Understanding Train Speed Problems

This problem involves calculating the speed of a train based on the time it takes to pass a stationary object (a man) and a stationary length (a platform). We need to use the fundamental relationship between distance, speed, and time: Speed = Distance / Time.

Key Concepts for Train Problems

• Passing a man/pole: When a train passes a man or a pole, the distance covered by the train is equal to its own length.
• Passing a platform/bridge/tunnel: When a train passes a platform, bridge, or tunnel, the total distance covered by the train is the sum of its own length and the length of the platform/bridge/tunnel.

Step-by-Step Calculation

Step 1: Define Variables

Let's define the unknown quantities:

• Let the length of the train be $L$ meters.
• Let the speed of the train be $S$ meters per second (m/s).

Step 2: Formulate Equations Based on Given Information

We are given two scenarios:

• Scenario 1: Train passing a man
  • Time taken = 4 seconds
  • Distance covered = Length of the train = $L$ meters
  • Using Speed = Distance / Time, we get: $S = L / 4$
  • From this, we can express the length of the train in terms of speed: $L = 4S$
• Scenario 2: Train passing a platform
  • Time taken = 12 seconds
  • Length of the platform = 500 meters
  • Distance covered = Length of the train + Length of the platform = $(L + 500)$ meters
  • Using Speed = Distance / Time, we get: $S = (L + 500) / 12$

Step 3: Solve for the Speed of the Train (in m/s)

Now, we can substitute the expression for $L$ from Scenario 1 into the equation from Scenario 2:

$$ S = (4S + 500) / 12 $$

To solve for $S$, we can cross-multiply:

$$ 12S = 4S + 500 $$

Subtract $4S$ from both sides:

$$ 12S - 4S = 500 $$

$$ 8S = 500 $$

Divide by 8:

$$ S = 500 / 8 $$

$$ S = 125 / 2 = 62.5 \text{ m/s} $$

So, the speed of the train is 62.5 meters per second.

Step 4: Convert Speed from m/s to km/h

The options are given in kilometers per hour (km/h). To convert speed from meters per second (m/s) to kilometers per hour (km/h), we multiply by the conversion factor $(18/5)$.

Speed in km/h = Speed in m/s $\times (18/5)$

Speed in km/h = $62.5 \times (18/5)$

Speed in km/h = $(125/2) \times (18/5)$

Speed in km/h = $(125 \times 18) / (2 \times 5)$

Speed in km/h = $(125 \times 9) / 5$

Speed in km/h = $25 \times 9$

Speed in km/h = 225 km/h

Final Answer

The calculated speed of the train is 225 km/h.