Two trains are each 50 m long moving parallel towards each other at speeds $10m{{s}^{-1}}$and $15m{{s}^{-1}}$ respectively. After what time will they pass each other?
A. $5\sqrt{\dfrac{2}{3}}$ sec
B. 4 sec
C. 2 sec
D. 6 sec

Hint:Here the concept of relative velocity is used. By applying the concept of relative velocity, we are able to find the resultant velocity and also length of train is given. Using the relation between length, velocity and time we are able to find the time at which they pass each other.

Formula used:
Relative velocity is the vector sum of velocity of both the trains. If ${{v}_{A}}$ is the velocity of one train and another train has velocity ${{v}_{B}}$, then relative velocity is given as:
${{v}_{AB}}={{v}_{A}}-{{v}_{B}}$
And we know that time,
$t=\dfrac{s}{{{v}_{AB}}}$
where s= distance covered and t=time taken

Complete step by step solution:
In most of the cases, the concept of relative velocity is applied to solve the problems related to dynamics. Relative velocity is defined as the velocity of an object with respect to another observer. Mathematically, we can quote that relative velocity is the rate of change of relative position of one particle with respect to another.

Here in the question, it is given that both of the trains are approaching each other in opposite directions, that is one train has negative velocity with respect to each other. Therefore, the equation for relative velocity becomes,
${{v}_{AB}}={{v}_{A}}+{{v}_{B}}$
Therefore, the resultant velocity,
${{v}_{AB}}=10+15=25\,m{{s}^{-1}}$
Total distance covered= sum of length of both the train
Therefore, the distance covered, s=100m
We know that time taken by the train to pass each other is:
$\therefore t=\dfrac{s}{{{v}_{AB}}}=\dfrac{100}{25}=4$ sec

Hence, the correct answer is option B.

Note: Here total distance taken is the sum of length of both the trains because the question asks the time they pass each other. And also remember in this question we have to add velocities since both trains are moving opposite to each other. If they were moving in the same direction then we had to subtract the velocities to get relative velocity.