Explain the concept of relative displacement and its various cases.

(N/A) Consider two particles $A$ and $B$ moving along the $X$-axis with uniform velocities $v_{A}$ and $v_{B}$ respectively.
Let $x_{OA}$ and $x_{OB}$ be their initial displacements from the origin $O$ at time $t = 0$.
If $x_{A}$ and $x_{B}$ are their position coordinates at any time $t$,then:
$x_{A} = x_{OA} + v_{A} t$
$x_{B} = x_{OB} + v_{B} t$
At time $t$,the displacement of particle $B$ with respect to particle $A$ is given by:
$x_{BA} = x_{B} - x_{A} = (x_{OB} - x_{OA}) + (v_{B} - v_{A}) t$
Here,$(x_{OB} - x_{OA})$ is the initial relative displacement at $t = 0$,and $(v_{B} - v_{A}) = v_{BA}$ is the relative velocity of $B$ with respect to $A$.
Cases:
$1$. If $v_{A} = v_{B}$,then $v_{BA} = 0$. The equation becomes $x_{B} - x_{A} = x_{OB} - x_{OA}$. This means the distance between the two particles remains constant over time.
$2$. If $v_{A} \neq v_{B}$,the relative displacement changes linearly with time. If $v_{B} > v_{A}$,the distance between them increases,and if $v_{A} > v_{B}$,the distance decreases until they meet.