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A person is walking parallel to a railway line, at a speed of $5\,{\rm{km/h}}$. A train travelling in the opposite direction at $49\,{\rm{km/h}}$, passes him in 12 seconds. Find the length of the train.
A.$160\,{\rm{m}}$
B.$130\,{\rm{m}}$
C.$120\,{\rm{m}}$
D.$180\,{\rm{m}}$

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Last updated date: 22nd Mar 2024
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MVSAT 2024
Answer
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Hint:
In the solution, first we have to calculate the relative speed of the person and the train. Since the train and the person are moving in the opposite direction to each other, we have to add the given speed of the train with the speed of the person. This will give the relative speed. After that we have to apply the formula of the speed$ s = \dfrac{d}{t}$, where $d$ is the distance, $t$ is the time and $s$ is the speed. By substituting the values, we have to calculate the length of the train.

Complete step by step solution:
Let the length of the train be ‘$l$’.
The time taken by the train to pass the person is ‘$t$’.
Now, calculating the relative speed.
Here, the person is moving in one direction with a speed $5\,{\rm{km/h}}$ and the train is moving in the opposite direction with a speed of $49\,{\rm{km/h}}$.
So, the relative speed $\begin{array}{l} = \left( {49 + 5} \right){\rm{ km/h}}\\{\rm{ = }}54{\rm{ km/h}}\end{array}$
Converting the relative speed ${\rm{km/h}}$into ${\rm{m/s}}$.
$\begin{array}{l} = 54{\rm{ km/h}}\\{\rm{ = }}\dfrac{{{\rm{54 km}}}}{{1\,{\rm{h}}}}\\{\rm{ = }}\dfrac{{{\rm{54}} \times {\rm{1000 m}}}}{{60 \times 60\,{\rm{s}}}}\\ = \dfrac{{{\rm{540 m}}}}{{36\,{\rm{s}}}}\\ = 15\,{\rm{m/s}}\end{array}$
So, the relative speed comes out to be$15\,{\rm{m/s}}$.
It is known that the speed of an object is$s = \dfrac{d}{t}$, where $d$ is the distance, $t$ is the time and $s$ is the speed.
Given that the train passes the person in 12 seconds.
Substituting the values $15\,{\rm{m/s}}$ for$s$, $12\,{\rm{s}}$ for $t$ and $l$ for $d$ in the above formula, we get
$\begin{array}{c} \Rightarrow s = \dfrac{l}{t}\\ \Rightarrow l = s \times t\\ \Rightarrow l = 15 \times 12\,\\ \Rightarrow l = 180\;{\rm{meters}}\end{array}$
Hence, the required length of the train is 180 meter.

Note:
Relative speed is a speed where one object is moving with respect to the other. If two objects are moving in the same direction, then their relative speed will be the difference of the speed of the objects. Similarly, when they move in the opposite direction, addition of their speed determines the relative speed. Here we have to determine the total length of the train. Since the speed of the train and the person is given, we can calculate their relative speed. Once we get the relative speed of the person and the train, substituting the values in the formula of speed, we can calculate the length of the train easily.