Solving for Train Length When Crossing a Platform

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

Understanding the Physics Concept

When a train crosses a platform, the total distance the train travels is equal to the sum of its own length and the length of the platform. This is because the crossing starts when the front of the train enters the platform and ends when the rear of the train leaves the platform.

The fundamental formula we'll use is:

$Distance = Speed × Time$

Step 1: Convert Speed to Meters per Second

The train's speed is given as $90$ kmph (kilometers per hour). Since the platform length is in meters and the time is in seconds, we must convert the speed to meters per second (m/s).

The conversion factor is: $1$ kmph = $\frac{5}{18}$ m/s.

So, the speed of the train is:

Speed = $90 \text{ kmph} \times \frac{5}{18} \text{ m/s/kmph}$

Speed = $5 \times 5$ m/s

Speed = $25$ m/s

Step 2: Define the Total Distance

Let the length of the train be $L$ meters.

The length of the platform is given as $250$ meters.

The total distance covered by the train while crossing the platform is the length of the train plus the length of the platform:

Total Distance = Length of Train + Length of Platform

Total Distance = $L + 250$ meters

Step 3: Apply the Speed-Distance-Time Formula

We know:

• Total Distance = $L + 250$ m
• Speed = $25$ m/s
• Time = $26$ seconds

Using the formula Distance = Speed × Time:

$L + 250 \text{ m} = 25 \text{ m/s} \times 26 \text{ s}$

Step 4: Calculate the Length of the Train

First, calculate the total distance traveled:

$25 \times 26 = 650$ meters

Now, substitute this back into the equation:

$L + 250 = 650$

To find the length of the train ($L$), subtract the platform length from the total distance:

$L = 650 - 250$

$L = 400$ meters

Therefore, the length of the train is $400$ meters.