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Finding Average Velocity

A particle tr...

Question

A particle travelling in a straight path in the same direction covers the first half of the distance with a velocity $v_{1}$ and the second half of the distance with two different velocities $v_{2}$ and $v_{3}$ in equal intervals of time. The average velocity of the particle for the entire journey is

A

\frac{(v_{1} + v_{2}) v_{3}}{v_{1} + v_{2} + v_{3}}

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B

\frac{(v_{1} + v_{3}) v_{2}}{v_{1} + v_{2} + v_{3}}

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C

\frac{2 v_{1} (v_{2} + v_{3})}{v_{1} + v_{2} + v_{3}}

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D

\frac{2 v_{1} (v_{2} + v_{3})}{2 v_{1} + v_{2} + v_{3}}

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Solution

The correct option is D $\frac{2 v_{1} (v_{2} + v_{3})}{2 v_{1} + v_{2} + v_{3}}$
Let the total displacement of the body be $2 S$ and let $t_{1}$ be the time taken to cover the first half.
During the second half, let the time of travel be $2 t$ , so that it travels with $v_{2}$ and $v_{3}$ velocities in time $t$ each.
Then, average velocity $= \frac{2 S}{t_{1} + 2 t}$
Here, $t_{1} = \frac{S}{v_{1}}$ and $(v_{2} + v_{3}) t = S$
i.e $t = \frac{S}{(v_{2} + v_{3})}$

$∴ v_{a v} = \frac{2 S}{\frac{S}{v_{1}} + 2 (\frac{S}{v_{2} + v_{3}})} = \frac{2}{\frac{1}{v_{1}} + \frac{2}{v_{2} + v_{3}}}$
$= \frac{2 v_{1} (v_{2} + v_{3})}{2 v_{1} + v_{2} + v_{3}}$

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