There are two trains running on two parallel tracks. Length of each train is 120 m. When they are running in opposite directions, they cross each other in 4 seconds and when they are running 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 of finding the speed of the faster train, we can follow these steps:

### Step 1: Understand the Problem
We have two trains, each of length 120 meters. When they are moving in opposite directions, they cross each other in 4 seconds. When they are moving in the same direction, they cross each other in 12 seconds. We need to find the speed of the faster train.

### Step 2: Set Up the Equations
1. **When trains are moving in opposite directions:**
- The total distance covered when they cross each other is the sum of their lengths:
\[ L_1 + L_2 = 120 + 120 = 240 \text{ meters} \]
- Let \( V_1 \) be the speed of the first train and \( V_2 \) be the speed of the second train. The relative speed when moving in opposite directions is \( V_1 + V_2 \).
- The formula for time is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Therefore, we have:
\[ 4 = \frac{240}{V_1 + V_2} \]
- Rearranging gives us:
\[ V_1 + V_2 = \frac{240}{4} = 60 \text{ m/s} \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
2. **When trains are moving in the same direction:**
- The total distance is still 240 meters.
- The relative speed when moving in the same direction is \( V_1 - V_2 \).
- Using the time formula again:
\[ 12 = \frac{240}{V_1 - V_2} \]
- Rearranging gives us:
\[ V_1 - V_2 = \frac{240}{12} = 20 \text{ m/s} \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( V_1 + V_2 = 60 \)
2. \( V_1 - V_2 = 20 \)

We can solve these equations simultaneously.

- Adding both equations:
\[ (V_1 + V_2) + (V_1 - V_2) = 60 + 20 \]
\[ 2V_1 = 80 \]
\[ V_1 = \frac{80}{2} = 40 \text{ m/s} \]

- Now, substituting \( V_1 \) back into Equation 1 to find \( V_2 \):
\[ 40 + V_2 = 60 \]
\[ V_2 = 60 - 40 = 20 \text{ m/s} \]

### Step 5: Identify the Faster Train
From our calculations, we have:
- Speed of the first train \( V_1 = 40 \text{ m/s} \)
- Speed of the second train \( V_2 = 20 \text{ m/s} \)

Thus, the faster train is the first train with a speed of 40 m/s.

### Step 6: Convert to km/h
To convert the speed from meters per second to kilometers per hour, we use the conversion factor \( \frac{18}{5} \):
\[ V_1 = 40 \text{ m/s} \times \frac{18}{5} = 40 \times 3.6 = 144 \text{ km/h} \]

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

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