Two trains whose respective lengths are 200 m and 250 m cross each other in 18 s when they are travelling in opposite direction and in 1 minute when they are travelling in the same direction.What is the speed of the faster train(in km/h)?

Options

A. 38.5
B. 48.5
C. 54
D. 58.5

Answer

To solve the problem step by step, we will follow the given information and apply the formulas for speed, distance, and time.

### Step 1: Understand the Problem
We have two trains with lengths 200 m and 250 m. They cross each other in two scenarios:
1. When traveling in opposite directions (taking 18 seconds).
2. When traveling in the same direction (taking 1 minute).

### Step 2: Calculate the Total Distance
The total distance when the two trains cross each other is the sum of their lengths:
\[ \text{Total Distance} = 200 \, \text{m} + 250 \, \text{m} = 450 \, \text{m} \]

**Hint:** Remember that when two objects cross each other, the distance covered is the sum of their lengths.

### Step 3: Set Up the Equations
Let \( x \) be the speed of the first train (in m/s) and \( y \) be the speed of the second train (in m/s).

#### For Opposite Directions:
When the trains are moving in opposite directions, their relative speed is \( x + y \). The equation for distance is:
\[ 450 = (x + y) \times 18 \]
From this, we can derive:
\[ x + y = \frac{450}{18} = 25 \, \text{m/s} \quad \text{(Equation 1)} \]

#### For Same Directions:
When the trains are moving in the same direction, their relative speed is \( x - y \). The equation for distance is:
\[ 450 = (x - y) \times 60 \]
From this, we can derive:
\[ x - y = \frac{450}{60} = 7.5 \, \text{m/s} \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + y = 25 \)
2. \( x - y = 7.5 \)

We can solve these equations simultaneously.

Adding both equations:
\[ (x + y) + (x - y) = 25 + 7.5 \]
\[ 2x = 32.5 \quad \Rightarrow \quad x = \frac{32.5}{2} = 16.25 \, \text{m/s} \]

Now, substitute \( x \) back into Equation 1 to find \( y \):
\[ 16.25 + y = 25 \quad \Rightarrow \quad y = 25 - 16.25 = 8.75 \, \text{m/s} \]

### Step 5: Convert Speed to km/h
To convert the speed of the faster train \( x \) from m/s to km/h, we use the conversion factor \( \frac{18}{5} \):
\[ \text{Speed in km/h} = 16.25 \times \frac{18}{5} = 16.25 \times 3.6 = 58.5 \, \text{km/h} \]

### Final Answer
The speed of the faster train is \( 58.5 \, \text{km/h} \).

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