Question

# A person travels along a straight line for the first half time with velocity ${{v}_{1}}$ and the second half time with velocity ${{v}_{2}}$. The mean velocity $\overline{v}$ is given by: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)$\overline{v}=\sqrt{\dfrac{{{v}_{2}}}{{{v}_{1}}}}$

Hint: Write down the mean velocity formula and try to frame all the values in terms of known velocity and time. Also, the mean velocity is equal to the total distance travelled by the body divided by the total time taken by the body to reach the final point. Don’t get confused between average velocity and mean velocity.

Formula used:
\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 ${{v}_{1}},{{v}_{2}}$.
\begin{align} & {{t}_{{}}}=\dfrac{{{s}_{1}}}{{{v}_{1}}} \\ & {{s}_{1}}={{v}_{1}}t \\ \end{align}
\begin{align} & t=\dfrac{{{s}_{2}}}{{{v}_{2}}} \\ & {{s}_{2}}={{v}_{2}}t \\ \end{align}
The total distance taken by the boy will be ${{s}_{1}}+{{s}_{2}}=({{v}_{1}}+{{v}_{2}})t$
\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}