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Finding Average Velocity

A particle tr...

Question

A particle traversed half of the distance with a velocity $v_{0}$ . The remaining part of the distance was covered with velocity $v_{1}$ for half the time and with velocity $v_{2}$ for the other half of the time. The mean velocity of the particle 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_{0} + 2 v_{1} + 2 v_{2}}{3}

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

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Solution

The correct option is D $\frac{2 v_{0} (v_{1} + v_{2})}{(2 v_{0} + v_{1} + v_{2})}$
Let $2 S$ be the total distance travelled
For first half of the journey
$S = v_{0} \times t_{1}$ (where $t_{1}$ is the time taken to complete half of the distance)
For the remaining part of the journey we can say,
$v_{1} t + v_{2} t = S$

$∴ t = \frac{S}{v_{1} + v_{2}}$
Total time $= t_{1} + 2 t$
Mean velocity $= \frac{Total displacement}{Total time}$
$= \frac{2 S}{(\frac{S}{v_{0}}) + 2 (\frac{S}{v_{1} + v_{2}})}$
$= \frac{2}{(\frac{1}{v_{0}}) + (\frac{2}{v_{1} + v_{2}})}$
$= \frac{2 v_{0} (v_{1} + v_{2})}{(2 v_{0} + v_{1} + v_{2})}$

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