A and B are moving in two circular orbits with angular velocity 2$\mathrm{\omega }$ and $\mathrm{\omega }$ respectively. Their positions are as shown at t = 0 . Find the time when they will meet for the first time.

For same sense ( let say both anticlockwise) :

Ball A-

First time A reaches to common point , ${\mathrm{t}}_1=\frac{2\mathrm{\pi }}{2\mathrm{\omega }}=\frac{\mathrm{\pi }}{\mathrm{\omega }}$

Second time A reaches to common point , ${\mathrm{t}}_2=\frac{\mathrm{\pi }}{\mathrm{\omega }}+\frac{\mathrm{\pi }}{\mathrm{\omega }}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

So, the time series is $\frac{\mathrm{\pi }}{\mathrm{\omega }} , \frac{2\mathrm{\pi }}{\mathrm{\omega }}, \frac{3\mathrm{\pi }}{\mathrm{\omega }}, \frac{4\mathrm{\pi }}{\mathrm{\omega }}.....$

Ball B :

First time B reaches to common point , ${\mathrm{t}}_1=\frac{{\displaystyle \raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\mathrm{\omega }}=\frac{1.5\mathrm{\pi }}{\mathrm{\omega }}$

Second time B reaches to common point , ${\mathrm{t}}_2=\frac{1.5\mathrm{\pi }}{\mathrm{\omega }}+\frac{2\pi }{\omega }=\frac{3.5\mathrm{\pi }}{\mathrm{\omega }}$

So, the time series is $\frac{1.5\mathrm{\pi }}{\mathrm{\omega }} , \frac{3.5\mathrm{\pi }}{\mathrm{\omega }}, \frac{5.5\mathrm{\pi }}{\mathrm{\omega }}, \frac{7.5\mathrm{\pi }}{\mathrm{\omega }}.....$

There is no common time when A and B meet.

For opposite sense ( let say A anticlockwise ad B clockwise) :

Ball A-

First time A reaches to common point , ${\mathrm{t}}_1=\frac{2\mathrm{\pi }}{2\mathrm{\omega }}=\frac{\mathrm{\pi }}{\mathrm{\omega }}$

Second time A reaches to common point , ${\mathrm{t}}_2=\frac{\mathrm{\pi }}{\mathrm{\omega }}+\frac{\mathrm{\pi }}{\mathrm{\omega }}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

So, the time series is $\frac{\mathrm{\pi }}{\mathrm{\omega }} , \frac{2\mathrm{\pi }}{\mathrm{\omega }}, \frac{3\mathrm{\pi }}{\mathrm{\omega }}, \frac{4\mathrm{\pi }}{\mathrm{\omega }}.....$

Ball B :

First time B reaches to common point , ${\mathrm{t}}_1=\frac{{\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\mathrm{\omega }}=\frac{0.5\mathrm{\pi }}{\mathrm{\omega }}$

Second time B reaches to common point , ${\mathrm{t}}_2=\frac{0.5\mathrm{\pi }}{\mathrm{\omega }}+\frac{2\pi }{\omega }=\frac{2.5\mathrm{\pi }}{\mathrm{\omega }}$

So, the time series is $\frac{0.5\mathrm{\pi }}{\mathrm{\omega }} , \frac{2.5\mathrm{\pi }}{\mathrm{\omega }}, \frac{4.5\mathrm{\pi }}{\mathrm{\omega }}, \frac{6.5\mathrm{\pi }}{\mathrm{\omega }}.....$

Again, there is no common time when A and B meet.