In the previous section, we learned about Acceleration. In this section we will see the Graphical representations related to Motion. We have already discussed about graphs in our maths classes. Details here.
Consider an object in motion. We are going to make a distance-time graph for that object. For making that graph, we need some readings. So let us get those readings first:
1. The line in fig.1.10(a) below shows the path on which the object moves.
2. In that path, mark a point ‘O’ on the ground. This is the first mark. It is shown in fig.1.10(b).
3. Make an number of markings A, B, C etc., at convenient distances.
4. One person should be present at each of the marked points. And each of them should have a stop watch. When these arrangements are complete, the object can begin it’s journey along the path.
5. The object begins it's journey at some distance before O. At the instant when the object passes O, all persons should start their stop watches. The experiment has begun. Now, all that we have to do is this:
6. Record the exact time instant at which the object passes each of the markings.
This is a simple process which can be done as follows:
■ Consider the person at ‘A’.
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘A’, he stops the watch.
(iii) He records the reading of the watch. Let it be 3 minutes. He has done his work.
(iv) We can infer that the object took 3 minutes to travel from O to A.
■ All the other stop watches are still running. Consider the person at ‘B’
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘B’, he stops the watch.
(iii) He records the reading of the watch. Let it be 7 minutes. He has done his work.
(iv) We can infer that the object took 7 minutes to travel from O to B.
■ All other stop watches beyond B are still running. Consider the person at ‘C’
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘C’, he stops the watch.
(iii) He records the reading of the watch. Let it be 12 minutes. He has done his work.
(iv) We can infer that the object took 12 minutes to travel from O to C.
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In this way we will obtain the time at all the points.
7. Now we need distances. For that, measure OA, AB, BC, CD, . . .
From this, we can calculate the distance of each point from the origin ‘O’. This is done as follows:
OA = OA
OB = OA + AB
OC = OB + BC
OD = OC + CD
So on..
Note that all the distances are to be measured along the path. For this experiment, we do not want the straight line distances from ‘O’.
8. Now we tabulate the results. It is shown in table 1.1 below:
9. Using the above table, we can plot the graph.
• Time is taken along the x-axis and distance is taken along the y-axis
• The resulting graph is shown below:
10. All the points are plotted on the graph paper. When those points are joined together, a straight line is obtained.
• In other words, all the points obtained from the experiment lie on a straight line.
• This line is shown in yellow colour.
• The reader may plot the above graph on a fresh graph paper. Any convenient scale can be used. The following scale is also adequate:
♦ X axis: 1 cm represents 1 minute
♦ Y axis: 1 cm represents 50 m
• What if we want to know some other distances?
• For example, What distance did the object travel in 15 seconds after passing 'O'
The answer can be obtained as follows:
1. Make a mark on 15 in the x-axis
2. Draw a vertical line (vertical means parallel to y-axis) through this mark
3. Let this vertical line intersect the graph at 'P'. The vertical line is shown in purple colour in fig.1.12 below
4. Draw a horizontal line (horizontal means parallel to x-axis) through 'P'
5. Mark the point where this horizontal line intersects the the y-axis
6. Take this reading on the y-axis. This reading is the required distance
7. In the above graph, the horizontal line meets the y-axis at 450. So we can confirm that, the object travelled 450 m in 15 seconds after it passed 'O'
Explanation:
• 'P' is like any other point on the yellow line.
• It's x coordinate will be the time since the object passed 'O'
• It's y coordinate will be the distance from 'O'
• In this problem, P has coordinates (15,450).
• So 450 m is the required answer
In the next section, we will see another application.
Consider an object in motion. We are going to make a distance-time graph for that object. For making that graph, we need some readings. So let us get those readings first:
1. The line in fig.1.10(a) below shows the path on which the object moves.
 |
| Fig.1.10 |
3. Make an number of markings A, B, C etc., at convenient distances.
4. One person should be present at each of the marked points. And each of them should have a stop watch. When these arrangements are complete, the object can begin it’s journey along the path.
5. The object begins it's journey at some distance before O. At the instant when the object passes O, all persons should start their stop watches. The experiment has begun. Now, all that we have to do is this:
6. Record the exact time instant at which the object passes each of the markings.
This is a simple process which can be done as follows:
■ Consider the person at ‘A’.
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘A’, he stops the watch.
(iii) He records the reading of the watch. Let it be 3 minutes. He has done his work.
(iv) We can infer that the object took 3 minutes to travel from O to A.
■ All the other stop watches are still running. Consider the person at ‘B’
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘B’, he stops the watch.
(iii) He records the reading of the watch. Let it be 7 minutes. He has done his work.
(iv) We can infer that the object took 7 minutes to travel from O to B.
■ All other stop watches beyond B are still running. Consider the person at ‘C’
(i) He sees the object coming towards him from a distance. So he gets ready.
(ii) At the instant when the object passes the marking at ‘C’, he stops the watch.
(iii) He records the reading of the watch. Let it be 12 minutes. He has done his work.
(iv) We can infer that the object took 12 minutes to travel from O to C.
- - -
- - -
In this way we will obtain the time at all the points.
7. Now we need distances. For that, measure OA, AB, BC, CD, . . .
From this, we can calculate the distance of each point from the origin ‘O’. This is done as follows:
OA = OA
OB = OA + AB
OC = OB + BC
OD = OC + CD
So on..
Note that all the distances are to be measured along the path. For this experiment, we do not want the straight line distances from ‘O’.
8. Now we tabulate the results. It is shown in table 1.1 below:
 |
| Table.1.1 |
• Time is taken along the x-axis and distance is taken along the y-axis
• The resulting graph is shown below:
 |
| Fig.1.11 |
• In other words, all the points obtained from the experiment lie on a straight line.
• This line is shown in yellow colour.
• The reader may plot the above graph on a fresh graph paper. Any convenient scale can be used. The following scale is also adequate:
♦ X axis: 1 cm represents 1 minute
♦ Y axis: 1 cm represents 50 m
So we have successfully drawn the graph. We will now see some of the uses of a distance-time graph:
• We know the distances travelled by the object in 3 seconds, 7 seconds, 12 seconds etc., after passing 'O'. They were recorded in the field, and are available in table 1.1 above• What if we want to know some other distances?
• For example, What distance did the object travel in 15 seconds after passing 'O'
The answer can be obtained as follows:
1. Make a mark on 15 in the x-axis
2. Draw a vertical line (vertical means parallel to y-axis) through this mark
3. Let this vertical line intersect the graph at 'P'. The vertical line is shown in purple colour in fig.1.12 below
 |
| Fig.1.12 |
5. Mark the point where this horizontal line intersects the the y-axis
6. Take this reading on the y-axis. This reading is the required distance
7. In the above graph, the horizontal line meets the y-axis at 450. So we can confirm that, the object travelled 450 m in 15 seconds after it passed 'O'
Explanation:
• 'P' is like any other point on the yellow line.
• It's x coordinate will be the time since the object passed 'O'
• It's y coordinate will be the distance from 'O'
• In this problem, P has coordinates (15,450).
• So 450 m is the required answer
In the next section, we will see another application.
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