In the previous section, we completed a discussion on units and measurements. In this chapter we will see motion of objects.
• We have already discussed the basics of the topic of 'objects in motion' in our previous classes. (Details here)
• The reader might want to revisit those details before taking up the present discussion.
■ In this chapter, we will deal with motion of objects along a single straight line only.
• Figs.3.1(a), (b) and (c) below, shows examples of motion along a single straight line.
■ Motion of objects along a straight line is called rectilinear motion.
• Motions indicated by figs.3.1(d) and (e) are not rectilinear motion.
■ For our present discussion, the objects under motion are considered as point objects.
• That is., size of those objects are neglected.
• But can we simply neglect the size of an object?
• Consider fig.3.2 below:
• A 3 m long car travels from A to B.
♦ The distance from A to B is 1 km (1000 m).
♦ Compared to the distance AB, the size of the car is very small.
♦ It appears as a point. It can be seen only by 'zooming in'
• Consider fig.3.3 below:
• The same 3 m long car travels from P to Q.
♦ The distance from P to Q is 10 m.
♦ This time the size of the car cannot be neglected.
♦ It does not appear as a point when compared to the distance PQ.
■ In most of the real life situations, the objects under consideration can be taken as point objects.
1. Consider a rectangular room in our school or home.
• Let one wall of the room be painted with yellow colour.
• Let an adjacent wall be painted with white colour.
♦ The white wall should be adjacent to the yellow wall
♦ The white wall should not be opposite to the yellow wall
• Let the floor be painted with brown colour.
This is shown in fig.3.4(a) below:
2. Consider the portion where the floor meets the yellow wall.
• They meet along a line. This line is an edge of the floor.
• The x axis can be assumed to lie along this edge.
3. Consider the portion where the yellow wall meets the white wall.
• They meet along a line. This line is a vertical edge of the room.
• The y axis can be assumed to lie along this edge.
4. Consider the portion where the floor meets the white wall.
• They meet along a line. This line is an edge of the floor.
• The z axis can be assumed to lie along this edge.
5. Thus we get the three axes: x, y and z
■ The point where all the three axes meet is called origin.
• It is denoted by the letter 'O'.
• It is a corner of the floor.
■ Also note:
• Any two perpendicular plane surfaces will meet along a line
♦ This line will be common to both the planes
• But if there are 3 mutually perpendicular planes, they can intersect only at a point.
♦ There will not be a line common to three such planes
■ It is very important to note that, the three axes are mutually perpendicular.
• This can be explained as follows:
Take any two axes randomly. The angle between them should be 90o.
• This 'perpendicular orientations' can be easily visualised, if we look carefully at a bottom corner of the rectangular room.
So now we know how to setup a frame of reference. Let us see how it can be put to practical use. We will write it in steps:
1. Consider any point object anywhere within the room.
• The object is at rest
2. Let someone ask us to give the position of that object
• We can easily write it's position with the help of the three axes. It is done as follows:
Case 1: The object is on an axis.
• This is shown in fig.3.4(b) above. The object is denoted by the magenta dot. It is on the x axis.
• It is at a distance of 2 m from the origin
• This distance should be measured along the x axis or parallel to the x axis
• So we can write the answer. The answer must contain the following 4 statements:
(i) The object is on the x axis
(ii) It is at a distance of 2 m from O
(iii) O is at the point of intersection of yellow wall, white wall and the floor
(iv) The x axis lies along the intersection of yellow wall and floor
The answer will not be complete if we do not write all the above four statements
• However, a simplified way to write the answer using two statements:
(i) The coordinates of the object is: (2,0,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• Note that, in the coordinates the values corresponding to y and z are zero. Because:
♦ The 'distance of object from origin' measured along y axis is zero
♦ The 'distance of object from origin' measured along z axis is zero
• In fact, we do not need any help from the y and z axes to write the position of a point on the x axis
• We can ignore those axes
In a similar way,
• If the object is on the y axis, at a distance of 'y' m from the origin, we write:
(i) The coordinates of the object is: (0,y,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• If the object is on the z axis, at a distance of 'z' m from the origin, we write:
(i) The coordinates of the object is: (0,z,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• The second and third statements are given in order to specify the exact position of the reference frame
• In this, we need to mention about the x axis only
• Because, once 'O' and 'x axis' are fixed, the other two axes also get fixed immediately. We will learn about it in later chapters
Case 2: The object is on a plane.
This is shown in fig.3.5(a) below. The object is denoted by the magenta dot. It is on the yellow wall.
• The yellow wall is bounded by the x and y axes
• So we can say: The object is on the 'xy plane'
• When measured parallel to the x axis, it is at a distance of 1.8 m from O
• When measured parallel to the y axis, it is at a distance of 0.5 m from O
• So we can write the answer. The answer must contain the following 3 statements:
(i) The coordinates of the object is: (1.8, 0.5, 0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
In a similar way,
• If the object is on the white wall (yz plane), we write the following 3 statements:
(i) The coordinates of the object is: (0, y, z)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• If the object is on the floor (xz plane), we write the following 3 statements:
(i) The coordinates of the object is: (x, 0, z)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
Case 3: The object is in 3-dimensional space. It is away from the axes and planes
• This is shown in fig.3.5(b) above. The object is denoted by the magenta dot. It is in 3-dimensional space.
