Two trains are moving in the same direction at the speed of 42 km/hr and 84 km/hr and their lengths are 320 metres and 380 metres respectively. What is the time taken (in seconds) by faster train to cross the slower train?

Options

A. 60
B. 80
C. 90
D. 120

Answer

To solve the problem of how long it takes for the faster train to cross the slower train, we can follow these steps:

### Step 1: Determine the speeds of the trains
- The speed of the slower train is 42 km/hr.
- The speed of the faster train is 84 km/hr.

### Step 2: Calculate the relative speed
Since both trains are moving in the same direction, we find the relative speed by subtracting the speed of the slower train from the speed of the faster train.

\[ \text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train} = 84 \text{ km/hr} - 42 \text{ km/hr} = 42 \text{ km/hr} \]

### Step 3: Convert the relative speed from km/hr to m/s
To convert the speed from kilometers per hour to meters per second, we use the conversion factor \( \frac{5}{18} \).

\[ \text{Relative Speed in m/s} = 42 \text{ km/hr} \times \frac{5}{18} = \frac{42 \times 5}{18} = \frac{210}{18} = 11.67 \text{ m/s} \quad (\text{approximately}) \]

### Step 4: Calculate the total distance to be covered
The total distance that the faster train needs to cover to completely cross the slower train is the sum of their lengths.

- Length of the slower train = 320 meters
- Length of the faster train = 380 meters

\[ \text{Total Distance} = 320 \text{ m} + 380 \text{ m} = 700 \text{ m} \]

### Step 5: Calculate the time taken to cross
Using the formula for time, which is:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

We can substitute the total distance and the relative speed:

\[ \text{Time} = \frac{700 \text{ m}}{11.67 \text{ m/s}} \approx 60 \text{ seconds} \]

### Final Answer
The time taken by the faster train to cross the slower train is approximately **60 seconds**.

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