Question

A particle covers half of its total distance with speed ${v_1}$ and the rest half distance with speed ${v_2}$. Its average speed during the complete journey is:
A. $\dfrac{{{v_1}{v_2}}}{{{v_1} + {v_2}}}$
B. $\dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}$
C. $\dfrac{{2v_1^2v_2^2}}{{v_1^2 + v_2^2}}$
D. $\dfrac{{{v_1} + {v_2}}}{2}$

Answer 1

First of all, we will find the time required for the two half journeys separately and then will find the total distance for the complete journey. Then we will apply the formula which relates speed, time and distance and use the total distance and total time in it. We will manipulate accordingly and obtain the result.

Complete step by step answer:
In the given problem, we are supplied the following data:
A particle covers half of its distance with a given speed ${v_1}$ and the rest half of the distance is covered with speed ${v_2}$ .
We are asked to find the average speed of the particle during the complete journey.

To begin with, let us first take the total distance be $2x$ .
From the above assumption, we can say that the half distance be $x$ and the other half be also $x$.
Let us take the time taken to cover the first half be ${t_1}$ and with a velocity ${v_1}$ .
We know, the relation between speed, time and distance is given,
${v_1} = \dfrac{x}{{{t_1}}}$
$\Rightarrow {t_1} = \dfrac{x}{{{v_1}}}$ …… (1)
Where,
${v_1}$ indicates the velocity of the particle in the first half.
$x$ indicates the first half distance.
${t_1}$ indicates the time taken in the first half.

Let us again take the time taken to cover the first half be ${t_2}$ and with a velocity ${v_2}$ .
We know, the relation between speed, time and distance is given,
${v_2} = \dfrac{x}{{{t_2}}}$
$\Rightarrow {t_2} = \dfrac{x}{{{v_2}}}$ …… (2)
Where,
${v_2}$ indicates the velocity of the particle in the second half.
$x$ indicates the first half distance.
${t_2}$ indicates the time taken in the first half.

Now, we calculate the total distance:
$x + x = 2x$

Again, we calculate the total time taken for the journey:
$\Rightarrow {t_1} + {t_2} = \dfrac{x}{{{v_1}}} + \dfrac{x}{{{v_2}}}$
Now, we calculate the average speed during the complete journey, which is given by the total distance over total time taken.
Mathematically, we can write:
${\text{Average velocity}} = \dfrac{{{\text{total distance}}}}{{{\text{total time}}}}$ …… (3)
Now, we substitute the required values in the equation (3) and we get:

$$\begin{aligned} \Rightarrow {\text{Average velocity}} &= \dfrac{{2x}}{{\left( {\dfrac{x}{{{v_1}}}} \right) + \left( {\dfrac{x}{{{v_2}}}} \right)}} \\\ \Rightarrow {\text{Average velocity}} &= \dfrac{{2x}}{{\dfrac{{x{v_2} + x{v_1}}}{{{v_1}{v_2}}}}} \\\ \Rightarrow {\text{Average velocity}} &= \dfrac{{2x{v_1}{v_2}}}{{x\left( {{v_1} + {v_2}} \right)}} \\\ \Rightarrow {\text{Average velocity}} &= \dfrac{{2{v_1}{v_2}}}{{\left( {{v_1} + {v_2}} \right)}} \\\ \end{aligned}$$

Hence, the average velocity of the particle during the complete journey is found to be $\dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}$ .
The correct option is B.

Note: While solving this problem, many students tend to make mistakes while finding the average velocity. They simply add the two velocities in the numerator and put $2$ in the denominator, which is completely wrong. Try finding the result in a more systematic way.