Question

A body covers one-third of the distance with a velocity $v_1$, the second one-third of the distance with a velocity $v_2$ and the remaining distance with a velocity $v_3$. The average velocity is

• A. $\frac{v_1+v_2+v_3}{3}$
• B. $\frac{3v_1{v}_2{v}_3}{v_1{v}_2+v_2{v}_3+v_3{v}_1}$
• C. $\frac{v_1{v}_2+v_2{v}_3+v_3{v}_1}{3}$
• D. $\frac{v_1{v}_2{v}_3}{3}$

Answer 1

The correct option is B $\frac{3v_1{v}_2{v}_3}{v_1{v}_2+v_2{v}_3+v_3{v}_1}$

The average velocity is given by:
$\overline{v}=\frac{\text{total distance}}{\text{total time}}$

Let us assume that each $1/3^{rd}$ distance be $D$ and time taken by the body to cover each distance be $t_1$, $t_2$ and $t_3$ respectively.

So, $\overline{v}=\frac{3D}{t_1+t_2+t_3}$

$\Rightarrow \overline{v}=\frac{3D}{\frac{D}{v_1}+\frac{D}{v_2}+\frac{D}{v_3}}$

$\Rightarrow \overline{v}=\frac{3\mathrm{D/}}{\mathrm{D/}(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3})}$

$\Rightarrow \overline{v}=\frac{3v_1{v}_2{v}_3}{v_1{v}_2+v_2{v}_3+v_3{v}_1}$