Two trains one of length `l_(1)=630m` and other of length `l_(2)=120m` move uniformly in two parallel paths in opposite direction with speed `mu_(1)=48 km//h`and `mu_(2)=102 km//h` respectively.

Options

A. The relative speed of trains is `54 km//hr`.
B. The relative speed of trains is `50km//hr`
C. Time taken by one train to completely pass another is `15 s`.
D. Time taken by one train to completely pass another is `18 s`.

Explanation

To solve the problem, we need to find the time taken for the two trains to completely pass each other. Here are the steps involved in the solution:

### Step 1: Convert the speeds from km/h to m/s
The speeds of the trains are given in km/h. We need to convert these speeds to m/s for consistency with the lengths given in meters.

- Speed of train 1, \( \mu_1 = 48 \text{ km/h} \)
- Speed of train 2, \( \mu_2 = 102 \text{ km/h} \)

To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ \mu_1 = 48 \times \frac{5}{18} = \frac{240}{18} = 13.33 \text{ m/s} \]

\[ \mu_2 = 102 \times \frac{5}{18} = \frac{510}{18} = 28.33 \text{ m/s} \]

### Step 2: Calculate the relative speed of the trains
Since the trains are moving in opposite directions, we add their speeds to find the relative speed.

\[ \text{Relative speed} = \mu_1 + \mu_2 = 13.33 + 28.33 = 41.66 \text{ m/s} \]

### Step 3: Calculate the total distance to be covered
The total distance that needs to be covered for one train to completely pass the other is the sum of their lengths.

- Length of train 1, \( l_1 = 630 \text{ m} \)
- Length of train 2, \( l_2 = 120 \text{ m} \)

\[ \text{Total distance} = l_1 + l_2 = 630 + 120 = 750 \text{ m} \]

### Step 4: Calculate the time taken to pass each other
Using the formula for time, which is distance divided by speed, we can find the time taken for the trains to completely pass each other.

\[ \text{Time} = \frac{\text{Total distance}}{\text{Relative speed}} = \frac{750 \text{ m}}{41.66 \text{ m/s}} \approx 18 \text{ seconds} \]

### Final Answer
The time taken for the two trains to completely pass each other is approximately **18 seconds**.

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