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Velocity in 2D

A body cover ...

Question

A body cover one-third of the distance with a velocity $v_{1}$ the second one-third of the distance with a velocity $v_{2}$ and the last one-third of the distance with a velocity $v_{3}$ .The average velocity is

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

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

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

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

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Solution

The correct option is C $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$
Total distance = D
Total time, T = $t_{1} + t_{2} + t_{3} = \frac{D / 3}{v_{1}} + \frac{D / 3}{v_{2}} + \frac{D / 3}{v_{3}} = \frac{D}{3} (\frac{1}{v_{1}} + \frac{1}{v_{3}} + \frac{1}{v_{3}})$
Thus, average velocity = D/T= $\frac{D}{\frac{D}{3} (\frac{1}{v_{1}} + \frac{1}{v_{3}} + \frac{1}{v_{3}})} = \frac{3}{(\frac{1}{v_{1}} + \frac{1}{v_{3}} + \frac{1}{v_{3}})}$
= $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

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