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Q.

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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a

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

b

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

c

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

d

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

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

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A point traversed half of the distance with a velocity v0. The half of remaining part of the distance was covered with velocity v1 for half the time and with velocity v2 for the rest half, then the mean velocity of the point, averaged over the whole time of motion is