Question

A person travels along a straight road for first half length with a velocity $v_1$ and the second half length with a velocity $v_2$. Then the mean velocity $v$ is given by

• A. $v=\frac{v_1+v_2}{2}$
• B. $v=\sqrt{v_1{v}_2}$
• C. $v=\sqrt{\frac{v_1}{v_2}}$
• D. $\frac{2}{v}=\frac{1}{v_1}+\frac{1}{v_2}$

Answer 1

The correct option is D $\frac{2}{v}=\frac{1}{v_1}+\frac{1}{v_2}$

Let the total distance be d.
Time taken for the first half half distance is $t_1=\frac{d}{2v_1}$

Time taken for the second half of the distance is $t_2=\frac{d}{2v_2}$

Mean velocity $=\frac{\text{total distance}}{\text{total time}}$

$v_{mean}=\frac{d}{\frac{d}{2v_2}+\frac{d}{2v_1}}$

$v_{mean}=\frac{2v_1{v}_2}{v_1+v_2}$

$\Rightarrow \frac{2}{v_{mean}}=\frac{1}{v_1}+\frac{1}{v_2}$

$\text{Given,}{v}_{mean}=v$

$\Rightarrow \frac{2}{v}=\frac{1}{v_1}+\frac{1}{v_2}$