Distance–Time Graphs
Introduction


Information about movement can be presented in a number of ways. Data for the total distance moved at different times during movement can be recorded in a table. Alternatively, the same information can be presented in a graph.


Time / s Distance from
start / m
0 0
2 20
4 40
6 60
8 80



The table and graph both present information about an object's position at certain times. In this unit we will be looking at how such data can be interpreted and how additional quantities can be determined from graphs.

Motion sensors
A number of different techniques can be used for recording the way distance changes over time. One method commonly used in the school laboratory is the ultrasonic motion
sensor
Sensors used in electronics produce a change in their resistance when some feature of their surrounding environment changes. The resistance of a thermistor changes as the surrounding temperature alters.
sensor
.

Figure 1.   Using a motion sensor.
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This device emits a pulse of ultrasound which travels through the air at 330 ms−1 before rebounding off a target. The distance of the reflecting object from the sensor can be found by measuring the time interval between the emitted and reflected pulses and using the following equation:

How long will a pulse of ultrasound take to travel from the emitter to the trolley if the trolley is 1.65 m away from the motion sensor?
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What is the approximate delay between emitted and reflected pulses when the trolley is 1.65 m away from the emitter?
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The distance can be measured at regular time intervals, as the trolley moves, and a graph plotted. This type of measurement is best done by a computer where the rapid timings and calculations can be done quickly and results displayed on the monitor.

Other techniques are available for detecting distance and speed. A 'radar gun' emits a pulse of
electromagnetic
Electromagnetic waves, such as light, are made up from oscillating electric and magnetic fields. Because of this, they are self-propagating and can travel through a vacuum. All types of electromagnetic wave travel at the same speed in a vacuum, 3 × 108 ms−1.
electromagnetic
radiation with a certain
frequency
The wave frequency f is the number of complete waves passing any point each second. Frequency is measured in hertz, Hz.
frequency
. A passing vehicle will reflect the pulse back to a detector on the gun. The frequency of the reflected radiation will be dependent upon the speed of the moving reflector.

The speed of the moving reflector can be determined by measuring differences between the emitted and reflected pulses.
Distance–time graphs
The graph in Fig.2 shows a set of distance and time axes. Its y-axis shows the horizontal distance of the helicopter from its starting position. (This is indicated on the animation by the length of the line between the take-off and landing positions).

Start the helicopter in Fig.2 flying and drag it horizontally towards the landing pad.
Figure 2.   Helicopter motion
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As the helicopter flies towards the landing pad, the distance from its starting point increases and so the line on the graph moves from the bottom left-hand side of the axes towards the top right.

Use the animation in Fig.2 to answer the following questions.

The horizontal distance between the take-off and landing pads is approximately …
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If the helicopter moves quickly from the take-off position to the landing pad the line on the graph has …
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If during the journey the helicopter returns partly back towards the starting point the graph for that part of the journey has …
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Figure 3.   Graph for a helicopter moving at constant speed.
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A line drawn through the points on a graph, such as that shown in Fig.3, allows us to see trends. We can also use the line in the graph to estimate values that would lie between data points given in a results table, so a graph of Fig.3 contains more information than the table below.


Time / s Distance from start / m
0 0
2 20
4 40
6 60
8 80

Figure .  Distance versus time data.



Clearly in Fig.3 the helicopter is moving away from the starting point so the graph has a positive gradient. Additionally, the distance from the reference point increases in equal steps as time passes. This means that the graph is a straight line and therefore has a constant or uniform gradient.

The size of the gradient shows how quickly the distance travelled is changing with time. In this example the distance is changing at a rate of 10 m every second. Consequently we can say that this graph records the motion of an object moving with a constant speed of 10 ms−1.

The balls shown in Fig.5 are moving with different speeds but the speed of each is constant.


Click on the figure below to interact with the model.

 Figure 5.  Bouncing balls.

Which ball in Fig.5 is moving faster?
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Complete the following statements to summarize the behaviour of the balls in Fig.5.

  • The fact that the graph for each ball is a line indicates that the speed of each ball is . The faster ball can be identified by the line with the gradient.
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A straight line on a distance–time graph indicates that an object is moving at a steady speed. The steeper the line, the faster the speed. Both balls in the simulation of Fig.5 are moving with a steady speed and in a fixed direction. Therefore the velocities as well as the speeds are constant. However, in order to state the
velocity
An object's velocity states both the speed and direction of motion relative to a fixed reference point.
velocity
we would have to indicate the direction of movement in addition to the speed.

Grab the ball in the simulation of Fig.6 and drag it to the top of its range of motion. Release the ball and observe its motion and the distance versus time graph as it falls.


Click on the figure below to interact with the model.

 Figure 6.  Dropping ball.

When the ball is falling towards the ground, the distance values plotted on the graph are …
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The distance values in the graph of Fig.6 decrease as time passes because the ground is being used as the point from which distances are measured. The ball falling in Fig.6 is moving towards the reference point, so the measured distances are decreasing with time.

