A person travelling in a straight line moves with a constant...

For distance $x$, the person moves with constant velocity $\mathrm{v}_1$ and for another $\mathrm{x}$ distance, he moves with constant velocity of $\mathrm{v}_2$, then Total distance travelled, $\mathrm{D}=\mathrm{x}+\mathrm{x}=2 \mathrm{x}$ Total time-taken, $\mathrm{T}=\mathrm{t}_1+\mathrm{t}_2$ $$=\frac{\mathrm{x}}{\mathrm{v}_1}+\frac{\mathrm{x}}{\mathrm{v}_2} \quad\left[\because \mathrm{t}=\frac{\text{ Distance }}{\text{ Velocity }}\right]$$ The average velocity, $\mathrm{V}_{\mathrm{av}}=\frac{\text{ total distance }}{\text{ total time }}=\frac{\mathrm{D}}{\mathrm{T}}$ \begin{aligned} \mathrm{v} & =\frac{2 \mathrm{x}}{\frac{\mathrm{x}}{\mathrm{v}_1}+\frac{\mathrm{x}}{\mathrm{v}_2}}=\frac{2}{\frac{\mathrm{l}}{\mathrm{v}_1}+\frac{\mathrm{l}}{\mathrm{v}_2}} \quad\left[\because \mathrm{v}_{\mathrm{av}}=\mathrm{v}\right] \\ \Rightarrow \frac{1}{\mathrm{v}_1}+\frac{\mathrm{l}}{\mathrm{v}_2} & =\frac{2}{\mathrm{v}} \end{aligned}