibrium at a distance of 2.4 m from the wall by a horizontal force of magnitude 10 N acting on \(P\) (see diagram). Both strings are taut, and the 10 N force acts in the plane \(A P B\) which is perpendicular to the wall. Find the tensions in the strings. [6]

6 A particle \(P\) of mass 0.2 kg is released from rest at a point 7.2 m above the surface of the liquid in a container. \(P\) falls through the air and into the liquid. There is no air resistance and there is no instantaneous change of speed as \(P\) enters the liquid. When \(P\) is at a distance of 0.8 m below the surface of the liquid, \(P\) 's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only force on \(P\) due to the liquid is a constant resistance to motion of magnitude \(R \mathrm {~N}\).

1. Find the deceleration of \(P\) while it is falling through the liquid, and hence find the value of \(R\). The depth of the liquid in the container is \(3.6 \mathrm {~m} . P\) is taken from the container and attached to one end of a light inextensible string. \(P\) is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of \(R \mathrm {~N}\) continues to act. The particle reaches the surface 4 s after leaving the bottom of the container.
2. Find the tension in the string.

7 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-4_333_1001_264_573} A light inextensible string of length 5.28 m has particles \(A\) and \(B\), of masses 0.25 kg and 0.75 kg respectively, attached to its ends. Another particle \(P\), of mass 0.5 kg , is attached to the mid-point of the string. Two small smooth pulleys \(P _ { 1 }\) and \(P _ { 2 }\) are fixed at opposite ends of a rough horizontal table of length 4 m and height 1 m . The string passes over \(P _ { 1 }\) and \(P _ { 2 }\) with particle \(A\) held at rest vertically below \(P _ { 1 }\), the string taut and \(B\) hanging freely below \(P _ { 2 }\). Particle \(P\) is in contact with the table halfway between \(P _ { 1 }\) and \(P _ { 2 }\) (see diagram). The coefficient of friction between \(P\) and the table is 0.4 . Particle \(A\) is released and the system starts to move with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the part \(A P\) of the string is \(T _ { A } \mathrm {~N}\) and the tension in the part \(P B\) of the string is \(T _ { B } \mathrm {~N}\).

1. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(a\).
2. Show by considering the motion of \(P\) that \(a = 2\).
3. Find the speed of the particles immediately before \(B\) reaches the floor.
4. Find the deceleration of \(P\) immediately after \(B\) reaches the floor. \end{document}

2 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_262_1004_760_575} The tops of each of two smooth inclined planes \(A\) and \(B\) meet at a right angle. Plane \(A\) is inclined at angle \(\alpha\) to the horizontal and plane \(B\) is inclined at angle \(\beta\) to the horizontal, where \(\sin \alpha = \frac { 63 } { 65 }\) and \(\sin \beta = \frac { 16 } { 65 }\). A small smooth pulley is fixed at the top of the planes and a light inextensible string passes over the pulley. Two particles \(P\) and \(Q\), each of mass 0.65 kg , are attached to the string, one at each end. Particle \(Q\) is held at rest at a point of the same line of greatest slope of the plane \(B\) as the pulley. Particle \(P\) rests freely below the pulley in contact with plane \(A\) (see diagram). Particle \(Q\) is released and the particles start to move with the string taut. Find the tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_487_696_1537_721} Each of three light inextensible strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(O\). Two of the strings pass over fixed smooth pegs and the particles hang freely in equilibrium. The weights of the particles and the angles between the sloping parts of the strings and the vertical are as shown in the diagram. It is given that \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that \(W \cos \alpha = 3.8\) and find the value of \(W \sin \alpha\).
2. Hence find the values of \(W\) and \(\alpha\).

4 A particle \(P\) starts from rest and moves in a straight line for 18 seconds. For the first 8 seconds of the motion \(P\) has constant acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently \(P\) 's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after the motion started, is given by $$v = - 0.1 t ^ { 2 } + 2.4 t - k ,$$ where \(8 \leqslant t \leqslant 18\) and \(k\) is a constant.

1. Find the value of \(v\) when \(t = 8\) and hence find the value of \(k\).
2. Find the maximum velocity of \(P\).
3. Find the displacement of \(P\) from its initial position when \(t = 18\).

5 A box of mass 8 kg is on a rough plane inclined at \(5 ^ { \circ }\) to the horizontal. A force of magnitude \(P \mathrm {~N}\) acts on the box in a direction upwards and parallel to a line of greatest slope of the plane. When \(P = 7 X\) the box moves up the line of greatest slope with acceleration \(0.15 \mathrm {~ms} ^ { - 2 }\) and when \(P = 8 X\) the box moves up the line of greatest slope with acceleration \(1.15 \mathrm {~ms} ^ { - 2 }\). Find the value of \(X\) and the coefficient of friction between the box and the plane.

