A passenger train of length 60 m travels at a speed of 80 km / hr. Another freight train , the ratio of times taken by the train to completely cross the freight train when :(i) they are moving in direction , and (ii) in the opposite directions is :

Options

A. `(11)/(5)`
B. `(5)/(2)`
C. `(3)/(2)`
D. `(25)/(11)`

Answer

To solve the problem, we need to find the ratio of the time taken by a passenger train to completely cross a freight train when they are moving in the same direction and when they are moving in opposite directions.

### Given Data:
- Length of the passenger train (L1) = 60 m
- Length of the freight train (L2) = 120 m
- Speed of the passenger train (V1) = 80 km/h
- Speed of the freight train (V2) = 30 km/h

### Step 1: Convert Speeds from km/h to m/s
To work with the lengths in meters, we should convert the speeds from kilometers per hour to meters per second.

\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{1000 \, \text{m}}{3600 \, \text{s}} = \text{Speed in km/h} \times \frac{5}{18} \]

- Speed of the passenger train (V1) in m/s:
\[ V1 = 80 \times \frac{5}{18} = \frac{400}{18} \approx 22.22 \, \text{m/s} \]

- Speed of the freight train (V2) in m/s:
\[ V2 = 30 \times \frac{5}{18} = \frac{150}{18} \approx 8.33 \, \text{m/s} \]

### Step 2: Calculate Time Taken When Moving in the Same Direction
When both trains are moving in the same direction, the relative speed (V_rel) is given by:
\[ V_{\text{rel}} = V1 - V2 = 22.22 - 8.33 = 13.89 \, \text{m/s} \]

The total distance to be covered by the passenger train to completely cross the freight train is the sum of their lengths:
\[ \text{Total Distance} = L1 + L2 = 60 + 120 = 180 \, \text{m} \]

Now, we can calculate the time taken (T1) to cross the freight train:
\[ T1 = \frac{\text{Total Distance}}{V_{\text{rel}}} = \frac{180}{13.89} \approx 12.95 \, \text{s} \]

### Step 3: Calculate Time Taken When Moving in the Opposite Direction
When both trains are moving in opposite directions, the relative speed is given by:
\[ V_{\text{rel}} = V1 + V2 = 22.22 + 8.33 = 30.55 \, \text{m/s} \]

Again, the total distance to be covered remains the same (180 m). Now we can calculate the time taken (T2) to cross the freight train:
\[ T2 = \frac{\text{Total Distance}}{V_{\text{rel}}} = \frac{180}{30.55} \approx 5.89 \, \text{s} \]

### Step 4: Calculate the Ratio of Times
Now we can find the ratio of the time taken when moving in the same direction (T1) to the time taken when moving in the opposite direction (T2):
\[ \text{Ratio} = \frac{T1}{T2} = \frac{12.95}{5.89} \approx 2.20 \]

To express it in a simpler form, we can convert it to a fraction:
\[ \text{Ratio} = \frac{T1}{T2} = \frac{180/13.89}{180/30.55} = \frac{30.55}{13.89} \approx 2.20 \]

### Final Ratio
Thus, the final ratio of the times taken by the passenger train to cross the freight train in the same direction to the time taken in the opposite direction is approximately:
\[ \text{Ratio} = 11:5 \]