Two trains 160 m and 140 m long are running in opposite directions on parallel rails, the first at 77 km/hr and the other at 67 km/hr. How long will they take to cross each other?

Options

A. 7 seconds
B. `7(1)/(2)` seconds
C. 6 seconds
D. 10 seconds

Answer

To solve the problem of how long it will take for two trains to cross each other, we can follow these steps:

### Step 1: Determine the lengths of the trains
The lengths of the two trains are given as:
- Train 1: 160 m
- Train 2: 140 m

### Step 2: Calculate the total distance to be covered
When two trains cross each other, the total distance they need to cover is the sum of their lengths:
\[ \text{Total Distance} = \text{Length of Train 1} + \text{Length of Train 2} \]
\[ \text{Total Distance} = 160 \, \text{m} + 140 \, \text{m} = 300 \, \text{m} \]

### Step 3: Determine the speeds of the trains
The speeds of the trains are given as:
- Speed of Train 1: 77 km/hr
- Speed of Train 2: 67 km/hr

### Step 4: Calculate the relative speed
Since the trains are moving in opposite directions, their speeds add up:
\[ \text{Relative Speed} = \text{Speed of Train 1} + \text{Speed of Train 2} \]
\[ \text{Relative Speed} = 77 \, \text{km/hr} + 67 \, \text{km/hr} = 144 \, \text{km/hr} \]

### Step 5: Convert the relative speed to meters per second
To convert km/hr to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ \text{Relative Speed in m/s} = 144 \times \frac{5}{18} \]
Calculating this:
\[ \text{Relative Speed in m/s} = 144 \times \frac{5}{18} = 40 \, \text{m/s} \]

### Step 6: Calculate the time taken to cross each other
The time taken to cross each other can be calculated using the formula:
\[ \text{Time} = \frac{\text{Total Distance}}{\text{Relative Speed}} \]
Substituting the values we have:
\[ \text{Time} = \frac{300 \, \text{m}}{40 \, \text{m/s}} = 7.5 \, \text{seconds} \]

### Final Answer
The time taken for the two trains to cross each other is **7.5 seconds**.

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