Position, Path Length And Displacement

Position

The position is where an object is located in space relative to a reference point. It’s like saying, “I am 5 meters to the right of the tree.” That tree is your reference point.

Imagine you’re playing a game of hide and seek. You need to describe where you are without being seen. In physics, the position is that description—it tells us where an object is located in space relative to a chosen reference point.

Think of it like the address of a house. The house (object) has a specific location on a street (space), and the street’s name and house number (coordinates) tell us the exact position of the house.

In a one-dimensional space, like a straight road, a position can be described using just one number along with a direction, like “30 meters north of the school.” In two or three dimensions, like on a map or in a room, you need two or three numbers, respectively, to describe the position.

How to Describe Position

• Reference Point: Choose a fixed point to compare other locations too, like a landmark or a corner of a room. A reference point is a fixed place that we use to determine the position of other objects. It’s like the “You Are Here” marker on a map.

Example: In an experiment to measure the speed of sound, the starting point of the sound wave (where the sound is produced) is the reference point. All measurements of the sound’s position are taken from this point.

• Coordinates: Use numbers to describe how far and in what direction the object is from the reference point. They are like the latitude and longitude on a map that pinpoint your location.

Example: In a particle accelerator, the position of a subatomic particle is described using coordinates within the vacuum chamber. These coordinates help physicists track the particle’s path as it moves at high speeds.

• Axis: In more complex scenarios, like a city grid or a graph, we use axes (like the x and y-axis) to define position. An axis is a straight line that objects can move along or be measured against. In a graph, it’s one of the lines that define the chart area.

Example: In a graph showing a car’s motion, the x-axis could represent time, and the y-axis could represent the car’s speed. The position of the car at any given time is plotted against these axes to show how its speed changes over time.

If we say a point is at A(x, y, z), it means it’s at a certain spot according to these lines. The place where these lines meet is called the reference point or origin.

We can also add a clock to this system to keep track of time. If any of the numbers (coordinates) change over time, we know the object is moving. If they stay the same, the object isn’t moving.

For motion in one direction, like walking in a straight line, we only need one set of lines, say the X-axis. The position of the object can then be measured from the starting point, usually called the origin.

When someone walks to the right from the origin, the values are positive, and when they walk to the left, the values are negative.

Understanding the position is the first step in describing motion. If we know where something starts and where it ends up, we can begin to describe how it got there—that’s where path length and displacement come in, which we’ve discussed earlier.

Path Length

Path Length is the total distance traveled by an object, regardless of its direction. It’s like tracking every step you take during a walk.

Imagine you’re walking through a park. You start at the entrance, walk around the pond, follow the winding path through the garden, and finally reach the park bench. The path length is the total distance you walked, including all the twists and turns along the way.

Key Points:

• Path Length is a scalar quantity, which means it only has magnitude (how much), not direction.
• It’s like the reading on a pedometer or a car’s odometer; it only goes up as you move, regardless of the direction.
• Path Length is always positive or zero; it never decreases.
• When calculating path length in one-dimensional motion, like running back and forth on a straight track, you simply add up all the distances you’ve covered, even if you’ve retraced your steps.

When solving physics problems, you might be asked to calculate the path length of an object moving along a specific trajectory. For example, if a roller coaster car travels along the track, the path length would be the length of the track itself.

The Path length is different from displacement, which is the straight-line distance from the starting point to the ending point, considering the direction. So, if you walked in a big circle and ended up where you started, your path length would be the entire distance you walked, but your displacement would be zero because you didn’t end up any distance from your starting point.

Calculating Distance in One-dimensional Motion

In one-dimensional motion, like walking back and forth on a straight path, you add up all the distances you’ve walked, even if you turn around and walk back over the same path. In one-dimensional motion, the path length is simply the sum of all distances covered during the motion. Let’s consider a few examples to illustrate this:

• Athletics Track: If an athlete runs a 100-meter dash and then returns to the start line, the path length is \(\displaystyle 100 \text{m} + 100 \text{m} = 200 \text{m}\), even though the displacement (change in position) is 0.
• Particle in a Collider: A subatomic particle moving back and forth in a linear accelerator covers a certain distance each time it oscillates. The path length is the sum of all these individual distances.
• Light in a Fiber Optic Cable: Light traveling through a straight fiber optic cable bounces back and forth off the walls of the cable. The path length is the actual distance the light travels, not just the length of the cable.

Displacement

Displacement is the straight-line distance from the starting point to the ending point, along with the direction. It’s like drawing a straight line from the start to the finish. Displacement is a vector quantity that represents the change in position of an object. It’s not just about how far the object has traveled, but also the direction in which it moved.

Imagine you’re at home and you decide to walk to a friend’s house. Your journey takes you 5 blocks east and 3 blocks north. The displacement is not the total blocks you walked, but the straight line from your home to your friend’s house, in a northeast direction.

Formula: The formula for displacement is:

\(\displaystyle \Delta x = x_f – x_i \)

where: (∆ x ): Displacement ; (x_f): Final position ; (x_i): Initial position

Key Points:

• Direction Matters: Displacement includes both magnitude and direction. If you return to your starting point, your displacement is zero because there’s no change in position.
• Vector Representation: Displacement is often represented by an arrow on a graph, pointing from the starting position to the final position.
• Path Independent: Unlike path length, displacement doesn’t care about the route taken. It’s only concerned with the start and end points.

Also Read: Frame of Reference

Distinguish between Path Length and Displacement

A simple comparison between path length and displacement

Path Length	Displacement
It is the actual distance traveled by an object.	It is the shortest distance between the initial and final positions of the object.
It can be any value equal to or greater than zero.	It can be zero if the object returns to its initial position, otherwise, it is always positive.
It is a scalar quantity.	It is a vector quantity.
It does not depend on direction.	It depends on both magnitude and direction.
It can never be negative.	It can be positive, zero, or negative.
It is measured along the actual path taken by the object.	It is measured along a straight line joining the initial and final positions of the object.
It is the total length of the path covered by the object.	It is the change in position of the object.
Its SI unit is meter (m).	Its SI unit is also meter (m).

Different Graph Relations

Position-Time Graph

A position-time graph shows an object’s position relative to time. The x-axis represents time, and the y-axis represents position. If the line on the graph is:

• Horizontal, the object is stationary.
• Sloping upwards, the object is moving away from the reference point.
• Sloping downwards, the object is moving towards the reference point.

Path Length-Time Graph

Since path length is a scalar quantity and always increases, a path length-time graph will always be a curve that never decreases. It typically moves upwards as time progresses, indicating that the object is covering distance.

Displacement-Time Graph

A displacement-time graph is similar to a position-time graph but focuses on the change in position (displacement). If the line on the graph is:

• Horizontal, the displacement is not changing; the object may be at rest or moving at a constant speed in a loop.
• Sloping upwards, the displacement increases; the object moves away from the starting point.
• Sloping downwards, the displacement decreases; the object moves towards the starting point.

Velocity-Time Graph

Additionally, a velocity-time graph can show how the velocity (which includes direction) changes over time. The area under the curve in a velocity-time graph represents the displacement.

Example: Let’s say a car moves in a straight line, first accelerating and then decelerating back to its starting point. On a position-time graph, the line would slope upwards and then downwards, returning to the starting position. The path length-time graph would show a continuous increase, while the displacement-time graph would rise and then fall back to zero.

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