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Finding Average Velocity

A point trave...

Question

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

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

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

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\frac{v_{o} v_{1} v_{2}}{2 v_{1} v_{2} + v_{0} v_{1} + v_{0} v_{2}}

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\frac{4 v_{o} v_{1} v_{2}}{2 v_{1} v_{2} + v_{0} v_{1} + v_{0} v_{2}}

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Solution

The correct option is D $\frac{4 v_{o} v_{1} v_{2}}{2 v_{1} v_{2} + v_{0} v_{1} + v_{0} v_{2}}$
Let the total distance be $d$ . Then for first half distance, time $= \frac{d}{2 v_{o}}$ , next distance time taken is $t_{2} = \frac{d}{4 v_{1}}$ and last half distance time taken is $t_{3} = \frac{d}{4 v_{2}}$
Now, Average speed
$t = \frac{d}{\frac{d}{2 v_{o}} + \frac{d}{4 (v_{1})} + \frac{d}{4 (v_{2})}} = \frac{4 v_{o} v_{1} v_{2}}{2 v_{1} v_{2} + v_{0} v_{1} + v_{0} v_{2}}$

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