A body travels the first half of the total distance with velocity $v_{1}$ and the second half with velocity $v_{2}$ . Calculate the average velocity.

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

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\sqrt{v_{1} v_{2}}

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\sqrt{2 v_{1} v_{2}}

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

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Solution

The correct option is D $\frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$

From above diagram,

total distance $= 2 d$

Body travels $d$ distance from $A$ to $C$ in $t_{1}$ time with velocity $v_{1}$ .

$∴ t_{1} = \frac{d}{v_{1}}$

Similarly, body travels $d$ distance from $C$ to $B$ in $t_{2}$ time with velocity $v_{2}$ .

$∴ t_{2} = \frac{d}{v_{2}}$

So, Average velocity $(< v >)$ is defined as,

$< v > = \frac{total displacement}{total time} = \frac{2 d}{t_{1} + t_{2}}$

$\Rightarrow< v > = \frac{2 d}{\frac{d}{v_{1}} + \frac{d}{v_{2}}} = \frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$

Hence, option $(d)$ is correct.

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