6 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Particles \(P\) and \(Q\) have a total mass of 1 kg . The particles are attached to opposite ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest and \(Q\) hangs freely, with both straight parts of the string vertical. Both particles are at a height of \(h \mathrm {~m}\) above the floor (see Fig. 1). \(P\) is released from rest and the particles start to move with the string taut. Fig. 2 shows the velocity-time graphs for \(P\) 's motion and for \(Q\) 's motion, where the positive direction for velocity is vertically upwards. Find

1. the magnitude of the acceleration with which the particles start to move and the mass of each of the particles,
2. the value of \(h\),
3. the greatest height above the floor reached by particle \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-4_668_848_260_653} A small block of mass 3 kg is initially at rest at the bottom \(O\) of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). A force of magnitude 35 N acts on the block at an angle \(\beta\) above the plane, where \(\sin \beta = 0.28\) and \(\cos \beta = 0.96\). The block starts to move up a line of greatest slope of the plane and passes through a point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O A\) is 12.5 m (see diagram).

1. For the motion of the block from \(O\) to \(A\), find the work done against the frictional force acting on the block.
2. Find the coefficient of friction between the block and the plane. At the instant that the block passes through \(A\) the force of magnitude 35 N ceases to act.
3. Find the distance the block travels up the plane after passing through \(A\). \end{document}

1. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{3a8395fd-6e44-48a1-8c97-3365a284956a-02_404_755_312_577} Figure 1 shows the speed-time graph of a cyclist moving on a straight road over a 7 s period. The sections of the graph from \(t = 0\) to \(t = 3\), and from \(t = 3\) to \(t = 7\), are straight lines. The section from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis. State what can be deduced about the motion of the cyclist from the fact that

1. the graph from \(t = 0\) to \(t = 3\) is a straight line,
2. the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
3. Find the distance travelled by the cyclist during this 7 s period.

2. \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-04_269_1175_296_375}

2 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_714_1048_1231_552} A particle starts from the point A and travels in a straight line. The diagram shows the ( \(\mathrm { t } , \mathrm { v }\) ) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leqslant t \leqslant 290\).

1. Find the value of ther which the distance of the particle from A is greatest.
2. Find the displacement of the particle from A when \(\mathrm { t } = 290\).
3. Find the total distance travelled by the particle during the interval \(0 \leqslant \mathrm { t } \leqslant 290\).

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10 \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT

1 A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the particle in the interval \(0 < t < 6\).
2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
3. When \(t = 0\) the particle is at A . Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\).

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] \end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).

1 The velocity-time graph shown in Fig. 1 represents the straight line motion of a toy car. All the lines on the graph are straight. \begin{figure}[h] \end{figure} The car starts at the point A at \(t = 0\) and in the next 8 seconds moves to a point B .

1. Find the distance from A to B . \(T\) seconds after leaving A, the car is at a point C which is a distance of 10 m from B .
2. Find the value of \(T\).
3. Find the displacement from A to C .

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7(ii) (b

(ii) (b)

\section*{OCR} RECOGNISING ACHIEVEMENT

1 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h] \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.

1. \begin{figure}[h] \end{figure} Figure 1 shows the speed-time graph for the journey of a car moving in a long queue of traffic on a straight horizontal road. At time \(\mathrm { t } = 0\), the car is at rest at the point A .
The car then accelerates uniformly for 5 seconds until it reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) For the next 15 seconds the car travels at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then decelerates uniformly until it comes to rest at the point B.
The total journey time is 30 seconds.

1. Find the distance AB .
2. Sketch a distance-time graph for the journey of the car from A to B .

2 The graph shows how the velocity of a particle varies during a 50 -second period as it moves forwards and then backwards on a straight line. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-2_615_1312_1007_301}

1. State the times at which the velocity of the particle is zero.
2. Show that the particle travels a distance of 75 metres during the first 30 seconds of its motion.
3. Find the total distance travelled by the particle during the 50 seconds.
4. Find the distance of the particle from its initial position at the end of the 50 -second period.

2 The graph shows how the velocity of a train varies as it moves along a straight railway line. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-04_574_1595_402_203}

1. Find the total distance travelled by the train.
2. Find the average speed of the train.
3. Find the acceleration of the train during the first 10 seconds of its motion.
4. The mass of the train is 200 tonnes. Find the magnitude of the resultant force acting on the train during the first 10 seconds of its motion.