Question

Two trains, each 50m long are traveling in the opposite direction with velocity 10m/s and 15m/s. The time of crossing is
${\text{A}}{\text{. }}2s$
${\text{B}}{\text{. }}4s$
${\text{C}}{\text{. 2}}\sqrt 3$
${\text{D}}{\text{. 4}}\sqrt 3$

Answer 1

-Each train will cover the total length of the two trains.
-Find the relative velocity of one train w.r.t another train.
-Calculate the time required from the relative velocity and the total distance covered by each train.

Formula used:
Total distance covered by each train, $l = {l_1} + {l_2}$
${l_1} =$ length of one train.
${l_2} =$ length of another train.
The relative velocity of one train w.r.t another train, $v = {v_1} - {v_2}$
${v_1} =$ the velocity of one train.
${v_2} =$ the velocity of another train.
Require time,$t = \dfrac{l}{v}$ .

Complete step by step answer:
When two trains cross each other with certain velocities, each of them covers the distance which is the sum of both of their lengths.
Here in the problem, it is given that two trains coming from opposite directions cross each other with certain velocities. So, we have to calculate the relative velocity which is the same for both of the trains.
Now, let
${l_1} =$ the length of one train.
${l_2} =$ length of another train.
So,the Total distance covered by each train, $l = {l_1} + {l_2}$
Given, ${l_1} = 50m$
${l_2} = 50m$
$\therefore l = {l_1} + {l_2} = 100m$.
If, ${v_1} =$ the velocity of one train and,
${v_2} =$ the velocity of another train.
The relative velocity of one train w.r.t another train, $v = {v_1} - {v_2}$
Given, ${v_1} = 10m/s$ and ${v_2} = 15m/s$
Since the second train is coming from the opposite direction, the velocity of the second train has to be taken negatively.
$\therefore v = {v_1} - ( - {v_2})$
$\Rightarrow v = 10 - ( - 15)$
$\Rightarrow v = 25m/s$
So, the time required for the first train to cross another train, $t = \dfrac{l}{v}$
We find, $l = 100$ and $v = 25$
$\therefore t = \dfrac{l}{v} = \dfrac{{100}}{{25}}$
$\Rightarrow t = 4s$

Hence the correct option is $\left( {\text{B}} \right)$

Additional information:
If we calculate the relative velocity for the second train with w.r.t the first train, it will be written as,
$v = {v_2} - {v_1}$
And the velocity of the second train has to be taken negatively.
$\therefore v = {v_2} - ( - {v_1})$
$\Rightarrow v = 15 - ( - 10)$
$\Rightarrow v = 25m/s$
So, we can see that the relative velocity is always the same for two objects.

Note: In the problem the lengths of the two trains are equal. The relative velocity is always the same since here we have to calculate the difference of velocities.
Hence, the time required for both of the trains will be the same for this particular problem.
But if the lengths are not the same the required time also differs for the two trains.