3.02c Interpret kinematic graphs: gradient and area

1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$

1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the total distance travelled by the particle during the 15 seconds.

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

7 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] \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 \(3 \leqslant t \leqslant 4\).

7 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q . Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h] \end{figure} Use model A to calculate the following.

1. The acceleration of the ring when \(t = 0.5\).
2. The displacement of the ring from Q when
   (A) \(t = 2\),
   (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B .
4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .

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.

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 \(4 \mathrm {~ms} ^ { - 1 }\).

6. A parachutist drops from a helicopter \(H\) and falls vertically from rest towards the ground. Her parachute opens 2 s after she leaves \(H\) and her speed then reduces to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 2 s her motion is modelled as that of a particle falling freely under gravity. For the next 5 s the model is motion with constant deceleration, so that her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of this period. For the rest of the time before she reaches the ground, the model is motion with constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Sketch a speed-time graph to illustrate her motion from \(H\) to the ground.
   (3 marks)
2. Find her speed when the parachute opens.
   (2 marks)
   A safety rule states that the helicopter must be high enough to allow the parachute to open and for the speed of a parachutist to reduce to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before reaching the ground. Using the assumptions made in the above model,
3. find the minimum height of \(H\) for which the woman can make a drop without breaking this safety rule.
   (5 marks)
   Given that \(H\) is 125 m above the ground when the woman starts her drop,
4. find the total time taken for her to reach the ground.
5. State one way in which the model could be refined to make it more realistic.
   (1 mark)

1. At time \(t = 0\), a parachutist falls vertically from rest from a helicopter which is hovering at a height of 550 m above horizontal ground.

The parachutist, who is modelled as a particle, falls for 3 seconds before her parachute opens.
While she is falling, and before her parachute opens, she is modelled as falling freely under gravity. The acceleration due to gravity is modelled as being \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Using this model, find the speed of the parachutist at the instant her parachute opens. When her parachute is open, the parachutist continues to fall vertically.
   Immediately after her parachute opens, she decelerates at \(12 \mathrm {~ms} ^ { - 2 }\) for 2 seconds before reaching a constant speed and she reaches the ground with this speed. The total time taken by the parachutist to fall the 550 m from the helicopter to the ground is \(T\) seconds.
2. Sketch a speed-time graph for the motion of the parachutist for \(0 \leqslant t \leqslant T\).
3. Find, to the nearest whole number, the value of \(T\). In a refinement of the model of the motion of the parachutist, the effect of air resistance is included before her parachute opens and this refined model is now used to find a new value of \(T\).
4. How would this new value of \(T\) compare with the value found, using the initial model, in part (c)?
5. Suggest one further refinement to the model, apart from air resistance, to make the model more realistic.

1. A train travels along a straight horizontal track from station \(P\) to station \(Q\).

In a model of the motion of the train, at time \(t = 0\) the train starts from rest at \(P\), and moves with constant acceleration until it reaches its maximum speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The train then travels at this constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before finally moving with constant deceleration until it comes to rest at \(Q\). The time spent decelerating is four times the time spent accelerating.
The journey from \(P\) to \(Q\) takes 700 s .
Using the model,

1. sketch a speed-time graph for the motion of the train between the two stations \(P\) and \(Q\). The distance between the two stations is 15 km .
   Using the model,
2. show that the time spent accelerating by the train is 40 s ,
3. find the acceleration, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of the train,
4. find the speed of the train 572s after leaving \(P\).
5. State one limitation of the model which could affect your answers to parts (b) and (c).

1. \begin{figure}[h] \end{figure} Two children, Pat \(( P )\) and Sam \(( S )\), run a race along a straight horizontal track.
Both children start from rest at the same time and cross the finish line at the same time.
In a model of the motion:
Pat accelerates at a constant rate from rest for 5 s until reaching a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and then maintains a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\) until crossing the finish line. Sam accelerates at a constant rate of \(1 \mathrm {~ms} ^ { - 2 }\) from rest until reaching a speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains a constant speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until crossing the finish line. Both children take 27.5 s to complete the race.
The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together. Using the model,

1. explain why the areas under the two graphs are equal,
2. find the acceleration of Pat during the first 5 seconds,
3. find, in metres, the length of the race,
4. find the value of \(X\), giving your answer to 3 significant figures.

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 .

1. The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground.

The results are given in the table below with the time in seconds and the speed in \(\mathrm { ms } ^ { - 1 }\). Using all of this information,

1. estimate the length of runway used by the jet to take off. Given that the jet accelerated smoothly in these 25 seconds,
2. explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.

