Question

Two trains can run at a speed of 54 km/hr and 36 km/hr on parallel tracks. When To solve this question we will use the concept of relative speed. First we will calculate the relative speed of the trains in both cases when trains are running in opposite directions and when they move in the same direction. Then we calculate the length of train by using the basic formula of speed-distance relation which is given by
 $ \text{speed=}\dfrac{\text{distance}}{\text{time}} $

Answer 1

Hint: To solve this question we will use the concept of relative speed. First we will calculate the relative speed of the trains in both cases when trains are running in opposite directions and when they move in the same direction. Then we calculate the length of the train by using the basic formula of speed-distance relation which is given by
$\text{speed=}\dfrac{\text{distance}}{\text{time}}$


Complete step by step answer:
We have been given that two trains can run at a speed of 54 km/hr and 36 km/hr on parallel tracks.
We have to find the length of the trains.
Let us assume that the speed of the faster train is 54 km/hr and the speed of the slower train is 36 km/hr.
Now, as given in the question, when the trains are running in opposite directions they pass each other in 10 s. So in this case the relative speed of the trains will be
 $ \begin{align}
  & \Rightarrow 54+36\text{ km/hr} \\
 & \Rightarrow 90\text{ km/hr} \\
\end{align} $
 $ \begin{align}
  & \Rightarrow 90\times \dfrac{5}{18}\text{ m/sec} \\
 & \Rightarrow 25\text{ m/sec} \\
\end{align} $
We have given time to cross each other is 10 sec.
So, the length of train will be
 $ \text{speed=}\dfrac{\text{distance}}{\text{time}} $
Or
 $ \begin{align}
  & \Rightarrow \text{length=speed}\times \text{time} \\
 & \Rightarrow \text{length=25}\times \text{10} \\
 & \Rightarrow \text{length=250 m} \\
\end{align} $
Also, given in the question when trains move in the same direction a person sitting in the faster train crosses the other train in 30 s.
Now, we know that the person sitting in the faster train must have travelled the distance equivalent to the length of the slower train.
So, the relative speed of the trains will be
 $ \begin{align}
  & \Rightarrow 54-36\text{ km/hr} \\
 & \Rightarrow 18\text{ km/hr} \\
\end{align} $
 $ \begin{align}
  & \Rightarrow 18\times \dfrac{5}{18}\text{ m/sec} \\
 & \Rightarrow 5\text{ m/sec} \\
\end{align} $
Now, the length of the slower train will be
 $ \text{speed=}\dfrac{\text{distance}}{\text{time}} $
Or
 $ \begin{align}
  & \Rightarrow \text{length=speed}\times \text{time} \\
 & \Rightarrow \text{length=5}\times 3\text{0} \\
 & \Rightarrow \text{length=150 m} \\
\end{align} $
Now, the length of the faster train will be $ 250-150=100\text{ m} $
So, we have the length of the trains 100 m and 150 m faster and slower respectively.
Option A is the correct answer.

Note:
 It is necessary to convert the units because different units lead to an incorrect answer. Also, keep in mind that relative speed is the speed of a moving object with respect to another object. The possibility of mistake in this question is that students can take 250 as the length of the slower train.