If a train, A, 250 m long , travelling at a speed of 20 m/s takes 30 s to overtake another train B travelling at a speed of 18 km/h in the same direction. then find the length of the train B.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the speed of Train B from km/h to m/s
Given that Train B travels at a speed of 18 km/h, we need to convert this speed into meters per second (m/s).

\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]

Calculating:

\[ 18 \text{ km/h} = 18 \times \frac{5}{18} = 5 \text{ m/s} \]

### Step 2: Determine the relative speed of Train A with respect to Train B
Since both trains are moving in the same direction, the relative speed is calculated by subtracting the speed of Train B from the speed of Train A.

\[ \text{Relative Speed} = \text{Speed of Train A} - \text{Speed of Train B} \]

Calculating:

\[ \text{Relative Speed} = 20 \text{ m/s} - 5 \text{ m/s} = 15 \text{ m/s} \]

### Step 3: Calculate the total distance covered when Train A overtakes Train B
When Train A overtakes Train B, it covers its own length plus the length of Train B. Let \( L \) be the length of Train B. Therefore, the total distance covered is:

\[ \text{Total Distance} = \text{Length of Train A} + \text{Length of Train B} = 250 \text{ m} + L \]

### Step 4: Use the formula for distance to find the length of Train B
The distance covered can also be calculated using the formula:

\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]

Given that the time taken to overtake is 30 seconds, we can write:

\[ 250 + L = \text{Relative Speed} \times \text{Time} \]

Substituting the values:

\[ 250 + L = 15 \text{ m/s} \times 30 \text{ s} \]

Calculating:

\[ 250 + L = 450 \]

### Step 5: Solve for the length of Train B
Now, we can isolate \( L \):

\[ L = 450 - 250 \]

Calculating:

\[ L = 200 \text{ m} \]

### Final Answer
The length of Train B is **200 meters**.

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