Introduction
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 si is the initial position of an object and sf is the final position, then the displacement of this object is:
s = si - sf
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.
FAQs on Position, Path Length and Displacements
1. What are position, path length, and displacement in Physics?
In Physics, position refers to the specific location of an object with respect to a point of reference or origin. Path length is the total distance an object covers along its actual route of travel. In contrast, displacement is the shortest straight-line distance between the object's initial and final positions, and it is a vector quantity, meaning it also has a specific direction.
2. How is the position of an object specified in a frame of reference?
The position of an object is specified using a frame of reference. This system includes a fixed reference point, known as the origin (0,0,0), and a set of coordinate axes (typically x, y, and z). An object's location is precisely described by its coordinates along these axes at a given moment in time, which is measured by a clock within the same frame.
3. What are the key differences between path length and displacement?
The key differences between path length and displacement are:
- Nature: Path length is a scalar quantity, having only magnitude. Displacement is a vector quantity, having both magnitude and direction.
- Value: For a moving body, path length is always positive. Displacement can be positive, negative, or zero.
- Path Dependence: Path length depends on the entire route taken between two points. Displacement is independent of the path and only depends on the initial and final positions.
- Magnitude: The magnitude of displacement is always less than or equal to the path length.
4. Why is displacement considered a vector quantity while path length is a scalar?
Displacement is a vector because it defines the change in an object's position, which inherently involves both magnitude (how far the start and end points are from each other) and direction (the specific orientation of the straight line from start to end). Path length is a scalar because it only accumulates the total distance travelled, like the reading on a car's odometer, without any regard for the direction of motion.
5. Under what condition is the magnitude of an object's displacement equal to its path length?
The magnitude of displacement is equal to the path length only under one specific condition: when the object travels along a straight line and does not change its direction of motion. In all other cases, such as moving along a curved path or changing direction, the path length will be greater than the magnitude of the displacement.
6. If a car travels from point A to point B (5 km east) and then returns to point A, what is its path length and displacement?
In this example:
- The path length is the total distance covered, which is 5 km (A to B) + 5 km (B to A) = 10 km.
- The displacement is zero. This is because the car's initial position (A) and final position (A) are the same, resulting in no net change in position.
7. Can the path length of a moving object ever be zero or negative? What about displacement?
No, the path length of a moving object can never be zero (as it is moving) or negative; it is a measure of total distance and is always a positive value. However, displacement can be zero if the object returns to its starting point. It can also be negative, where the sign typically indicates its direction relative to a chosen positive axis in the coordinate system.
8. How is displacement calculated, and what are its standard units and dimensions?
Displacement (represented by Δx) is calculated by finding the difference between an object's final position (x_f) and its initial position (x_i). The formula is Δx = x_f - x_i. The standard SI unit for displacement is the meter (m). Its dimensional formula is [M⁰L¹T⁰].
