Suppose the two persons \(A\) and \(B\) start moving simultaneously from the points \(P\) and \(Q\)  respectively. They meet at a point \(R\) after a time \(T\)  and take \({t_A}\) and \({t_B}\) time to reach the point \(Q\) and \(P\) respectively. 

Let the distances \(PR\) and \(RQ\) are \(x\) and \(y\) and speeds of \(A\) and \(B\) are \({V_A}\) and \({V_B},\) then

\(x = {V_a} \times T\) (1)

\(y = {V_b} \times T\) (2)

\(x = {t_B} \times {V_B}\) (3) 

\(y = {t_A} \times {V_A}\) (4)

Now from the above equations, 

\(\therefore \) \(\frac{{{V_A}}}{{{V_B}}} = \sqrt {\frac{{{t_B}}}{{{t_A}}}} \) and \(T = \sqrt {{t_A}{t_B}} \)

In the above case, \(A\) and \(B\) start at the same time and after meeting they take unequal times to reach their destination. Note that if two persons start at the different times but after meeting at a point, they take equal times to reach their destinations, then same formula can be applied. 

Example 13: A train starts from \(A\) to \(B\) and another form \(B\) to \(A\) at the same time. After crossing each other they complete their journey in \(3\frac{1}{2}\) and \(2\frac{4}{7}\) hours respectively. If the speed of the first train is 60 km/h, then find the speed of the second train.  

Solution: Suppose speed of the second train is \(u\), then \(\frac{u}{{60}} = \sqrt {\frac{{3\frac{1}{2}}}{{2\frac{4}{7}}}}  = \sqrt {\frac{{49}}{{36}}}  = \frac{7}{6}\)

\( \Rightarrow u = {\rm{ }}70\) km/hr