Question

A train passes a station $270$m long in $32$ seconds and a man standing on the station in $14$ seconds. Find
i) The speed of the train in km/hr
ii) The length of the train.

Answer 1

We know that, the relation between speed, distance and time is
${\text{Distance} = \text{speed} \times \text{time}}$.Equating the speed of the train using the formula from both the condition we will get the speed of the train.

Complete step-by-step answer:
It is given that; the length of the station is $270$m. It passes the station in $32$ seconds and a man standing on the station in $14$ seconds.
We have to find the speed of the train and the length of the train.
Let us consider, the length of the train is $l$.
When the train passes the station, it means it covers the total length of the station and its own length.
So, in $32$ seconds the train covers $l + 270$m
Again, when it passes a man, it means it covers only its own length.
So, in $14$ seconds the train covers $l$m.
We know that, the relation between speed, distance and time is
Distance = speed $\times$ time
When, the train covers $l + 270$m in in $32$ seconds, its speed is $\dfrac{{l + 270}}{{32}}$ m/sec
When, the train covers $l$m in in $14$ seconds, its speed is $\dfrac{l}{{14}}$ m/sec
As speed of train is constant so equating both the speeds,we get
$\dfrac{{l + 270}}{{32}} = \dfrac{l}{{14}}$
Simplifying we get,
$14l + 270 \times 14 = 32l$
Simplifying we get,
$270 \times 14 = 18l$
Simplifying we get,
$l = \dfrac{{270 \times 14}}{{18}}$
Solving we get,
$l = 210$
Now, substitute the length of the train we get, the speed of the train is $\dfrac{{210}}{{14}}$ m/sec
Converting into km/hr we get, $\dfrac{{210 \times 60 \times 60}}{{14 \times 1000}}$km/hr
Simplifying we get, the speed of the train is $54$km/hr.
Hence,
i) The speed of the train is $54$km/hr.
ii) The length of the train is $210$m.

Note: When, the train passes the station, it means it covers the total length of the station and its own length. And when a man passes, it means it covers only its own length.