Question

A boy goes to school from his home at a speed of x km/hr and comes back at the speed of y km/hr then the average speed of the boy is
A. \[\dfrac{{x + y}}{2}\,{\text{km/hr}}\]
B. \[\sqrt {xy} \,{\text{km/hr}}\]
C. \[\dfrac{{2xy}}{{x + y}}\,{\text{km/hr}}\]
D. \[\dfrac{{x + y}}{{2xy}}\,{\text{km/hr}}\]

Answer 1

Hint: Express the time taken by the boy to go to school with speed x and also the time taken by the boy to come back to home with speed y. The total distance travelled by the boy is the twice the distance between school and home. Use the relation between velocity, distance and time to express the average velocity of the boy.

Formula used:
\[v = \dfrac{d}{t}\]
Here, v is the speed, d is the distance and t is the time.

Complete step by step answer:
We have given that the boy goes to school at a speed \[{v_1} = x\,{\text{km/hr}}\] and the boy comes back to the home at the speed \[{v_2} = y\,{\text{km/hr}}\]. Let the time taken by the boy to reach the school is \[{t_1}\] and the time taken by the boy to come back home from the school is \[{t_2}\].

Let’s express the time taken by the boy to reach the school as,
\[{t_1} = \dfrac{d}{{{v_1}}}\] …… (1)

Here, d is the distance between the school and the home.

Let’s express the time taken by the boy to come back home from the school as,
\[{t_2} = \dfrac{d}{{{v_1}}}\] …… (2)

We have the average speed of the boy within his journey from the home to the school and then back to the home,
\[{v_{avg}} = \dfrac{{2d}}{{{t_1} + {t_2}}}\]

Here, 2d is the total distance travelled by the boy and \[{t_1} + {t_2}\] is the total time taken by the boy for his journey.

Using equation (1) and (2) in the above equation, we get,
\[{v_{avg}} = \dfrac{{2d}}{{\dfrac{d}{{{v_1}}} + \dfrac{d}{{{v_2}}}}}\]
\[ \Rightarrow {v_{avg}} = \dfrac{2}{{\dfrac{1}{{{v_1}}} + \dfrac{1}{{{v_2}}}}}\]
\[ \Rightarrow {v_{avg}} = \dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}\]

Substituting \[{v_1} = x\,{\text{km/hr}}\] and \[{v_2} = y\,{\text{km/hr}}\] in the above equation, we get,
\[{v_{avg}} = \dfrac{{2xy}}{{x + y}}\,{\text{km/hr}}\]

So, the correct answer is “Option C”.

Note:
Remember, to calculate the average velocity, you should take the total distance travelled. In this case, the total distance travelled is 2d, where, d is the distance between the school and the home. We have seen that, if the distance is the same in the two intervals, the average velocity does not depend on the distance.