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: Firstly we will assume the length of the train and speed of the train to be any variable. Then we will convert the speed of the persons into \[m/s\]. We will then form two equations based on the given condition and by using the distance, speed and time formula. Then by solving these two equations we will get the required value of the length of the train.

Complete step by step solution:
Let \[x\] meters be the length of the train and \[y\,\]\[m/s\] be the speed of the train.
It is given that the speed of the two persons who are walking in the same direction in which the train is going at the rate of 2 kmph and 4 kmph.
Firstly we will convert the given speed of the two persons from kmph to \[m/s\].
We know that to convert kmph into \[m/s\], we need to multiply the speed with \[\dfrac{5}{{18}}\]. Therefore, we get
Speed of first person \[ = 2kmph = 2 \times \dfrac{5}{{18}}m/s\]
Multiplying the terms, we get
Speed of first person \[ = \dfrac{5}{9}m/s\]
Now,
Speed of second person \[ = 4kmph = 4 \times \dfrac{5}{{18}}m/s\]
Multiplying the terms, we get
Speed of second person \[ = \dfrac{{10}}{9}m/s\]
Now it is given that the train passes them completely in 9 and 10 seconds. We know that ratio of the distance to the speed is equals to the time. Therefore, we get
Time taken by train to cross first person \[ = 9\sec \]
\[ \Rightarrow \dfrac{{{\rm{distance}}}}{{{\rm{relative}}\,{\rm{speed}}}} = 9\sec \]
Substituting the values in above equation, we get
\[ \Rightarrow \dfrac{x}{{y - \dfrac{5}{9}}} = 9\sec \]
Simplifying the equation, we get
\[ \Rightarrow 9y - 5 = x\]
We will multiply both sides by 10. Therefore the equation becomes
\[ \Rightarrow 90y - 50 = 10x\]……………… \[\left( 1 \right)\]
Time taken by train to cross second person \[ = 10\sec \]
\[ \Rightarrow \dfrac{{{\rm{distance}}}}{{{\rm{relative}}\,{\rm{speed}}}} = 10\sec \]
\[ \Rightarrow \dfrac{x}{{y - \dfrac{{10}}{9}}} = 10\sec \]
Simplifying the equation, we get
\[ \Rightarrow 90y - 100 = 9x\]………………\[\left( 2 \right)\]
Now we will subtract equation \[\left( 2 \right)\] from the equation \[\left( 1 \right)\] to get the value of \[x\]. Therefore, we get
\[ \Rightarrow \left( {90y - 50} \right) - \left( {90y - 100} \right) = 10x - 9x\]
Rewriting the above equation, we get
\[ \Rightarrow 90y - 50 - 90y + 100 = x\]
Adding and subtracting the like terms, we get
\[ \Rightarrow x = 50m\]
Hence, the length of the train is \[50m\].
So, option D is the correct option.

Note: We should know that the speed is equal to the ratio of the distance to the time taken to travel that distance. In this type of question we will consider a person as a point and length of train as the distance covered by the train. We will take the relative speed between the train and the person.
We need to keep in mind that when the train and the person are going in the same direction then their relative speed is equal to the difference between their speeds. When the person and the train are going in opposite directions their relative speed is equal to the sum of their speed.