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Relative Motion in 1D

A particle mo...

Question

A particle moves half the time of its journey with $u$ . The rest of the half time it moves with two velocities $V_{1}$ and $V_{2}$ such that half the distance it covers with $V_{1}$ and the other half with $V_{2}$ . Find the net average velocity. Assume straight line motion.

A

\frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}

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B

\frac{2 u (V_{1} + V_{2})}{16 u + V_{1} + V_{2}}

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C

\frac{u (V_{1} + V_{2})}{2 V_{1}}

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D

\frac{2 V_{1} V_{2}}{u + V_{1} + V_{2}}

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Solution

The correct option is A $\frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}$
Let total time be $2 t$
$t = \frac{d}{v_{1}} + \frac{d}{v_{2}} = d \frac{v_{1} + v_{2}}{(v_{1} v_{2})}$
$\frac{d}{t} = \frac{(v_{1} v_{2})}{v_{1} + v_{2}}$
For first half of time
Distance $= u t$
$v_{a v g} = \frac{T o t a l d i s t a n c e}{T o t a l t i m e} = \frac{u t + 2 d}{2 t}$
$\Rightarrow \frac{u (v_{1} + v_{2}) + 2 v_{1} v_{2}}{2 (v_{1} + v_{2})}$

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