Calculating Train Length Using Relative Speed

This problem involves two trains of equal length moving in the same direction on parallel lines. We are given their speeds and the time it takes for the faster train to completely pass the slower train. We need to find the length of each train.

Key Concepts

• Relative Speed (Same Direction): When two objects move in the same direction, their relative speed is the difference between their individual speeds.
• Passing Distance: For one train to completely pass another, the total distance covered relative to each other is the sum of their lengths.

Step-by-Step Solution

1. Calculate Relative Speed

The speeds of the two trains are \(v_1 = 39\) km/h and \(v_2 = 21\) km/h. Since they are moving in the same direction, the relative speed (\(v_{rel}\)) is:

\( v_{rel} = v_1 - v_2 \)

\( v_{rel} = 39 \text{ km/h} - 21 \text{ km/h} = 18 \text{ km/h} \)

2. Convert Speed to Meters per Second (m/s)

The time is given in seconds, so we need the speed in m/s. The conversion factor is \(\frac{5}{18}\) m/s per km/h.

\( v_{rel} \text{ (m/s)} = 18 \times \frac{5}{18} \)

\( v_{rel} = 5 \text{ m/s} \)

3. Calculate Total Distance Covered

The faster train passes the slower train in \(t = 26\) seconds. The total distance (\(D\)) covered during this time is the product of the relative speed and the time.

\( D = v_{rel} \times t \)

\( D = 5 \text{ m/s} \times 26 \text{ s} \)

\( D = 130 \text{ meters} \)

4. Determine the Length of Each Train

Let the length of each train be \(L\). Since the trains have equal length, the total distance \(D\) is the sum of their lengths:

\( D = L + L = 2L \)

We know \(D = 130\) meters, so:

\( 2L = 130 \text{ meters} \)

\( L = \frac{130}{2} \)

\( L = 65 \text{ meters} \)

Therefore, the length of each train is 65 meters.