A person travels x distance with velocity v _ 1 and then x distance...

The time $\left({t}_{1}\right)$ taken by the person to travel distance $x$ with velocity ${v}_{1}$ is given by

${t}_{1} = \dfrac{x}{{v}_{1}} . . . \left(1\right)$

The time \left({t}_{2}\right)$taken by the person to travel distance$x$with velocity${v}_{2}$is given by</p><p>${t}_{2} = \dfrac{x}{{v}_{2}} . . . \left(2\right)$</p><p>The formula to calculate the average speed \left(v\right)$ of the person is given by

$v = \dfrac{2x}{{t}_{1} + {t}_{2}} . . . \left(3\right)$

Substitute the expressions from equations (1) and (2) into equation (3) to obtain the required relation.

$\begin{cases}v & = & \dfrac{2x}{\left(\dfrac{x}{{v}_{1}} + \dfrac{x}{{v}_{2}}\right)} \\ & ⇒ & \dfrac{x}{{v}_{1}} + \dfrac{x}{{v}_{2}} = \dfrac{2x}{v} \\ & ⇒ & \dfrac{2}{v} = \dfrac{1}{{v}_{1}} + \dfrac{1}{{v}_{2}}\end{cases}$