A body covers one-third of its journey with speed $u$ , next one-third with speed $v$ and the last one-third with speed $w$ . Calculate the average speed of the body over the entire journey.

\frac{u v w}{u v + v w + u w}

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\frac{3 u v w}{u v + v w + u w}

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\frac{2 u v w}{u v + v w + u w}

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\frac{4 u v w}{u v + v w + u w}

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Solution

The correct option is B $\frac{3 u v w}{u v + v w + u w}$
Let the distance travelled in the whole journey be $D$
Part 1: (One-third distance with speed $u$ )
Distance = $D / 3$ ; Speed = $u$
Time = Distance/Speed = $D / 3 u$
Part 2: (One third with speed $v$ )
Distance = $D / 3$ ; Speed = $v$
Time = Distance/Speed = $D / 3 v$
Part 3: (One third with speed w)
Distance = $D / 3$ ; Speed = $w$
Time = Distance/Speed= $D / 3 w$ .
Average speed over the entire journey $= \frac{Total Distance}{Total time}$
$= \frac{D}{\frac{D}{3 u} + \frac{D}{3 v} + \frac{D}{3 w}} = \frac{3 u v w}{u v + v w + u w}$

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