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Question

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a)$\overline{v}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2}$

b)$\dfrac{2}{v}=\dfrac{1}{{{v}_{1}}}+\dfrac{1}{{{v}_{2}}}$

c)$\overline{v}=\sqrt{{{v}_{1}}{{v}_{2}}}$

d)$\overline{v}=\sqrt{\dfrac{{{v}_{2}}}{{{v}_{1}}}}$

Answer
Verified

$\begin{align}

& \overline{v}=\dfrac{2s}{t} \\

& {{v}_{1}}=\dfrac{s}{{{t}_{1}}} \\

& {{v}_{2}}=\dfrac{s}{{{t}_{2}}} \\

\end{align}$

Let us assume the total distance travelled by the boy as ${{s}_{1}}+{{s}_{2}}$. The time taken to travel in two cases is $t$. Now, the speed of the boy in two cases will be ${{v}_{1}},{{v}_{2}}$.

Let us now calculate the distance travelled for first half time

$\begin{align}

& {{t}_{{}}}=\dfrac{{{s}_{1}}}{{{v}_{1}}} \\

& {{s}_{1}}={{v}_{1}}t \\

\end{align}$

The distance travelled for the second half time will be,

$\begin{align}

& t=\dfrac{{{s}_{2}}}{{{v}_{2}}} \\

& {{s}_{2}}={{v}_{2}}t \\

\end{align}$

The total distance taken by the boy will be ${{s}_{1}}+{{s}_{2}}=({{v}_{1}}+{{v}_{2}})t$

Now, the mean velocity is given by,

$\begin{align}

& \overline{v}=\dfrac{{{s}_{1}}+{{s}_{2}}}{2t} \\

& \overline{v}=\dfrac{({{v}_{1}}+{{v}_{2}})t}{2t} \\

& \overline{v}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2} \\

& \\

& \\

\end{align}$