Question

A particle goes from point \(\mathrm{A}\) to \(\mathrm{B}\). Its displacement is \(\mathrm{X}\) and path length is \(\mathrm{y}\). So \(\mathrm{x} / \mathrm{y} \ldots\) \((\mathrm{A})>1\) (B) \(<1\) (C) \(\geq 1\) (D) \(\leq 1\)

Short answer

The correct option is (D) \(\mathrm{x} \leq \mathrm{y}\), as the displacement is either equal to or less than the path length, accounting for both straight line and curved trajectory scenarios.

Step-by-step solution

Understanding Displacement and Path Length

Displacement is the shortest distance between the initial and final positions of a particle, and it is a vector quantity. Path length, on the other hand, is the total distance traveled by the particle along its trajectory and is a scalar quantity.

Comparing Displacement and Path Length

Since displacement is the shortest distance between two points, it can never be greater than the path length (the total distance traveled), as path length takes into account all the curves and turns in the trajectory. In the special case when the particle moves in a straight line along its trajectory, the displacement and path length would be equal.

Analyzing the given options

Now, let's analyze the given options in relation to the displacement and path length: (A) \(\mathrm{x} > \mathrm{y}\): This option suggests that the displacement is greater than the path length, which is not possible, as we concluded previously. (B) \(\mathrm{x} < \mathrm{y}\): This option suggests that the displacement is less than the path length, which is true when the particle's trajectory is not a straight line. (C) \(\mathrm{x} \geq \mathrm{y}\): This option suggests that the displacement is either equal to or greater than the path length, which is only true when the particle moves in a straight line along its trajectory. (D) \(\mathrm{x} \leq \mathrm{y}\): This option suggests that the displacement is either equal to or less than the path length, which covers both the cases when the particle moves in a straight line or along a curved trajectory.

Selecting the correct option

Considering all the analysis, we can conclude that the correct option is: (D) \(\mathrm{x} \leq \mathrm{y}\), as in this case the displacement is either equal to or less than the path length, accounting for both straight line and curved trajectory scenarios.

Key concepts

Path Length

When a particle moves from one position to another, it can follow any path that connects these two points. The complete journey taken by the particle is known as the path length. Path length is a total measure of the distance traveled, taking into account every twist, turn, or loop along the way.
This means that even if two journeys start and end at the same points, different paths can result in different path lengths.

• Path length is a scalar quantity, which means it only considers the magnitude (how much distance was covered) and not the direction.
• You can think of it as what you would read on your car's odometer while driving from one location to another.
• It always remains positive, since it sums the space traveled.

As the path length includes every segment of a journey, it's often longer than the shortest route possible between two points, which brings us to compare it with another important concept: Displacement.

Vector and Scalar Quantities

Vector and scalar quantities are fundamental distinctions in physics, as they determine how we describe and measure different physical phenomena.

• Vector quantities are characterized by both magnitude and direction. For instance, displacement is a vector because it describes not just the distance between two points, but also the direction from the start point to the endpoint.
• Other examples of vector quantities include velocity and force.
• Scalar quantities, however, include only magnitude. Path length, as already mentioned, is a scalar quantity because it only focuses on the size of the journey, not the direction.
• Distance, speed, and mass are all examples of scalar quantities.

Understanding whether a quantity is a vector or scalar is crucial for solving physics problems correctly. Misidentifying them can lead to errors in calculation and analysis.

Trajectory Analysis

In understanding how a particle moves from one point to another, trajectory analysis helps us visualize the path taken. This analysis is crucial for problems where path length and displacement differ significantly.
The trajectory is essentially the route that the particle takes; it can be straight or curved.

• When a particle moves straight from start to finish without deviation, the trajectory is a straight line, making its displacement equal to its path length.
• If the trajectory curves or loops around, however, the path length increases, but the displacement remains the straight distance between the initial and final positions.
• Through trajectory analysis, we can determine scenarios where the ratio of displacement to path length falls into different categories, such as \( x \leq y \), meaning displacement is lesser than or equal to the path length.

This kind of analysis is essential in physics problems as it simplifies the complexity of real-world paths into workable mathematical problems.