Train Length Calculation: Crossing a Platform Problem

This problem involves calculating the length of a train based on its speed, the time it takes to cross a platform, and the relationship between the train's length and the platform's length.

Problem Analysis

• Train Speed: The train is moving at a constant speed of 25 m/s.
• Platform Length: The platform's length is one-third the length of the train.
• Crossing Time: The time taken for the train to completely cross the platform is 20 seconds.
• Objective: Determine the length of the train.

Key Concepts

When a train crosses a platform, the total distance covered by the train is the sum of its own length and the length of the platform. The relationship between distance, speed, and time is given by the formula: Distance = Speed × Time.

Step-by-Step Calculation

1. Define variables:

   • Let the length of the train be '$L_T$' meters.
   • Let the length of the platform be '$L_P$' meters.

2. Express the platform length in terms of the train length:

   According to the problem, '$L_P = \frac{1}{3} L_T$'.

3. Calculate the total distance covered:

   The total distance '$D$' the train travels to cross the platform is the sum of its length and the platform's length:

   $D = L_T + L_P$

   Substitute the expression for '$L_P$':

   $D = L_T + \frac{1}{3} L_T$

   Combine the terms:

   $D = \frac{3L_T + L_T}{3} = \frac{4}{3} L_T$

4. Use the speed and time to find the distance:

   The train's speed is 25 m/s, and the time taken is 20 seconds.

   $D = \text{Speed} \times \text{Time}$

   $D = 25 \text{ m/s} \times 20 \text{ s}$

   $D = 500 \text{ m}$

5. Equate the two expressions for distance and solve for '$L_T$':

   We have two expressions for the distance '$D$': '$D = \frac{4}{3} L_T$' and '$D = 500$ m'.

   $\frac{4}{3} L_T = 500$

   To find '$L_T$', multiply both sides by $\frac{3}{4}$:

   $L_T = 500 \times \frac{3}{4}$

   $L_T = \frac{1500}{4}$

   $L_T = 375 \text{ m}$

Conclusion

The length of the train is 375 meters. This calculation confirms that when a train moving at 25 m/s crosses a platform that is one-third its length in 20 seconds, its length must be 375 m.