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Finding Average Velocity

A particle mo...

Question

A particle moves half the time of its journey with velocity $u$ . 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.

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

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

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

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

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Solution

The correct option is A $\frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}$

Average velocity:
$¯ v = \frac{total displacement}{total time}$

$¯ v = \frac{\frac{u T}{2} + 2 d}{T}$

Also, $\frac{T}{2} = \frac{d}{V_{1}} + \frac{d}{V_{2}}$

$\Rightarrow d = \frac{T V_{1} V_{2}}{2 (V_{1} + V_{2})}$

So, $¯ v = \frac{\frac{u T}{2} + 2 \frac{T V_{1} V_{2}}{2 (V_{1} + V_{2})}}{T}$

$¯ v = \frac{u}{2} + \frac{V_{1} V_{2}}{(V_{1} + V_{2})}$

$\Rightarrow ¯ v = \frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}$

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