A body traversed half of the distance with a velocity $V_{0}$ . The remaining part $V_{1}$ for half of the time and with the velocity $V_{2}$ for the other half of the time. Find the average velocity of the body over the whole journey.
$\frac{2 V_{0} (V_{1} + V_{2})}{(2 V_{0} + V_{1} + V_{2})}$
$\frac{3 V_{0} (V_{1} + V_{2})}{(2 V_{0} + V_{1} + V_{2})}$
$\frac{2 V_{0} (V_{1} + V_{2})}{(3 V_{0} + V_{1} + V_{2})}$
$\frac{V_{0} (V_{1} + V_{2})}{(2 V_{0} + V_{1} + V_{2})}$

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Solution

The correct option is A $\frac{2 V_{0} (V_{1} + V_{2})}{(2 V_{0} + V_{1} + V_{2})}$
Let $s$ be total distance to be traveled
$t_{0}$ be the time for which it travels with speed $v_{0}$
$t_{1} / 2$ be the time for which it travels with $v_{1}$ and $v_{2}$
using distance=speed $\times$ time :
$\frac{s}{2} = v_{1} \frac{t_{1}}{2} + v_{2} \frac{t_{1}}{2}$
$t_{1} = \frac{s}{v_{1} + v_{2}} \dots (1)$
Average speed =Total distance traveled /Total time
$< v >= \frac{s}{t_{0} + t_{1}}$
$< v >= \frac{s}{\frac{s}{2 v_{0}} + \frac{s}{2 (v_{1} + v_{2})}}$
$< v >= \frac{2 v_{0} (v_{1} + v_{1})}{2 v_{0} + v_{1} + v_{2}}$

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