Question

Two trains 110m and 90m long, respectively, are running in opposite directions with velocities $36\,Km{h^{ - 1}}$ and $54\,Km{h^{ - 1}}$ . Find the time taken by the two trains to completely cross each other.

Answer 1

Hint: From the individual distance and the speed of the two trains, calculate the total distance covered and the relative speed of them. Substitute the values of this in the time formula given to find the time taken for the trains to completely cross each other.

Formula used:
The time taken is given by
$t = \dfrac{d}{v}$
Where $t$ is the time taken for the trains to cross each other, $d$ is the distance travelled by the trains and $v$ is the velocity of the trains.

Complete step by step solution:
The length of the first train, ${L_1} = 110\,m$
The length of the second train, ${L_2} = 90\,m$
Velocity of the first train, ${v_1} = 36\,km{h^{ - 1}}$
Velocity of the second train, ${v_2} = 54\,km{h^{ - 1}}$
It is clear that the total distance that these two trains need to travel is equal to that of the length of both trains.
$d = 110 + 90$
$d = 200\,m$
Hence the distance need to be travelled by both the trains are $200\,m$
Similarly the relative velocity of the train is the sum of the velocity of the trains.
$\Rightarrow v = {v_1} + {v_2}$
Substituting the known values,
$\Rightarrow v = 36 + 54 = 90\,km{h^{ - 1}}$
Converting the obtained unit into the standard system of units.
$\Rightarrow v = 90 \times \dfrac{5}{8}$
$\Rightarrow v = 25\,m{s^{ - 1}}$
Substituting this in the time formula,
$\Rightarrow$ \[t = \dfrac{d}{v}\]
Substituting the values known in the above step,
$\Rightarrow t = \dfrac{{200}}{{90}}$
$\Rightarrow t = 8\,s$

Hence the two trains take $8$ seconds to completely cross each other.

Note: The distance covered by two trains is taken as the sum of their length. This is because at a unit time, the two trains cover the distance of their length at the same time. Hence the distance covered by them must be equal to their sum of the length.