A point traversed half of the distance with a velocity $v_{0} .$ The half of remaining part of the distance was covered with velocity $v_{1}$ for half the time and with velocity $v_{2}$ for the rest half, then the mean velocity of the point, averaged over the whole time of motion is

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$\frac{2 v_{0} + v_{1} + v_{2}}{3}$

$\frac{v_{0} + v_{1} + v_{2}}{3}$

$\frac{v_{0} + .2 v_{1} + 2 v_{2}}{3}$

$\frac{2 v_{0} (v_{1} + v_{2})}{(2 v_{0} + v_{1} + v_{2})}$

answer is D.

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Detailed Solution

Let the total distance be d. Then for first half distance, time $= \frac{d}{2 v_{0}},$ next
distance, $= v_{1} t$ and last half distance $= v_{2} t$
$∴ v_{1} t + v_{2} t = \frac{d}{2}; t = \frac{d}{2 (v_{1} + v_{2})}$
Now average speed
$t = \frac{d}{\frac{d}{2 v_{0}} + \frac{d}{2 (v_{1} + v_{2})} + \frac{d}{2 (v_{1} + v_{2})}} = \frac{2 v_{0} (v_{1} + v_{2})}{(v_{1} + v_{2}) + 2 v_{0}}$