2. \begin{figure}[h] \end{figure} Figure 2 shows a speed-time graph for a model of the motion of an athlete running a \(\mathbf { 2 0 0 m }\) race in 24 s . The athlete

• starts from rest at time \(t = 0\) and accelerates at a constant rate, reaching a speed of \(10 \mathrm {~ms} ^ { - 1 }\) at \(t = 4\)
• then moves at a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\) from \(t = 4\) to \(t = 18\)
• then decelerates at a constant rate from \(t = 18\) to \(t = 24\), crossing the finishing line with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

Using the model,

1. find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,
2. find the distance covered by the athlete during the first 18s of the race,
3. find the value of \(U\).

9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Find the time, in seconds, for which the cyclist is decelerating.
2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
3. Find the time, in seconds, for which the cyclist is moving at constant velocity.

10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.

1. Sketch the velocity-time graph for Rory's run using this model.
2. Calculate \(T\).
3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
5. Sketch a velocity-time graph that represents Rory's run using this refined model.
6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)

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 \(t = 5\).

9 Two trains are travelling in the same direction on parallel straight tracks and train A overtakes train B . At time \(t\) seconds after the front of train A overtakes the front of train B the velocities of trains A and B are \(v _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(v _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. The velocity of train A is modelled by \(\mathrm { v } _ { \mathrm { A } } = 25 - 0.6 \mathrm { t }\). The velocity-time graph of train A is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-5_664_1399_550_242}

1. A student argues that the speed of train A changes by \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds so its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Comment on the validity of the student's argument.
2. When the front of train A overtakes the front of train B , train B has a velocity of \(10 \mathrm {~ms} ^ { - 1 }\). The acceleration of train \(B\) is constant and is modelled as \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Write down the equation for \(v _ { \mathrm { B } }\) in terms of \(t\) that models the velocity of train B .
3. Draw the velocity-time graph of train B on the copy of the diagram in the Printed Answer Booklet.
4. Determine the distance between the fronts of the trains at the time when the trains are travelling at the same velocity.
5. Explain why the model for train A would not be valid for large values of \(t\).

9 A car travelling in a straight line accelerates uniformly from rest to \(V \mathrm {~ms} ^ { - 1 }\) in \(T \mathrm {~s}\). It then slows down uniformly, coming to rest after a further \(2 T\) s.

1. Sketch a velocity-time graph for the car. The acceleration in the first stage of the motion is \(2.5 \mathrm {~ms} ^ { - 2 }\) and the total distance travelled is 240 m .
2. Calculate the values of \(V\) and \(T\).

7 A car is usually driven along the whole of a 5 km stretch of road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On one occasion, during a period of 50 seconds, the speed of the car is as shown by the speed-time graph in Fig. 7.
The rest of the 5 km is travelled at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h] \end{figure} How much more time than usual did the journey take on this occasion?
Show your working clearly.

14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds. In model A, the acceleration is assumed to be uniform.

1. Find an expression for the velocity of the car at time \(t\) using this model.
2. Explain why this model is not appropriate in the long term. Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
4. Calculate the displacement of the car in the first 20 seconds according to this refined model. In model B, the velocity of the car is given by $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8 \\ 17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
6. Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
7. Show that model B gives the same value as model A for the displacement at time 20 s .

5 A child is running up and down a path. A simplified model of the child's motion is as follows:

• he first runs north for 5 s at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\);
• he then suddenly stops and waits for 8 s ;
• finally he runs in the opposite direction for 7 s at \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Taking north to be the positive direction, sketch a velocity-time graph for this model of the child's motion.

Using this model,

calculate the total distance travelled by the child,

find his final displacement from his original position.

10 A ball is thrown upwards with a velocity of \(29.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the ball reaches its maximum height after 3 s .
2. Sketch a velocity-time graph for the first 5 s of motion.
3. Calculate the speed of the ball 5 s after it is thrown. A second ball is thrown at \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha ^ { \circ }\) above the horizontal. It reaches the same maximum height as the first ball.
4. Use this information to write down
   This second ball reaches its greatest height at a point which is 48 m horizontally from the point of projection.
5. Calculate the values of \(u\) and \(\alpha\).

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.

3 The diagram shows a velocity-time graph for a train as it moves on a straight horizontal track for 50 seconds. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-3_620_1221_408_358}

1. Find the distance that the train moves in the first 28 seconds.
2. Calculate the total distance moved by the train during the 50 seconds.
3. Hence calculate the average speed of the train.
4. Find the displacement of the train from its initial position when it has been moving for 50 seconds.
5. Hence calculate the average velocity of the train.
6. Find the acceleration of the train in the first 18 seconds of its motion.