The speeds of two trains A and B are 42 km/h and 33 km/h, respectively. Length of train B is 3/2 times the length of A. Train A takes 50 s to cross train B, if they are moving in the same direction. How long will they take to cross each other if they are moving in opposite directions ?

Options

A. 6s
B. 17s
C. 12 s
D. 9s

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Define Variables
Let:
- Length of Train A = \( L_A = 2k \) meters
- Length of Train B = \( L_B = \frac{3}{2} L_A = 3k \) meters

### Step 2: Calculate Relative Speed in Same Direction
The speeds of the trains are given as:
- Speed of Train A = 42 km/h
- Speed of Train B = 33 km/h

When the trains are moving in the same direction, the relative speed is:
\[ \text{Relative Speed} = \text{Speed of Train A} - \text{Speed of Train B} = 42 \text{ km/h} - 33 \text{ km/h} = 9 \text{ km/h} \]

### Step 3: Convert Relative Speed to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ \text{Relative Speed in m/s} = 9 \text{ km/h} \times \frac{5}{18} = \frac{45}{18} = 2.5 \text{ m/s} \]

### Step 4: Use Time to Cross Train B
It is given that Train A takes 50 seconds to cross Train B. The formula for time taken to cross is:
\[ \text{Time} = \frac{L_A + L_B}{\text{Relative Speed}} \]
Substituting the known values:
\[ 50 = \frac{2k + 3k}{2.5} \]
This simplifies to:
\[ 50 = \frac{5k}{2.5} \]

### Step 5: Solve for k
Multiplying both sides by 2.5:
\[ 50 \times 2.5 = 5k \implies 125 = 5k \implies k = 25 \]

### Step 6: Calculate Lengths of the Trains
Now we can find the lengths of the trains:
- Length of Train A:
\[ L_A = 2k = 2 \times 25 = 50 \text{ meters} \]
- Length of Train B:
\[ L_B = 3k = 3 \times 25 = 75 \text{ meters} \]

### Step 7: Calculate Relative Speed in Opposite Direction
When moving in opposite directions, the relative speed is:
\[ \text{Relative Speed} = \text{Speed of Train A} + \text{Speed of Train B} = 42 \text{ km/h} + 33 \text{ km/h} = 75 \text{ km/h} \]
Convert this to m/s:
\[ \text{Relative Speed in m/s} = 75 \times \frac{5}{18} = \frac{375}{18} = 20.83 \text{ m/s} \]

### Step 8: Calculate Time to Cross Each Other
Now, we can find the time taken to cross each other:
\[ \text{Time} = \frac{L_A + L_B}{\text{Relative Speed}} = \frac{50 + 75}{20.83} = \frac{125}{20.83} \approx 6 \text{ seconds} \]

### Final Answer
The time taken for the two trains to cross each other when moving in opposite directions is approximately **6 seconds**.

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