Contents

- 1 Derive Equations for Distance-Time Graphs and Other Types of Graphs
- 1.1 (i) Distance-Time Graph for Uniform Speed (or Constant Speed)
- 1.2 (ii) Distance-Time Graph for Non-uniform Speed (or Changing Speed)
- 1.3 (iii) Distance-Time Graph When the Object is Stationary (Not Moving)
- 1.4 How to Draw Distance-Time Graphs
- 1.5 Advantages of Distance-Time Graphs
- 1.6 Other Types of Graphs

Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.

## Derive Equations for Distance-Time Graphs and Other Types of Graphs

The motion of objects (like speed) can be represented in diagram form by drawing their distance-time graphs. A distance-time graph shows how the distance travelled by a moving object changes with time. This point will become clear from the following examples.

### (i) Distance-Time Graph for Uniform Speed (or Constant Speed)

When an object moves with a uniform speed (or constant speed), it travels ‘equal distances’ in ‘equal intervals of time’. So, the distance travelled is directly proportional to time. When the two things are directly proportional to each other, then the graph drawn between them is a straight line. Thus, the distance-time graph of an object moving with a uniform speed (or constant speed) is a straight line (as shown by the line OA in Figure).

The straight line graph OA in Figure is sloping upwards showing that as the time increases, the distance travelled by the object increases in the same proportion. So, if the distance-time graph of an object is a straight line, it indicates that the object is moving with a uniform speed (or constant speed). Its speed is not changing at all.

The slope (or slant) of a distance-time graph indicates the speed of the object. If the distance-time graph has a low slope (it is less steep), then the object has low speed. On the other hand, if the distance-time graph has a higher slope (it is more steep), then the object has a higher speed.

So, just by looking at the slopes of two distance-time graph lines drawn on the same paper, we can tell which of the two objects is moving faster and which is moving slower.

This is because greater the slope of a distance-time graph, the higher will be the speed. In Figure, two distance-time graphs OA and OB have been drawn for two moving objects. Now, graph OA has a smaller slope (it makes smaller angle with x-axis), so graph OA represents a lower speed.

The graph OB has a greater slope (it makes a greater angle with x-axis), therefore, graph OB represents a higher speed. So, we can now say that the object (say, a car) having distance-time graph OB is moving faster than another object (or another car) whose distance-time graph is OA (see Figure).

### (ii) Distance-Time Graph for Non-uniform Speed (or Changing Speed)

When an object moves with a non-uniform speed (or changing speed), it travels ‘unequal distances’ in ‘equal intervals of time’. In this case, the distance travelled by object is not directly proportional to time and hence the distance-time graph is not a straight line. The distance-time graph for an object moving with non-uniform speed (or changing speed) is a curved line (as shown by line OA in Figure).

Thus, a curved line graph between distance and time tells us that the object is moving with a non-uniform speed (or changing speed). The distance-time graph in the form of a curved line tells us that the object is moving with a speed which is not constant.

### (iii) Distance-Time Graph When the Object is Stationary (Not Moving)

After travelling a certain distance, when a moving object stops moving (or becomes stationary), then the distance travelled by it does not change with time, it remains the same (or constant). The distance-time graph for an object which stops moving (or becomes stationary) is a straight line parallel to the time-axis (or x-axis) (as shown by line AB in Figure).

When an object becomes stationary, its speed is 0 (zero). So, a straight line graph parallel to the time-axis shows that the speed of object is zero. We can see from Figure that the distance-time graph for a stationary object is a horizontal line (see line AB).

Thus, a straight-line graph parallel to the time-axis (or a horizontal line graph) tells us that the object has become stationary. It is not moving. The graph given in Figure shows the shape of the distance-time graph for a car parked on a side road (when it has stopped moving and become stationary).

Please note that before becoming stationary, the car has already travelled a certain distance (represented by OA in Figure).

We get the following conclusions from the above discussion on distance-time graphs for a moving object (say, a moving car) : –

- If the distance-time graph of an object is a straight line, then it is moving with a constant speed (The greater the slope of distance-time graph, the greater the speed).
- If the distance-time graph of an object is not a straight line (it is a curved line), then the speed of object is not constant. The speed is changing.
- If the distance-time graph of an object is a horizontal line parallel to the time-axis, then the speed of object is zero. The object is not moving. It is stationary.

