Train Speed Calculation Using Platform and Person Crossing Times

This problem involves calculating the speed of a train based on the time it takes to pass a stationary object (a person) and a stationary object of known length (a platform).

Given Information:

• Platform Length: 108 meters
• Time to cross the platform: 36 seconds
• Time to pass Ram (a person): 12 seconds

Understanding the Concepts:

To solve this, we need to relate speed, distance, and time using the formula:

$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$

When a train passes a stationary person, the distance covered is equal to the length of the train itself.

When a train passes a platform, the total distance covered is the sum of the length of the train and the length of the platform.

Setting up the Equations:

Let:

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

Scenario 1: Passing Ram

The distance the train covers is its own length, $L$.

The time taken is 12 seconds.

Using the formula, we get:

$$ S = \frac{L}{12} $$

From this, we can express the length of the train as:

$$ L = 12S \quad (Equation \, 1) $$

Scenario 2: Passing the Platform

The total distance the train covers is the length of the train plus the length of the platform, which is $L + 108$ meters.

The time taken is 36 seconds.

Using the formula, we get:

$$ S = \frac{L + 108}{36} \quad (Equation \, 2) $$

Solving for the Train Speed:

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

$$ S = \frac{(12S) + 108}{36} $$

To solve for $S$, we first multiply both sides by 36:

$$ 36S = 12S + 108 $$

Next, we isolate the terms with $S$ by subtracting $12S$ from both sides:

$$ 36S - 12S = 108 $$

$$ 24S = 108 $$

Finally, we find the speed $S$ by dividing 108 by 24:

$$ S = \frac{108}{24} \, \text{m/s} $$

$$ S = 4.5 \, \text{m/s} $$

Converting Speed to km/h:

The calculated speed is in meters per second (m/s), but the options are given in kilometers per hour (km/h). We need to convert the speed.

The conversion factor is: $1 \, \text{m/s} = \frac{18}{5} \, \text{km/h}$

So, the speed in km/h is:

$$ \text{Speed (km/h)} = 4.5 \times \frac{18}{5} $$

$$ \text{Speed (km/h)} = \frac{4.5 \times 18}{5} = \frac{81}{5} $$

$$ \text{Speed (km/h)} = 16.2 \, \text{km/h} $$

Conclusion:

The speed of the train is 16.2 km/h.