A person travels $x$ distance with velocity $v_{1}$ and then $x$ distance with velocity $v_{2}$ in the same direction. The average velocity of the person is $\mathrm{v}$, then the relation between $v, v_{1}$ and $v_{2}$ will be.

<p>The average velocity is defined as the total displacement divided by the total time. Here, the person travels the same distance $x$ twice, once with velocity $v_1$ and once with velocity $v_2$. </p> <p>The time to travel distance $x$ with velocity $v_1$ is $t_1 = \frac{x}{v_1}$, and the time to travel distance $x$ with velocity $v_2$ is $t_2 = \frac{x}{v_2}$. <br/><br/>The total time is then<br/><br/> $t = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2}$.</p> <p>The total displacement is $2x$. So, the average velocity $v$ is given by</p> <p>$ v = \frac{\text{total displacement}}{\text{total time}} = \frac{2x}{\frac{x}{v_1} + \frac{x}{v_2}} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} $</p> <p>Multiplying both sides by $2$, we get</p> <p>$ \frac{2}{v} = \frac{1}{v_1} + \frac{1}{v_2} $</p>