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A body travelling on a long straight line traversed one –third of the total distance with a velocity $ν_{1}$ . The remaining part of the distance was covered with a velocity $ν_{2}$ for half the time and with velocity $ν_{3}$ for the other half of time. The mean velocity averaged over the whole time of motion is:

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

$\frac{3 v_{1} (v_{2} + v_{3})}{4 v_{1} + v_{2} + v_{3}}$

$\frac{v_{1} (v_{2} + v_{3})}{4 v_{1} + v_{2} + v_{3}}$

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

answer is B.

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Detailed Solution

$t_{1} = \frac{S / 3}{v_{1}} = \frac{S}{3 v_{1}}$

$t_{2} = t_{3}$

$\frac{2 S}{3} = v_{2} t_{2} + v_{3} t_{2}$

$t_{2} = \frac{2 S}{3 (v_{2} + v_{3})}$

Average velocity
$= \frac{S}{t_{1} + 2 t_{2}} = \frac{S}{\frac{S}{3 v_{1}} + \frac{4 S}{3 (v_{2} + v_{3})}} = \frac{3 v_{1} (v_{2} + v_{3})}{v_{2} + v_{3} + 4 v_{1}}$

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