Finding the speed
Fig.7 shows the distance–time graph for an object speeding up as it moves away from its reference point. The curve of the line shows that the ball is getting faster as time passes. To find the speed at any instant during this motion we would have to draw a tangent to the curve and determine its gradient.

Figure 7.   Finding the speed at any instant during the motion.
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In Fig.7 the speed of the object 6 seconds after the start of the motion is approximately …
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When the object in Fig.7 is 35 m away from its starting position, it is travelling with a speed of approximately …
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The object in Fig.7 is moving at a speed of approximately 1 ms−1 after travelling for a time of …
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On distance–time graphs, bigger speeds are indicated by steeper lines. We can regard the curve in Fig.7 as being made up from a large number of short straight lines each having a slightly different gradient.

The curve in Fig.7 gets steeper as the distance increases, showing that the object is moving progressively faster as it moves away from the observation point. This object is accelerating away from its starting position.

When the mass in Fig.8 falls the trolley accelerates and the distance–time graph for the
acceleration
An object's acceleration is its rate of change of velocity.
acceleration
is plotted.

Figure 8.   An increasing acceleration.
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By clicking on the right-hand green button in Fig.8, you can switch from a trolley towed by a falling mass to one towed by a falling chain. The falling mass and the falling chain produce different distance–time graphs because they represent different types of acceleration.

When the trolley is being towed by the falling mass its speed increases uniformly. The falling mass produces a constant acceleration. As the chain falls, the towing force increases and the increase in the speed each second is not uniform. The falling chain produces an increasing acceleration.

Displacement–time graphs
In most of the situations we will meet, movement is in a straight line and so displacements from the starting point are easy to calculate. The label on the y-axis of Fig.9 shows that an increase in the y-axis value represents motion due north.

Figure 9.   Displacement–time graph.
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How far north does the object move in the first 2 seconds?
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Calculate the average speed of the moving object during the first 3 seconds.
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Three seconds after the beginning of the motion of Fig.9 the object is …
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How far is the object away from its starting position six seconds after the start of its motion?
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Figure 10.   Displacement–time graph.
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The object whose motion is represented by the graph in Fig.10 moves north at a steady speed of 3 ms−1 for 2 seconds. After this time the
displacement
An object's displacement quotes both its bearing and distances relative to a fixed reference point.
displacement
decreases as the object starts to move in the opposite direction. After 6 seconds the object is just 3 m north of its starting point even though it has travelled a total distance of 9 m.

The average speed of the object whose motion is represented by the graph of Fig.10 for its entire journey is …
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The object's velocity after 6 seconds is determined from the gradient of the line at that point. The size of the gradient is 0.75 ms−1, so after 6 seconds the velocity of the object is 0.75 ms−1 in a southerly direction.

Match graphs
In the set-up of Fig.11 the position of the trolley relative to the motion sensor is plotted. Move the trolley to match the motion described on the preset graph drawn and then answer the questions below. There are a number of preset graphs that you might want to investigate.

Figure 11.   Match graphs.
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Complete the following statements.

  • Moving the trolley to the left causes the measured distance to while motion to the right causes the distance to . Slow motion produces a line with a slope while fast movement produces a line. Motion to the left produces a line with a gradient and motion to the right results in a line with a gradient.
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Summary


Drawing a line on a graph shows the trends in data presented in a table. Additional information can be determined from the slope of the graph.

The size of the slope or the gradient of a distance versus time graph gives the speed at which an object is moving.

The gradient of a displacement versus time graph indicates an object's velocity.

Exercises
Figure 12.  
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1. Fig.12 shows the speed of a train as it travels along a certain section of track.

  • What time does the train take to travel along this section of track?
     s   (to the nearest whole number)

    What is the total length of this section of track?
     m   (to the nearest whole number)

    What is the average speed of the train during its journey along this section of track?
     ms−1   (to 1 d.p.)
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Figure 13.  
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2. Fig.13 shows the speed–time graph for an object starting from rest and increasing its speed as it travels.

  • What is the speed of the object 5 seconds after starting?
     ms−1   (to the nearest whole number)

    What is the average speed over the first 5 seconds?
     ms−1   (to 1 d.p.)

    What distance does the object travel during the first 5 seconds?
     ms−1   (to 1 d.p.)

    What is its average speed between times of 5 and 10 seconds?
     ms−1   (to the nearest whole number)
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Figure 14.  
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3. Fig.14 shows a distance–time graph for a toy car being pushed uphill towards an observer.

  • How far is the observer from the point where the car starts?
     m   (to the nearest whole number)

    How far does the car travel between 2 seconds and 6 seconds after release?
     m   (to the nearest whole number)
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Figure 15.  
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4. In an experiment, as shown in Fig.15, a ball, held directly underneath a motion sensor, is released. A pupil makes the following statements about the graph. Decide whether they are true or false.
  • At point P the ball is held directly underneath the motion sensor.
    Between Q and R the ball is in contact with the ground.
    Between S and T the ball is rising.
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