Question

A car moves a distance of $200 m$. It covers the first half of the distance at speed $40 km/h$ and the second half of the distance at speed v. If the average speed is $48 km/h$, then the value of v is-
A. $50 km/h$
B. $56 km/h$
C. $58 km/h$
D. $60 km/h$

Answer 1

Hint: Express the time taken by the car to cover the first half and then cover the second half in terms of total distance. The addition of these two times is the total time taken by the car to cover the total distance. Use the relation between distance, velocity and time to determine the speed of the car in the second half.

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

Complete step by step answer:
We assume the total distance covered by the car is x. We have given the speed of the car in the first half is \[{v_1} = 40\,{\text{km/hr}}\] and the average speed of the car in the total journey is \[{v_{avg}} = 48\,{\text{km/hr}}\].

Let’s express the time taken by the car to cover the first half as follows,
\[{t_1} = \dfrac{{\left( {x/2} \right)}}{{{v_1}}}\]
\[ \Rightarrow {t_1} = \dfrac{x}{{2{v_1}}}\] ……. (1)

Let’s express the time taken by the car to cover the second half as follows,
\[{t_2} = \dfrac{{\left( {x/2} \right)}}{v}\]
\[ \Rightarrow {t_2} = \dfrac{x}{{2v}}\] ……. (2)

We can express the average speed of the car as follows,
\[{v_{avg}} = \dfrac{x}{t}\] …… (3)
Here, t is the total time taken by the car to cover the distance x. The total time taken by the car is,
\[t = {t_1} + {t_2}\]

Using equation (1) and (2), we can write the above equation as,
\[t = \dfrac{x}{{2{v_1}}} + \dfrac{x}{{2v}}\]

Substituting the above equation in equation (3), we get,
\[{v_{avg}} = \dfrac{x}{{\dfrac{x}{{2{v_1}}} + \dfrac{x}{{2v}}}}\]
\[ \Rightarrow {v_{avg}} = \dfrac{{2{v_1}v}}{{{v_1} + v}}\]

Substituting \[{v_1} = 40\,{\text{km/hr}}\] and \[{v_{avg}} = 48\,{\text{km/hr}}\] in the above equation, we get,
\[48 = \dfrac{{2\left( {40} \right)v}}{{40 + v}}\]
\[ \Rightarrow 48\left( {40 + v} \right) = 80v\]
\[ \Rightarrow 1920 + 48v = 80v\]
\[ \Rightarrow 32v = 1920\]
\[ \Rightarrow v = \dfrac{{1920}}{{32}}\]
\[ \therefore v = 60\,{\text{km/hr}}\]

Therefore, the correct answer is option D.

Note: To solve such types of questions, we don’t require the total distance covered by the car. We can express the equations such that the term total distance will get cancelled. However, you can solve the question by taking the total distance term. Note that we didn’t convert the speed of the car into m/s as it does not require.