Question

A passenger train of length 60m travels at a speed of 80km/hr. Another freight train of length 120m travels at a speed of 30km/hr. The ratio of time taken by the passenger train to completely cross the freight train when: (1) they are moving in the same direction, and (2) in the opposite direction is ……………?
$
  (a){\text{ }}\dfrac{5}{2} \\
  (b){\text{ }}\dfrac{{25}}{{11}} \\
  (c){\text{ }}\dfrac{3}{2} \\
  (d){\text{ }}\dfrac{{11}}{5} \\
$

Answer 1

Hint: In this question use the concept of relativity that is if two objects travel with different velocity towards each other than the relative speed is the sum of the individual speeds and if two objects travel with different velocity in same direction with respect to each other than the relative speed is the magnitude of the difference of the individual speeds along with the relationship between distance speed and time that is ${\text{distance = speed}} \times {\text{time}}$.

Complete step-by-step answer:
Length of the passenger train = 60m,
Speed of the passenger train = 80 Km/hr.
Length of the freight train = 120m,
Speed of the freight train = 30 Km/hr.
Now according to the theory of relativity if two objects travel with different velocity towards each other than the relative speed is the sum of the individual speeds.
And if two objects travel with different velocity in the same direction with respect to each other then the relative speed is the magnitude of the difference of the individual speeds.
$\left( i \right)$ When they are moving in the same direction
So the relative speed of the two trains is
$V_{rel}' = \left( {80 - 30} \right) = 50$ km/hr.
Now let the time taken to completely cross the freight train by the passenger train be t’ hr.
Now as we know the relation between time, speed and distance, as time taken is the ratio of the distance covered to the speed.
Let the length of the freight train be x km.
\[ \Rightarrow t' = \dfrac{x}{V_{rel}'} = \dfrac{x}{50} hr\].................... (1)
$\left( {ii} \right)$ When they are moving in the opposite direction
So the relative speed of the two trains this time is
$ \Rightarrow {V_{rel}} = \left( {80 + 30} \right) = 110$ Km/hr.
Now let the time taken to completely cross the freight train by the passenger train be (t) hr.
\[ \Rightarrow t = \dfrac{x}{{{V_{rel}}}} = \dfrac{x}{{110}}hr\]................ (2)
Now we have to find the ratio of the times
So divide equation (1) from equation (2) we have,
$ \Rightarrow \dfrac{{t'}}{t} = \dfrac{{\dfrac{x}{{50}}}}{{\dfrac{x}{{110}}}} = \dfrac{{110}}{{50}} = \dfrac{{11}}{5}$
So this is the required ratio of the time taken in the first and second case.
Hence option (D) is the correct answer.

Note – In the concept of relativity we consider one object to come to stationary and its velocity is always given in exactly the opposite direction to the other object that is not being considered at rest. In this way the addition or subtraction sign is being used. For example suppose two trains coming towards each other then consider one train to be at rest and given it’s velocity to the other train but however in the direction opposite to which the train that is being brought to rest was moving.