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.