There are two trains runing on two parallel tracks length of each train I s 120 m when they are runing in opposite directions they cross each other in 4 seconds and when they are runing in the same direction they cross in 12 seconds.What is the speed of the faster train?

Options

A. 80 km/h
B. 72 km/h
C. 120 km/h
D. 144 km/h

Answer

To solve the problem, we need to determine the speeds of the two trains based on the information provided about how long it takes them to cross each other in different directions. Let's break it down step by step.

### Step 1: Understand the Problem
We have two trains, each 120 meters long. They cross each other in:
- 4 seconds when running in opposite directions.
- 12 seconds when running in the same direction.

### Step 2: Calculate the Total Distance When Crossing
When the trains cross each other, the total distance they cover is the sum of their lengths:
- Total distance = Length of Train 1 + Length of Train 2 = 120 m + 120 m = 240 m.

### Step 3: Calculate Relative Speed in Opposite Directions
When the trains are running in opposite directions, they cross each other in 4 seconds. The formula for speed is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Using this, we can find the relative speed (A + B):
\[ A + B = \frac{240 \text{ m}}{4 \text{ s}} = 60 \text{ m/s} \]

### Step 4: Calculate Relative Speed in Same Direction
When the trains are running in the same direction, they cross each other in 12 seconds. Using the same speed formula:
\[ A - B = \frac{240 \text{ m}}{12 \text{ s}} = 20 \text{ m/s} \]

### Step 5: Set Up the Equations
Now we have two equations:
1. \( A + B = 60 \) (1)
2. \( A - B = 20 \) (2)

### Step 6: Solve the Equations
To find A and B, we can add the two equations:
\[ (A + B) + (A - B) = 60 + 20 \]
\[ 2A = 80 \]
\[ A = 40 \text{ m/s} \]

Now, substitute A back into one of the equations to find B:
Using equation (1):
\[ 40 + B = 60 \]
\[ B = 60 - 40 = 20 \text{ m/s} \]

### Step 7: Identify the Faster Train
From our calculations:
- Speed of Train A (faster train) = 40 m/s
- Speed of Train B = 20 m/s

### Step 8: Convert Speed to km/h
To convert the speed from m/s to km/h, we multiply by 3.6:
\[ 40 \text{ m/s} \times 3.6 = 144 \text{ km/h} \]

### Final Answer
The speed of the faster train is **144 km/h**.

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