Question

A car covers the first half distance between two places at 40 km/h another half at 60 km/h, the average speed of car

(A) 40 km/h

(B) 48 km/h

(C) 50 km/h

(D) 60 km/h

Answer 1

Hint: Assume the distance between the two places is x km. Calculate the time taken by the car to cover the first half and then second half of the total distance in terms of x. Use the relation between velocity, distance and time to determine the average velocity of the car.

Formula used:

\[v = \dfrac{d}{t}\] 

Here, v is the velocity, d is the distance and t is the time.

Complete answer:

Since we don’t know the distance between the two places, we assume it is x km. We can determine the time taken by the car to cover the first half using the relation between distance, velocity and time as follows,

\[{t_1} = \dfrac{{\left( {x/2} \right)}}{{{v_1}}}\] 

Here, \[{v_1}\] is the velocity of the car in the first half.

Substituting 40 km/h for \[{v_1}\] in the above equation, we get,

\[{t_1} = \dfrac{{\left( {x/2} \right)}}{{40}}\]

\[ \Rightarrow {t_1} = \dfrac{x}{{80}}\]                                                              ……. (1)

Now, we can calculate the time taken by the car to cover the second half as follows,

\[{t_2} = \dfrac{{\left( {x/2} \right)}}{{{v_2}}}\] 

Here, \[{v_2}\] is the velocity of the car in the second half.

Substituting 60 km/h for \[{v_2}\] in the above equation, we get,

\[{t_2} = \dfrac{{\left( {x/2} \right)}}{{60}}\]

\[ \Rightarrow {t_2} = \dfrac{x}{{120}}\]                                                              ……. (2)

Now, the total time taken by the car to cover the total distance between the two places is,

\[t = {t_1} + {t_2}\] 

Using equation (1) and (2) in the above equation, we get,

\[t = \dfrac{x}{{80}} + \dfrac{x}{{120}}\]

\[ \Rightarrow t = \dfrac{x}{{48}}\,h\]

We can calculate the average velocity of the car between the two places as follows,

\[{v_{avg}} = \dfrac{x}{t}\] 

Substituting \[\dfrac{x}{{48}}\,h\] for t in the above equation, we get,

\[{v_{avg}} = \dfrac{x}{{\dfrac{x}{{48}}}}\]

\[ \therefore {v_{avg}} = 48\,km/h\]

Therefore, the average velocity of the car is 48 km/h.So, the correct answer is option (B).

Note: We know the formula for average velocity, \[{v_{avg}} = \dfrac{{{x_f} - {x_i}}}{{\Delta t}}\], where \[{x_f}\] is the final position and \[{x_i}\] is the initial position. In the given question, we have assumed the distance of the second place from the first is x km. Therefore, the total displacement of the car is\[x - 0 = x\]. \[\Delta t\] in the above equation is the total elapsed time and not the difference in the time.