Train Length Calculation Using Relative Speed

This problem involves calculating the length of the faster train using the concept of relative speed. We are given the speeds of two trains moving in the same direction and the time taken for the faster train to overtake a man in the slower train.

Relative Speed Calculation

When two objects move in the same direction, the relative speed of the faster object with respect to the slower object is the difference between their speeds.

• Speed of the faster train (\(v_1\)) = 40 km/h
• Speed of the slower train (\(v_2\)) = 20 km/h
• Relative speed (\(v_{rel}\)) = \(v_1 - v_2\) = 40 km/h - 20 km/h = 20 km/h

Converting Speed to Meters per Second

Since the time is given in seconds and the required length is in meters, we need to convert the relative speed from km/h to m/s. The conversion factor is \(\frac{5}{18}\).

• \(v_{rel}\) = 20 km/h = \(20 \times \frac{5}{18}\) m/s
• \(v_{rel}\) = \(\frac{100}{18}\) m/s = \(\frac{50}{9}\) m/s

Calculating the Length of the Faster Train

The faster train overtakes the man in the slower train in 10 seconds. The distance covered by the faster train relative to the man during this time is equal to the length of the faster train. We use the formula: Distance = Speed × Time.

• Time (\(t\)) = 10 seconds
• Length of faster train (\(L\)) = Relative Speed × Time
• \(L = v_{rel} \times t\)
• \(L = \frac{50}{9} \text{ m/s} \times 10 \text{ s}\)
• \(L = \frac{500}{9}\) meters
• \(L \approx 55.555...\) meters

Rounding the Answer

The question asks to round the answer to the nearest integer value.

• Rounding 55.555... to the nearest integer gives 56.

Therefore, the length of the faster train is 56 meters.