Question

An auto travels the first half time with a uniform speed $u$ and the next half time with a uniform speed $v$. Find its average speed.

• A. $2uv$
• B. $\frac{u+v}{2}$
• C. $uv$
• D. $\frac{uv}{2}$

Answer 1

The correct option is B $\frac{u+v}{2}$

Given, speed of the auto during the first half of the journey is $u$ and the next half of the journey is $v$
Let total time taken for the journey be $T$
$\therefore$Time taken for the first half of the journey be $\frac{T}{2}$ and the next half of the journey be $\frac{T}{2}$
Let the average speed be $V_{av}$
We know, $speed=\frac{Distance}{time}$
$\therefore$ $Distance=speed\times time$

During the first half of the journey, distance, $d_1=u\times \frac{T}{2}$
During the second half of the journey, distance, $d_2=v\times \frac{T}{2}$
Total distance $=d=d_1+d_2$
$=u\times$$\frac{T}{2}+v\times \frac{T}{2}$
$\Rightarrow d=\frac{T}{2}(u+v)$

Average speed,$V_{av}$ = $\frac{Totaldistance}{Totaltimetaken}$
$=\frac{d}{T}$
$=\frac{\frac{T}{2}(u+v)}{T}$
$=\frac{u+v}{2}$