Solving Train Overtaking Problem

This problem involves calculating the length of a faster train based on relative speed and overtaking time.

Convert Speeds to m/s

First, convert the given speeds from km/h to meters per second (m/s) because the time is in seconds and the length is in meters.

• Speed of faster train ($S_1$): $72 \text{ km/h} = 72 \times \frac{5}{18} \text{ m/s} = 20 \text{ m/s}$
• Speed of slower train ($S_2$): $36 \text{ km/h} = 36 \times \frac{5}{18} \text{ m/s} = 10 \text{ m/s}$

Calculate Relative Speed

Since the trains are moving in the same direction, the relative speed is the difference between their speeds.

• Relative Speed ($S_{rel}$) = $S_1 - S_2 = 20 \text{ m/s} - 10 \text{ m/s} = 10 \text{ m/s}$

Calculate Total Distance Covered

The total distance the faster train needs to cover to overtake the slower train is the sum of both trains' lengths. This distance is covered at the relative speed over the given time.

• Time ($t$) = 27 seconds
• Total Distance ($D$) = Relative Speed $\times$ Time
• $D = 10 \text{ m/s} \times 27 \text{ s} = 270 \text{ m}$

Determine Faster Train Length

The total distance covered during overtaking is the sum of the lengths of the two trains ($L_1$ = length of faster train, $L_2$ = length of slower train).

• Total Distance ($D$) = $L_1 + L_2$
• Given $L_2 = 180$ m
• $270 \text{ m} = L_1 + 180 \text{ m}$
• $L_1 = 270 \text{ m} - 180 \text{ m} = 90 \text{ m}$

Therefore, the length of the faster train is 90 meters.