A scooter is moving along a straight path covering one-third of the distance with a velocity${\text{V}}_1$ . The remaining distance is covered with a velocity${\text{V}}_2$ for half of the time and with a velocity${\text{V}}_3$ for the next half of the time. The average velocity of the scooter for the whole journey is

Given that

A scooter moving along straight line from A to D

Let AD = s

Covered AB i.e. $\frac{s}{3}$  with speed ${\text{v}}_{\text{1}}$

${\text{t}}_{\text{1}}=\frac{\text{s}}{\text{3v}}\left(\because \text{t}=\frac{\text{s}}{\text{v}}\right)$

Covered BC i.e. $\frac{s}{3}$  in $\frac{\text{t}}{\text{2}}$

With speed ${\text{v}}_1$

We get BC $={\text{v}}_{\text{1}}\frac{\text{t}}{\text{2}}$

Covered CD i.e. $\frac{s}{3}$  in $\frac{\text{t}}{\text{2}}$

With speed ${\text{v}}_2$

We get $\text{CD}={\text{v}}_{\text{2}}\frac{\text{t}}{\text{2}}$

$\frac{2\text{s}}{3}=\text{BD}=\text{BC}+\text{CD}=\left({\text{v}}_{\text{1}}+{\text{v}}_{\text{2}}\right)\frac{\text{t}}{\text{2}}$

$\therefore \text{t}=\frac{4\text{s}}{3\left({\text{v}}_{\text{1}}+{\text{v}}_{\text{2}}\right)}$                     

${\text{v}}_{\text{BD}}=\frac{{\text{v}}_{\text{1}}+{\text{v}}_{\text{2}}}{\text{2}}$

${\text{v}}_{\text{AD}}=\frac{\text{s}}{\frac{\text{s}}{{\text{3v}}_{\text{1}}}+\frac{\text{2s}}{\frac{3\left[{\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}\right]}{2}}}$

$=\frac{\text{s}}{\frac{\text{s}}{{\text{3v}}_{\text{1}}}+\frac{\text{4s}}{3\left[{\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}\right]}}$

$=\frac{\text{3s}}{\frac{\text{s}\left({\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}+{\text{4v}}_{\text{1}}\right)}{3\left[{\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}\right]}}$

$=\frac{{\text{3v}}_{\text{1}}\left({\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}\right)}{{\text{v}}_{\text{2}}+{\text{v}}_{\text{3}}+{\text{4v}}_{\text{1}}}$