Displacement-time graph interpretation or sketching

3 \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is \(s\) metres at time \(t\) seconds after leaving a fixed point \(O\). The particle starts from rest and passes through points \(P , Q\) and \(R\), at times \(t = 5 , t = 10\) and \(t = 15\) respectively, and returns to \(O\) at time \(t = 20\). The distances \(O P , O Q\) and \(O R\) are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from \(t = 0\) to \(t = 20\). The graph consists of two curved segments \(A B\) and \(C D\) and two straight line segments \(B C\) and \(D E\).

1. Find the speed of the particle between \(t = 5\) and \(t = 10\).
2. Find the acceleration of the particle between \(t = 0\) and \(t = 5\), given that it is constant.
3. Find the average speed of the particle during its motion. \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings \(A C\) and \(B C\), of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle \(A C B = 90 ^ { \circ }\). There is a horizontal force of magnitude \(F \mathrm {~N}\) acting on the block. The block is in equilibrium.

5 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).

1. Sketch the velocity-time graph for the journey.
2. Find the velocity at \(t = 80\).
3. Find the total distance for the journey.
4. Find the acceleration of the car for \(0 < t < 80\).

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] \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) \(t = 3.25\).

1 Fig. 1 shows the velocity-time graph of a cyclist travelling along a straight horizontal road between two sets of traffic lights. The velocity, \(v\), is measured in metres per second and the time, \(t\), in seconds. The distance travelled, \(s\) metres, is measured from when \(t = 0\). \begin{figure}[h] \end{figure}

1. Find the values of \(s\) when \(t = 4\) and when \(t = 18\).
2. Sketch the graph of \(s\) against \(t\) for \(0 \leqslant t \leqslant 18\).

5 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle \(P\) can move in a straight line on a horizontal surface. At time \(t\) seconds the displacement of \(P\) from a fixed point \(A\) on the line is \(x \mathrm {~m}\). The diagram shows the \(( t , x )\) graph for \(P\). In the interval \(0 \leqslant t \leqslant 10\), either the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\), or \(P\) is at rest.

1. Show by calculation that \(T = 1.75\).
2. State the velocity of \(P\) when
   1. \(t = 2\),
   2. \(t = 8\),
   3. \(t = 9\).
   4. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 10\). For \(t > 10\), the displacement of \(P\) from \(A\) is given by \(x = 20 t - t ^ { 2 } - 96\).
   5. Calculate the value of \(t\), where \(t > 10\), for which the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).

5 The graph shows displacement \(s m\) against time \(t \mathrm {~s}\) for a model of the motion of a bead moving along a straight wire. The points \(( 0,4 ) , ( 2,7 ) , ( 5,7 )\) and \(( 9 , - 7 )\) are the endpoints of the line segments. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-4_741_1301_404_239}

1. Find an expression for the displacement of the bead for \(0 \leqslant t \leqslant 2\).
2. Sketch the velocity-time graph for this model.
3. Explain why the model may not be suitable at \(t = 2\) and \(t = 5\).

1 A particle moves along a straight line. The displacement \(s \mathrm {~m}\) at time \(t \mathrm {~s}\) is shown in the displacementtime graph below. The graph consists of straight line segments joining the points \(( 0 , - 2 ) , ( 10,5 )\) and \(( 15,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-04_641_848_641_242}

1. Find the distance travelled by the particle in the first 15 s .
2. Calculate the velocity of the particle between \(t = 10\) and \(t = 15\).

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] \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) \(t = 3.25\).

\includegraphics{figure_1} The displacement of a particle at time \(t\) s after leaving a fixed point \(O\) is \(s\) m. The diagram shows a displacement-time graph which models the motion of the particle. The graph consists of 4 straight line segments. The particle travels 50 m in the first 10 s, then travels at \(2\) m s\(^{-1}\) for a period of 10 s. The particle then comes to rest for a period of 20 s, before returning to its starting point when \(t = 60\).

1. Find the velocity of the particle during the last 20 s of its motion. [2]
2. Sketch a velocity-time graph for the motion of the particle from \(t = 0\) to \(t = 60\). [3]

A sprinter runs a race of \(200 \text{ m}\). His total time for running the race is \(20 \text{ s}\). He starts from rest and accelerates uniformly for \(6 \text{ s}\), reaching a speed of \(12 \text{ m s}^{-1}\). He maintains this speed for the next \(10 \text{ s}\), before decelerating uniformly to cross the finishing line with speed \(V \text{ m s}^{-1}\).

1. Find the distance travelled by the sprinter in the first \(16 \text{ s}\) of the race. Hence sketch a displacement-time graph for the \(20 \text{ s}\) of the sprinter's race. [6]
2. Find the value of \(V\). [2]

A particle \(P\) moves in a straight line such that its displacement from a fixed point \(O\) at time \(t\) s is \(y\) metres. The graph of \(y\) against \(t\) is as shown.

1. Write down the velocity of \(P\) when
   1. \(t = 1\), \quad (ii) \(t = 10\). \hfill [2 marks]
2. State the total distance travelled by \(P\). \hfill [2 marks]
3. Write down a formula for \(y\) in terms of \(t\) when \(2 \leq t < 4\). \hfill [3 marks]
4. Sketch a velocity-time graph for the motion of \(P\) during the twelve seconds. \hfill [3 marks]
5. Find the maximum speed of \(P\) during the motion. \hfill [3 marks]

A man travels \(360\) m along a straight road. He walks for the first \(120\) m at \(1.5\) m s\(^{-1}\), runs the next \(180\) m at \(4.5\) m s\(^{-1}\), and then walks the final \(60\) m at \(1.5\) m s\(^{-1}\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \((t, x)\) graph for the journey, showing the values of \(t\) for which \(x = 120, 300\) and \(360\). [5]

A woman jogs the same \(360\) m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.

1. Draw a dotted line on your \((t, x)\) graph to represent the woman's journey. [1]
2. Calculate the value of \(t\) at which the man overtakes the woman. [5]

\includegraphics{figure_3} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s. The woman runs at a constant speed throughout. Fig. 1 shows the \((t, v)\) graph for the woman.

1. Find the total distance run by the woman. [3]
2. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), [3]

\includegraphics{figure_4} At time \(t = 0\), a child also starts to move, from \(A\), along \(AB\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s. Fig. 2 shows the \((t, v)\) graph for the child; it consists of two straight line segments.

1. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s. [3]
2. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant. [3]

A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}

A car, starting from rest at a point \(A\), travels along a straight horizontal road towards a point \(B\). The distance between points \(A\) and \(B\) is 1·9 km. Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of 26 ms\(^{-1}\). The car continues at 26 ms\(^{-1}\) for 1 minute, before decelerating at a constant rate of 0·75 ms\(^{-2}\) until it passes the point \(B\).

1. Show that the car travels 156 m while it is accelerating. [2]
2. 1. Work out the distance travelled by the car while travelling at a constant speed. [1]
   2. Hence calculate the length of time for which the car is decelerating until it passes the point \(B\). [5]
3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\). [3]