Position, Path Length and Displacements

The length of the path travelled by a car moving from one point to another point is called distance and the distance travelled is also known as the path length. However, if it takes the shortest path, it becomes the  displacement. We can also say that the difference between the initial and final position of a car is its displacement. 

But before we move forward let us know what motion is. If we look around, everywhere we can see objects moving. Kids play, birds fly, animals move in search of food, people walk or run, vehicles run on the road, river flow, etc. So basically, we can find motion everywhere in the universe. But what is meant by motion?

If you notice the above-mentioned instances they all move from one place to another and change their place. Therefore, motion is nothing but change in the position of a body with respect to time. If a body is at rest, it means that the body is not in motion, merely means that it is being described concerning a frame of reference. 

Position - Understanding with an Example

To describe the motion of an object, you must know and be able to describe its position.

Let us understand this with an example, let us say Ram moved from point R to S. This means Ram's initial/previous position was R from where he shifted to S after sometime. Now the question is how can we represent Ram's initial position? 

In physics, we specify a position with the help of a reference point and a set of three mutually perpendicular axes or  rectangular coordinate systems. They are the X, Y, and Z-axis. The reference point is known as origin; it is the intersection of the above three-axis (X, Y, and Z). So we take point R as the reference point or origin with coordinates (0, 0, 0) and S is represented by a set of coordinates on the three-axis (X, Y, Z).

Since we know that motion is the change in position with time, we install a clock in this system. The coordinate system along with the clock is the frame of reference. A frame of reference is an arbitrary set of axes from which the position and motion of the object are described. Thus, if one or more coordinates of a body change with time, the body is said to be in motion.

Path Length

The path length is the actual length of the path traversed by the body between the Initial and Final positions.

Displacement - Understanding with an Example

It is the shortest length i.e. Straight line distance between Initial and Final positions. Displacement is a vector quantity.

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Displacement formula:

If s_i is the initial position of an object and s_f is the final position, then the displacement of this object is:

s  = s_i - s_f

Here, s is a variable referred to as displacement.

Since displacement has magnitude and direction, it is a vector quantity and the path is a scalar because it has only and no direction.

Explanation with Examples

To have a proper understanding of the position, path length, and displacement and the difference between them. Follow are some examples given with explanation:

Let us take three examples here. In the first one, Ram starts travelling from point R of the square path RSTU with RS = 1 km. He travels through S, T, U, and comes back to R in 20 minutes. The distance travelled by him is 4 x 1 km = 4 km. But if you see the change in his position from the start to the end of the journey, it is nil (It has no change). Ram started at point R and came back at R.

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In the second example, 

Ram travels from Point R to S along the straight line in 60 minutes. The distance travelled by him is 5 km. And the total distance from the start to the endpoint travelled by Ram is also 5 km.

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Now in the third example, Ram travels through the triangular path. He starts from point R and reaches T through point S in 120 minutes. The distance that has been travelled by him is 3 km + 4 km = 7 km. But if we see how far he is from the point where he started his journey, it is 5 km.

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If you notice the above examples, the distance travelled and the change in position may or may not be the same.

The distance travelled by the body is known as the path length. Whereas the change in position, that is the difference between the initial and final positions of the body is called its displacement.

Hence, the path length is 4 km but the displacement is 0 in the first case.  The path length is the same as the displacement – 5 km in the second example and the third example, the path length is 7 km but the displacement is 5 km.

From the above text, we understand that the position of an object describes the point at which the object is standing at an instant. The change in position describes that an object is set into motion and this distance travelled is the path length. Also, if an object takes the shortest path, it is displacement.