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

Hint: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.