A train can cross a pole and a tunnel in `(1)/(1200)` hrs and 10 seconds respectively. If Difference between length of tunnel and length of train is 200 meters, then find speed of train.

Options

A. 70 m/s
B. 45 m/s
C. 35 m/s
D. 50 m/s

Answer

To solve the problem, we need to find the speed of the train based on the time it takes to cross a pole and a tunnel, along with the given difference in lengths. Let's break it down step by step.

### Step 1: Convert Time to Seconds
The time taken to cross the pole is given as \( \frac{1}{1200} \) hours. We need to convert this into seconds.

\[ \text{Time to cross pole (T1)} = \frac{1}{1200} \text{ hours} = \frac{1}{1200} \times 3600 \text{ seconds} = 3 \text{ seconds} \]

**Hint:** Remember that 1 hour = 3600 seconds.

### Step 2: Identify the Time to Cross the Tunnel
The time taken to cross the tunnel is given as 10 seconds.

\[ \text{Time to cross tunnel (T2)} = 10 \text{ seconds} \]

**Hint:** Note that the time to cross the tunnel is provided directly in seconds.

### Step 3: Define Variables
Let:
- \( L \) = Length of the train (in meters)
- \( T \) = Length of the tunnel (in meters)

From the problem, we know:
\[ T - L = 200 \text{ meters} \quad \text{(1)} \]

### Step 4: Set Up the Distance Equations
When the train crosses a pole, it covers a distance equal to its own length \( L \) in time \( T1 \):
\[ L = \text{Speed} \times T1 \quad \text{(2)} \]

When the train crosses the tunnel, it covers a distance equal to the sum of the lengths of the train and the tunnel \( (L + T) \) in time \( T2 \):
\[ L + T = \text{Speed} \times T2 \quad \text{(3)} \]

### Step 5: Substitute the Speed
Let the speed of the train be \( x \) meters per second. From equations (2) and (3), we can write:
\[ L = x \times 3 \quad \text{(from T1)} \]
\[ L + T = x \times 10 \quad \text{(from T2)} \]

### Step 6: Substitute \( L \) into the Tunnel Equation
Substituting \( L = 3x \) into the equation for the tunnel:
\[ 3x + T = 10x \]
This simplifies to:
\[ T = 10x - 3x = 7x \quad \text{(4)} \]

### Step 7: Substitute \( T \) into the Length Difference Equation
Now substitute equation (4) into equation (1):
\[ 7x - 3x = 200 \]
This simplifies to:
\[ 4x = 200 \]

### Step 8: Solve for Speed
Now, solve for \( x \):
\[ x = \frac{200}{4} = 50 \text{ meters per second} \]

### Conclusion
The speed of the train is \( 50 \) meters per second.

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