A car travelling on a straight track moves with a uniform velocity${\text{V}}_1$ for half of the time and with a uniform velocity${\text{V}}_{\text{2}}$ for rest of the time. The average velocity is

Given that a car is moving along straight track

Let the total time of journey from A to B is = t

During ${\text{t}}_{\text{1}}=\frac{\text{t}}{\text{2}}$  it travelled from A to C with speed $={\text{v}}_{\text{1}}$

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

During ${\text{t}}_{\text{2}}=\frac{\text{t}}{\text{2}}$  it travelled from C to B with speed $={\text{v}}_{\text{2}}$

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

We can have

Average speed $=\frac{\text{total}\text{distance}\text{covered}}{\text{total}\text{time}\text{taken}}$

$=\frac{\text{AB}}{\text{t}}=\frac{\text{AC}+\text{CB}}{2}$

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

$\therefore \text{Average}\text{speed}=\frac{{\text{v}}_{\text{1}}+{\text{v}}_{\text{2}}}{2}$