Question

A person covers a certain distance by car at the speed of 30km/hr and comes back with the speed of 40km/hr.
The average speed of the car is
[a] 34.3 km/hr
[b] 35 km/hr
[c] 37.5km/hr
[d] 32.8 km/hr

Answer 1

Hint: Assume that the distance covered by the car in each trip is x km/hr. Use the fact that the time taken to cover distance s with velocity v is given by $t=\dfrac{s}{v}$. Hence determine the time taken by car in covering the distance in the whole trip. Use the fact that the average speed is given by $v=\dfrac{{{s}_{total}}}{{{t}_{total}}}$, where ${{s}_{total}}$ is the total distance covered and ${{t}_{total}}$ is the total time taken.

Complete step-by-step answer:
Let the distance covered by the car in each direction of the trip be x.
We know that the time taken to cover distance s with velocity v is given by $t=\dfrac{s}{v}$.
Hence time taken in covering the distance x @ 30 km/hr is given by ${{t}_{1}}=\dfrac{x}{30}$
Similarly the time taken in covering the distance x @40 km/hr is given by ${{t}_{2}}=\dfrac{x}{40}$
Hence the total time is given by ${{t}_{total}}={{t}_{1}}+{{t}_{2}}=\dfrac{x}{30}+\dfrac{x}{40}$
Also the total distance covered by the car $=2x$
We know that the average speed is given by $v=\dfrac{{{s}_{total}}}{{{t}_{total}}}$, where ${{s}_{total}}$ is the total distance covered and ${{t}_{total}}$ is the total time taken.
Hence, we have
${{v}_{av}}=\dfrac{2x}{\dfrac{x}{30}+\dfrac{x}{40}}=\dfrac{2\times 40\times 30}{70}=34.3Km/hr$
Hence option [a] is correct.

Note: Alternatively, we can use the fact that if a distance x is covered at the speed u and the same distance is covered with speed v, then the average speed is given by
${{v}_{av}}=\dfrac{2uv}{u+v}$(The harmonic mean of u and v)
Hence, we have
${{v}_{av}}=\dfrac{2\times 40\times 30}{30+40}=34.3km/hr$, which is the same as obtained above.
Hence option [a] is correct.