### How to Draw Distance-Time Graphs

We use a graph paper to draw distance-time graphs. The graph paper has 1 centimetre squares marked on it. Each centimetre square has 100 smaller squares in it (which are millimetre squares). It is obvious that the side of bigger square on the graph paper is 1 cm and that of the smaller square is 1 mm (see Figure).

In order to draw the distance-time graph for a moving object, we need a graph paper, and the readings of ‘distances travelled by the object’ and the corresponding ‘time values’ which have been obtained experimentally. The following points should be kept in mind while drawing distance-time graphs on graph paper :

(i) We should draw a horizontal line OX on the graph paper to represent x-axis (see Figure). Label the x-axis by writing ‘Time’. The unit of time should be written in bracket such as Time (s), Time (min) or Time (h) as the case may be. An arrow should be put in front of the labelling of time and its unit like Time (min) → (see Figure).

(ii) Draw a vertical line OY from point O on the left side of the graph paper to represent y-axis (see Figure). Label the y-axis by writing the word ‘Distance’. The unit of distance should be written in bracket such as Distance (m), Distance (cm) or Distance (km) as the case may be. An arrow should be put in front of the labelling of distance and its unit like Distance (km) → (see Figure). Please note that the point of intersection of x-axis and y-axis is called ‘origin’ and marked as O.

(iii) We should choose suitable scales so as to represent the large values of ‘time’ and ‘distance’ conveniently on the small graph paper. The scales to be used depend on the range of time and distance values, and hence vary from question to question.

(iv) Take the first value of ‘time’and the first value of ‘distance’ from the data given in the question and mark one point (as pencil dot) on the graph paper where the graph lines representing these two values meet. Then take second, third, fourth and fifth sets of values of ‘time’ and ‘distance’ and mark corresponding points (as pencil dots) on the graph paper.

(v) Join all the marked points (or pencil dots) with a pencil line to obtain the required distance-time graph (see Figure).

The drawing of distance-time graphs from the given values of distance and time will become more clear from the following examples.

**Example Problem 1.**

The distance travelled by a car at various times is shown below:

Draw graph of distance against time. What conclusion do you get about the motion of the car from this graph ?

**Solution:**

(i) We take the graph paper and draw a horizontal line OX (x-axis) and a vertical line OY (y-axis) at right angles to each other (see above Figure).

(ii) Write ‘Time (mm)’ on x-axis and ‘Distance (km)’ on y-axis and also put arrows with them.

(iii) In this problem we have only small time values (0, 1, 2, 3, 4 and 5 minutes) to represent. So, the scale to be used for showing time can be: 1 min 1 cm. We mark the time values 0, 1, 2, 3, 4 and 5 on. the line OX as shown in Figure 8. Again, the distance values given in this problem are small (0, 1, 2, 3, 4 and 5 km). So, the scale to be used for representing distance values on graph can be : 1 km = 1 cm. We now mark the distance values 0, 1, 2, 3, 4 and 5 on the line OY (see Figure).

(iv) The first reading given in this problem is Time = 0 and Distance = 0. The point 0 (called origin) represents the 0 (zero) values both for time and distance. Thus, at point 0 on graph paper, time is 0 and distance is also 0. The second reading is Time = 1 min and Distance = 1 km.

Now, the vertical line above the 1 min mark on the graph paper and horizontal line on the right side of 1 km mark on graph paper cross at point A (see Figure). So, we put a pencil dot at point A. Similarly, the third, fourth, fifth and sixth readings of time and the corresponding readings of distance will give us points B, , D and E on the graph paper which are marked as pencil dots (see Figure).

(v) We join the point 0 and the dots at points A, B, C, D and E with a pencil line. We will get a straight line graph OE (see Figure). This is the required distance-time graph for the motion of the car.

Since the distance-time graph for the motion of the car is a straight line, we conclude that the car is moving with a constant speed (or uniform speed).

**Example Problem 2.**

A bus starts from rest at 8.00 AM. The distances covered by this bus at various instants of time are as follows :

(a) Draw the distance-time graph for the bus.

(b) What distance was covered by the bus at 9.45 AM ?