• When measured parallel to the x axis, it is at a distance of 1.8 m from O
• When measured parallel to the y axis, it is at a distance of 0.5 m from O
• When measured parallel to the z axis, it is at a distance of 0.7 m from O
• So we can write the answer. The answer must contain the following 3 statements:
(i) The coordinates of the object is: (1.8, 0.5, 0.7)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
3. The above 3 are the only possible cases.
• We can effectively use such reference frames to specify the position of any given point object.
• We used the corner of a room as the origin. For out door experiments, we can take this reference frame outside and fix it up in any convenient orientation that is suitable.
• For example, we can assume the 'bottom corner of a bridge pillar' to be O and a suitable line drawn from it as the x axis
• When O and x axis are fixed, other two axes get fixed immediately.
• In this chapter, we are dealing with rectilinear motion.
• So all objects that we consider will be moving along a single line. They will not change direction.
• So in this chapter, we can assume that the objects are moving along the x axis.
• But we cannot ask a car for example, to move along the x axis drawn by us. So what do we do?
Ans: We take our reference frame and orient it in such a way that the x axis coincides with 'the straight line path on which the car moves'
• As a result, the coordinates will not have y and z values
♦ That is., all coordinates will be of the form: (x,0,0)
• So we need not write the coordinate notation (x,y,z) in this chapter. We can use a simpler form which can be explained with the help of an example. We will write the steps:
1. Consider fig.3.6 below:
• Only the x axis is shown. We do not need the y and z axis and so they are ignored.
2. This x axis is placed in such a way that, it coincides with the path of an object
• After placing the x axis in this way, a suitable point is marked on it as the origin O
3. When the object is at P, some one asks us:
What is the position of the object?
• The answer should contain 2 statements:
(i) The object is on the positive side of the x axis
(ii) The object is at a distance of 360 m from O
4. A simplified way to write the answer containing only one statement:
■ The position coordinate of the object is +360
5. When the object is at Q, some one asks us:
What is the position of the object?
We can write the answer as:
■ The position coordinate of the object is +240
6. When the object is at R, some one asks us:
What is the position of the object?
We can write the answer as:
■ The position coordinate of the object is -120
• We have already discussed the basics of the topic of 'objects in motion' in our previous classes. (Details here)
• The reader might want to revisit those details before taking up the present discussion.
■ In this chapter, we will deal with motion of objects along a single straight line only.
• Figs.3.1(a), (b) and (c) below, shows examples of motion along a single straight line.
 |
| Fig.3.1 |
• Motions indicated by figs.3.1(d) and (e) are not rectilinear motion.
■ For our present discussion, the objects under motion are considered as point objects.
• That is., size of those objects are neglected.
• But can we simply neglect the size of an object?
• Consider fig.3.2 below:
 |
| Fig.3.2 |
♦ The distance from A to B is 1 km (1000 m).
♦ Compared to the distance AB, the size of the car is very small.
♦ It appears as a point. It can be seen only by 'zooming in'
• Consider fig.3.3 below:
 |
| Fig.3.3 |
♦ The distance from P to Q is 10 m.
♦ This time the size of the car cannot be neglected.
♦ It does not appear as a point when compared to the distance PQ.
■ In most of the real life situations, the objects under consideration can be taken as point objects.
Frame of reference
We will write the steps to describe a frame of reference:1. Consider a rectangular room in our school or home.
• Let one wall of the room be painted with yellow colour.
• Let an adjacent wall be painted with white colour.
♦ The white wall should be adjacent to the yellow wall
♦ The white wall should not be opposite to the yellow wall
• Let the floor be painted with brown colour.
This is shown in fig.3.4(a) below:
 |
| Fig.3.4 |
• They meet along a line. This line is an edge of the floor.
• The x axis can be assumed to lie along this edge.
3. Consider the portion where the yellow wall meets the white wall.
• They meet along a line. This line is a vertical edge of the room.
• The y axis can be assumed to lie along this edge.
4. Consider the portion where the floor meets the white wall.
• They meet along a line. This line is an edge of the floor.
• The z axis can be assumed to lie along this edge.
5. Thus we get the three axes: x, y and z
■ The point where all the three axes meet is called origin.
• It is denoted by the letter 'O'.
• It is a corner of the floor.
■ Also note:
• Any two perpendicular plane surfaces will meet along a line
♦ This line will be common to both the planes
• But if there are 3 mutually perpendicular planes, they can intersect only at a point.
♦ There will not be a line common to three such planes
■ It is very important to note that, the three axes are mutually perpendicular.
• This can be explained as follows:
Take any two axes randomly. The angle between them should be 90o.
• This 'perpendicular orientations' can be easily visualised, if we look carefully at a bottom corner of the rectangular room.
