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 is given by
A. $v = \dfrac{{{v_1} + {v_2}}}{2}$
B. $v = \sqrt {{v_1}{v_2}}$
C. $v = \sqrt {\dfrac{{{v_1}}}{{{v_2}}}}$
D. $\dfrac{2}{v} = \dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}$

Answer 1

Use the formula for displacement of an object. This formula gives the relation between the velocity of the object, displacement of the object and time. Using this formula determines the expression for the time required for the first half travel and second half travel. Then determine the time for the total travel. Using the formula for velocity, determine the expression for the mean velocity.

Formula used:
The velocity $v$ of an object is given by
$v = \dfrac{x}{t}$ …… (1)
Here, $x$ is the displacement of the object and $t$ is the time.

Complete step by step answer:
We have given that a person is travelling along the straight road. The velocity of the road for half-length of its total travel is ${v_1}$and the velocity for the remaining half length of the total travel is ${v_2}$.We have asked to calculate the mean velocity of the person.Let $d$ be the total displacement of the person.Hence, the displacement of the person after of its travel is $\dfrac{d}{2}$.

Let ${t_1}$ be the time required for travelling the first half length.Hence, the velocity ${v_1}$ of the person according to equation (1) is given by
${v_1} = \dfrac{{\dfrac{d}{2}}}{{{t_1}}}$
$\Rightarrow {t_1} = \dfrac{d}{{2{v_1}}}$
Let ${t_2}$ be the time required for travelling the first half length.Hence, the velocity ${v_2}$ of the person according to equation (1) is given by
${v_2} = \dfrac{{\dfrac{d}{2}}}{{{t_2}}}$
$\Rightarrow {t_2} = \dfrac{d}{{2{v_2}}}$

Thus, the total time $t$ required for the travel is given by
$t = {t_1} + {t_2}$
Substitute $\dfrac{d}{{2{v_1}}}$ for ${t_1}$ and $\dfrac{d}{{2{v_2}}}$ for ${t_2}$ in the above equation.
$t = \dfrac{d}{{2{v_1}}} + \dfrac{d}{{2{v_2}}}$
$\Rightarrow t = \dfrac{d}{2}\left( {\dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}} \right)$
Let us now calculate the mean velocity $v$ of the person.
Substitute $d$ for $x$ and $\dfrac{d}{2}\left( {\dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}} \right)$ for $t$ in equation (1).
$v = \dfrac{d}{{\dfrac{d}{2}\left( {\dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}} \right)}}$
$\Rightarrow \dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}} = \dfrac{2}{v}$
$\therefore \dfrac{2}{v} = \dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}$
This is the required expression for mean velocity of the person.

Hence, the correct option is D.

Note: The students may think that the mean velocity of the person is the average of the two velocities of the person in two travels of total displacement. But the students should keep in mind that this average of two velocities gives the average velocity and not mean velocity. Thus, the students should not get confused between the mean velocity and average velocity.