Question

A body travelling along a straight line one-third of the total distance with a velocity $4$m/s. The remaining part of the distance was covered with a velocity $2$m/s for the first half of the remaining journey and with a velocity $6$m/s for the other half of the remaining journey. The average velocity is
A. $5$m/s
B. $4$m/s
C. $4.5$m/s
D. $3.5$m/s

Answer 1

Hint: -We know that the formula for the average velocity of any object travelling through different distances in different time span is equal to the total distance travelled by that object divided by the total time taken.

Complete step-by-step answer:
$average $ \[ velocity\]= $\dfrac{{total{\text{ }}distance{\text{ }}travelled}}{{\;total{\text{ }}time{\text{ }}taken}}$
Now let us suppose that the total distance travelled=$d$
Let the time to cover first one-third of the total distance=${t_1}$
So ${t_1} = \dfrac{{\dfrac{1}{3}d}}{4}$
$ \Rightarrow {t_1} = \dfrac{d}{{12}}$-----equation (1)

Let the ${t_2}$ be the time for remaining two journeys,
It is given that the half the time of remaining the velocity of the body is $2$m/s and for the other half time the velocity is $6$m/s.
So, the remaining distance (after travelling the one-third journey) =$\dfrac{{2d}}{3}$
This distance is the sum of the distance covered in two journeys which is covered with two different velocities.
So, $\dfrac{{2d}}{3} = 2{t_2} + 6{t_2}$
$ \Rightarrow \dfrac{{2d}}{3} = 8{t_2}$
$ \Rightarrow \dfrac{{2d}}{{3 \times 8}} = {t_2}$
$ \Rightarrow {t_2} = \dfrac{d}{{12}}$------equation (2)
After combining all the equations for the time, we will get the total time taken=$t$
Hence now, with the help of the equation (1), equation (2),
Total time taken=$t = {t_1} + 2{t_2}$
$ \Rightarrow t = \dfrac{d}{{12}} + (2 \times \dfrac{d}{{12}})$
$ \Rightarrow t = \dfrac{{d + 2d}}{{12}}$
$ \Rightarrow t = \dfrac{{3d}}{{12}}$
$ \Rightarrow t = \dfrac{d}{4}$
So now,
Average velocity = $\dfrac{d}{t}$
$ = \dfrac{d}{{\dfrac{d}{4}}}$
$ = 4$
Hence the average velocity of the whole journey is $4$m/s.
So, the option (B) is the correct answer.

Note: Average velocity for any journey basically shows a common velocity with which the whole journey is being covered. Methods for finding average velocity are different for different given conditions, the above problem is solved with the help of distance and time and if the journey is covered with different variable velocities and initial velocity and final velocity is known for that particular journey then we will use a different method. Then the formula for the average velocity will be the sum of the initial velocity and final velocity and the whole sum is divided by 2.

Average velocity = $\dfrac{{initial{\text{ }}velocity + final{\text{ }}velocity}}{2}$