**Solution.**

(a) Let us choose the scale for the time readings first. The first time reading is 8.00 AM and the second time reading is 8.30 AM, which is after an interval of 30 minutes. So, let us choose the scale : 15 min = 1 cm (on graph paper). We mark the origin of graph at 8.00 AM and write other time readings at 2 cm apart from one another on the x-axis (as shown in Figure).

We will now choose the scale for distance readings. The distances given are in the range of 0 to 80 km. Now, the first distance reading is 0 km and the second reading is 20 km. So, we choose the scale : 10 km = 1 cm (on graph paper). The first reading of distance, 0 km, is marked at the origin and the other distance readings are marked at 2 cm apart from one another on the y-axis (as shown in Figure).

When we draw all the time readings and the corresponding distance readings on the graph paper, we get five points O, A, B, C and D on the graph paper (see Figure). On joining all these points, we get a straight line graph (OD) for the motion of the bus.

(b) In order to find the distance covered by the bus at time 9.45 AM, we mark a point P on the x-axis midway between the time readings of 9.30 AM and 10.00 AM. Point P represents the time of 9.45 AM. From point P, we draw a perpendicular line PQ which cuts the straight line graph at point Q (see Figure).

Now, from point Q we draw a line QR parallel to the x-axis which meets the y-axis at point R (see Figure). The point R is midway between the distance readings of 60 km and 80 km. So, point R will represent a distance which is exactly half way between 60 km and 80 km. We can see from the graph in Figure 9 that point R shows a distance of 70 km. Thus, the distance covered by the bus at 9.45 AM is 70 km.

### Advantages of Distance-Time Graphs

The presentation of experimentally obtained ‘distance’ and ‘time’ values for a moving object in the form of a distance-time graph has many advantages as compared to the presentation of this data in a table form. Some of the advantages of distance-time graphs are given below :

1. The variation of distance travelled by an object with time can be seen more easily from a distance¬time graph than from the distance and time values given in the table form. For example, a straight line distance-time graph tells us that the moving object covers equal distances in equal time intervals, so its speed is constant (or uniform).

On the other hand, a curved line distance-time graph tells us that the moving object covers unequal distances in equal time intervals and hence its speed is not constant (it is non-uniform). Again, a straight line distance-time graph parallel to the time-axis (or a horizontal line graph) tells us that the distance moved by the object does not change with time, so its speed is zero (it is stationary).

2. The data given in table form may give information about the distance moved by the object only at certain definite time intervals but from a distance-time graph we can find the distance moved by the object at any point of time. For example, the data given in table form may tell us the distance moved by an object at say, 9.30 AM and 10.00 AM (but not at 9.45 AM). But from the distance-time graph of this object, we can easily find the distance moved by the object even at 9.45 AM.

3. The speed of an object can be obtained from its distance-time graph. This is because by using distance-time graph, we can find the distance moved by the object between any two time readings. And if we divide this distance by time (given by the difference in the two time readings), we will obtain speed of the object.

### Other Types of Graphs

The distance-time graphs which we have studied so far are called line graphs’. These line graphs show the variation of distance travelled by a moving object (car, bus, truck, etc.) with time. A line graph can also be drawn to show the variation of any other two related quantities (such as the variation of weight of a man with age). In addition to line graphs, there are two other kinds of graphs. These are : bar graphs and pie chart.

A bar graph is a diagram which shows information as bars (thin rectangles) of different heights. In a bar graph, the positions and heights of the bars represent the values of the variable quantity about which information is being given. A bar graph showing the runs scored by a cricket team in six overs of a match is given in Figure.

This bar graph tells us that the 1st over has given only 1 run, the 2nd over has yielded 5 runs, the 3rd over has produced 3 runs, the 4th over has given 10 runs, the 5th over has yielded only 2 runs whereas the 6th over has produced 6 runs. Bar graphs are frequently used in newspapers, magazines and TV to present various types of information in an interesting way. A bar graph is also known as a ‘histogram’.

A pie chart is a kind of graph or diagram which shows the percentage composition of ‘something’ in the form of slices of a circle (the whole circle representing 100 per cent). A pie chart showing the percentage composition of air is given in Figure. This pie chart shows the composition of air to be : Nitrogen 78 per cent, Oxygen 21 per cent, and Other gases 1 percent.