Question

A train overtakes two persons who are walking in the same direction in which the train is going at the rate of $2$ kmph and $4$ kmph and passes them completely in $9$ and $10$ seconds respectively. The length of the train is:
(A) $20m$
(B) $30m$
(C) $40m$
(D) $50m$

Answer 1

Hint:
Start with assuming a variable for the speed and length of the train. Now write the expression for the relative speed of train and person, i.e. ${\text{Relative Speed}} = {\text{Spee}}{{\text{d}}_{train}} - {\text{Spee}}{{\text{d}}_{human}} = \dfrac{{{\text{Distance}}}}{{Time}}$. Make two equations for both the persons. Now combine these equations and find the unknown value of the length of the train.

Complete step by step solution:
Here in this problem, we are given a train that overtakes two persons going in the same direction. Both the people were walking at a rate of $2$ kmph and $4$kmph and the train passed them completely in $9$ and $10$ seconds respectively.
Since the given options are in meters, the speeds of two persons in the MKS system will be:
$ \Rightarrow 2\dfrac{{km}}{h} = 2 \times \dfrac{{1000{\text{ }}m}}{{60 \times 60{\text{ }}s}} = \dfrac{5}{9}{\text{ }}m/s$
Similarly, $4\dfrac{{km}}{h} = 4 \times \dfrac{{1000{\text{ }}m}}{{60 \times 60{\text{ }}s}} = \dfrac{{10}}{9}{\text{ }}m/s$
Before starting with a solution we must understand the concepts of speed and distance. Here the train and two persons are traveling in the same direction but at different speeds. According to the question, the train passes both persons in $9$ and $10$ seconds respectively.
So, as we know that the speed is the ratio of distance traveled by the time taken. We need to find the relative speed of the train and the human.
$ \Rightarrow $ Relative Speed of train and human$ = $ Speed of train $ - $ Speed of human
Let us assume that the train was traveling at $t{\text{ m/s}}$ and has a length of $l{\text{ m}}$
Therefore, the relative speed of train and first-person can be given as:
$ \Rightarrow {\text{Relative Speed}} = \dfrac{{{\text{Distance}}}}{{{\text{Time taken}}}} \Rightarrow t - \dfrac{5}{9} = \dfrac{l}{9}$
This equation can be simplified as:
$ \Rightarrow t - \dfrac{5}{9} = \dfrac{l}{9} \Rightarrow t = \dfrac{{l + 5}}{9}$ ……………(i)
Similarly, for the second person, the relative speed of train and human will be:
$ \Rightarrow {\text{Relative Speed}} = \dfrac{{{\text{Distance}}}}{{{\text{Time taken}}}} \Rightarrow t - \dfrac{{10}}{9} = \dfrac{l}{{10}}$
This equation can be easily simplified as:
 $ \Rightarrow t - \dfrac{{10}}{9} = \dfrac{l}{{10}} \Rightarrow t = \dfrac{{9l + 100}}{{90}}$ …………(ii)
Thus we get two expressions for the speed of the train, ‘t’ in equation (i) and (ii). Equating the RHS of these two equations, we get:
$ \Rightarrow \dfrac{{l + 5}}{9} = \dfrac{{9l + 100}}{{90}}$
Now this equation has only one unknown and can be easily solved as:
$ \Rightarrow \dfrac{{l + 5}}{9} = \dfrac{{9l + 100}}{{90}} \Rightarrow 10\left( {l + 5} \right) = 9l + 100 \Rightarrow 10l - 9l = 100 - 50 \Rightarrow l = 50$
Therefore, we get the length of the train (l) as $50m$

Hence, the option (D) is the correct answer.

Note:
In questions like this, always be careful with the unit system. Notice that the speed of humans is given in ‘kmph’ but the options are given in ‘meters’. Use the conversion $1km/h = \dfrac{5}{{18}}m/s$ to change speeds. An alternative approach can be taken by using relation (i) and (ii) to find the speed of the train and then substitute this in (i) to find the required length of the train.