Question

Two trains of length ${l_1}\,(and)\,{l_2}$ are moving on parallel tracks with speed ${v_1}\,(and)\,{v_2}$ such that $({v_1} > {v_2})$ with respect to the ground. The time taken to cross each other when they move in the same direction is ?

Answer 1

In order to solve this question, we will use the concept of relative velocity which means when two bodies are moving with their own velocities then the velocity of one body with respect to another body is the sum of each velocity when they move in opposite direction whereas difference of individual velocity when they move in same direction.

Formula used:
If two bodies are moving in same direction with velocities ${v_1}(and){v_2}$ the, velocity of one body with respect to another body is ${v_{12}} = {v_1} - {v_2}$ such that $({v_1} > {v_2})$.

Complete step by step answer:
According to the question, train of length ${l_1}$ is moving faster than train of length ${l_2}$ such that $({v_1} > {v_2})$ so, the relative velocity of train one with another will be ${v_{12}} = {v_1} - {v_2}$
Now, since total length covered by this relative velocity of train one will be the sum of lengths of both trains which is
$\text{Length} = {l_1} + {l_2}$
Now, using the general formula of time distance and speed we have,
$\text{time} = \dfrac{\text{Length}}{\text{speed}}$
$\therefore \text{time} = \dfrac{{{l_1} + {l_2}}}{{{v_1} - {v_2}}}$

Hence, the time taken by the train to cross each other is $time = \dfrac{{{l_1} + {l_2}}}{{{v_1} - {v_2}}}$.

Note: It should be remembered that, while solving such questions always check the directions of moving trains if they were moving in opposite direction then the relative velocity of one train with respect to another would became ${v_{12}} = {v_1} + {v_2}$, relative velocity concept is very useful in questions where two or more bodies move in different directions and we need to examine the distance or time variables.