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From Dwarka metro station, metro train starts at regular intervals of $10\,\min $ and runs towards Noida metro station with a constant speed of $80{\text{ }}kmph$ without any stoppage. At some point in time, all the trains simultaneously have to reduce their speed ${\text{50 }}kmph$ due to defects in rails. What will become the time intervals between arrivals of the trains at the Noida metro station during the defect in rails?

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Last updated date: 13th Jun 2024
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Answer
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Hint
Determine the distance between two trains using the total distance and the interval between the departure times of two trains. This same distance will have to be covered at a lower speed and the time to cover this distance will determine the time intervals between arrivals of the trains.
Formula used:
$v = \dfrac{d}{t}$ where $v$ is the velocity of an object, $d$ is the distance travelled by the object, and $t$ is the time taken to cover this distance.

Complete step by step answer
We’ve been given that the speed of the trains $ = {\text{ }}80{\text{ }}kmph$ and they arrive at regular intervals of $10{\text{ mins = 1/6 hour}}$. Then we can calculate the distance between 2 successive trains as:
$d = vt $
$ = 80 \times \dfrac{1}{6} $
$ = \dfrac{{40}}{3}km $
Now when the speed of all the trains is reduced to ${\text{v' = 50 }}kmph$, the distance between 2 successive trains remains the same however the time required by the trains to cover this distance will increase. As a result, the time between arrivals of trains will also increase. We can calculate this time interval $t'$ as:
$t' = \dfrac{d}{{v'}}$
$t' = \dfrac{{40}}{{3 \times 50}} = \dfrac{4}{{15}}hr$
Hence the time interval between the arrival of trails during the defect in the rails is $4/15\,hr$ or equivalently$\dfrac{4}{{15}} \times 60 = 16\,{\text{mins}}$.

Note
The tricky part in this question is breaking the problem down to a single formula as the distance between the trains remains constant because they all travel at the same speed. As a result, the time interval between the arrivals of 2 successive trains will only depend on the change in velocity.