So now we know how to setup a frame of reference. Let us see how it can be put to practical use. We will write it in steps:
1. Consider any point object anywhere within the room.
• The object is at rest
2. Let someone ask us to give the position of that object
• We can easily write it's position with the help of the three axes. It is done as follows:
Case 1: The object is on an axis.
• This is shown in fig.3.4(b) above. The object is denoted by the magenta dot. It is on the x axis.
• It is at a distance of 2 m from the origin
• This distance should be measured along the x axis or parallel to the x axis
• So we can write the answer. The answer must contain the following 4 statements:
(i) The object is on the x axis
(ii) It is at a distance of 2 m from O
(iii) O is at the point of intersection of yellow wall, white wall and the floor
(iv) The x axis lies along the intersection of yellow wall and floor
The answer will not be complete if we do not write all the above four statements
• However, a simplified way to write the answer using two statements:
(i) The coordinates of the object is: (2,0,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• Note that, in the coordinates the values corresponding to y and z are zero. Because:
♦ The 'distance of object from origin' measured along y axis is zero
♦ The 'distance of object from origin' measured along z axis is zero
• In fact, we do not need any help from the y and z axes to write the position of a point on the x axis
• We can ignore those axes
In a similar way,
• If the object is on the y axis, at a distance of 'y' m from the origin, we write:
(i) The coordinates of the object is: (0,y,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• If the object is on the z axis, at a distance of 'z' m from the origin, we write:
(i) The coordinates of the object is: (0,z,0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• The second and third statements are given in order to specify the exact position of the reference frame
• In this, we need to mention about the x axis only
• Because, once 'O' and 'x axis' are fixed, the other two axes also get fixed immediately. We will learn about it in later chapters
Case 2: The object is on a plane.
This is shown in fig.3.5(a) below. The object is denoted by the magenta dot. It is on the yellow wall.
 |
| Fig.3.5 |
• So we can say: The object is on the 'xy plane'
• When measured parallel to the x axis, it is at a distance of 1.8 m from O
• When measured parallel to the y axis, it is at a distance of 0.5 m from O
• So we can write the answer. The answer must contain the following 3 statements:
(i) The coordinates of the object is: (1.8, 0.5, 0)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
In a similar way,
• If the object is on the white wall (yz plane), we write the following 3 statements:
(i) The coordinates of the object is: (0, y, z)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
• If the object is on the floor (xz plane), we write the following 3 statements:
(i) The coordinates of the object is: (x, 0, z)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
Case 3: The object is in 3-dimensional space. It is away from the axes and planes
• This is shown in fig.3.5(b) above. The object is denoted by the magenta dot. It is in 3-dimensional space.
• When measured parallel to the x axis, it is at a distance of 1.8 m from O
• When measured parallel to the y axis, it is at a distance of 0.5 m from O
• When measured parallel to the z axis, it is at a distance of 0.7 m from O
• So we can write the answer. The answer must contain the following 3 statements:
(i) The coordinates of the object is: (1.8, 0.5, 0.7)
(ii) O is at the point of intersection of yellow wall, white wall and the floor
(iii) The x axis lies along the intersection of yellow wall and floor
3. The above 3 are the only possible cases.
• We can effectively use such reference frames to specify the position of any given point object.
• We used the corner of a room as the origin. For out door experiments, we can take this reference frame outside and fix it up in any convenient orientation that is suitable.
• For example, we can assume the 'bottom corner of a bridge pillar' to be O and a suitable line drawn from it as the x axis
• When O and x axis are fixed, other two axes get fixed immediately.
• In this chapter, we are dealing with rectilinear motion.
• So all objects that we consider will be moving along a single line. They will not change direction.
• So in this chapter, we can assume that the objects are moving along the x axis.
• But we cannot ask a car for example, to move along the x axis drawn by us. So what do we do?
Ans: We take our reference frame and orient it in such a way that the x axis coincides with 'the straight line path on which the car moves'
• As a result, the coordinates will not have y and z values
♦ That is., all coordinates will be of the form: (x,0,0)
• So we need not write the coordinate notation (x,y,z) in this chapter. We can use a simpler form which can be explained with the help of an example. We will write the steps:
1. Consider fig.3.6 below:
 |
| Fig.3.6 |
2. This x axis is placed in such a way that, it coincides with the path of an object
• After placing the x axis in this way, a suitable point is marked on it as the origin O
3. When the object is at P, some one asks us:
What is the position of the object?
• The answer should contain 2 statements:
(i) The object is on the positive side of the x axis
(ii) The object is at a distance of 360 m from O
4. A simplified way to write the answer containing only one statement:
■ The position coordinate of the object is +360
5. When the object is at Q, some one asks us:
What is the position of the object?
We can write the answer as:
■ The position coordinate of the object is +240
6. When the object is at R, some one asks us:
What is the position of the object?
We can write the answer as:
■ The position coordinate of the object is -120
Now we know how to specify positions. In the next section, we will see distances between these positions.
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