A pair of commuter trains are travelling along parallel tracks in the same direction. A passenger sitting in the slower train notices that the faster train takes 30 seconds to pass completely by their window. If the speed of the faster train is 18 km/h greater than that of the slower train, what is the length of the faster train in meters?

The correct answer is 150 (Option 2).

Step 1: Understand the Relative Speed

When two objects move in the same direction, their relative speed is the difference between their individual speeds.

Let the speed of the slower train be _vslower and the speed of the faster train be _vfaster.

We are given that the faster train's speed is 18 km/h greater than the slower train's speed:

$v_{\text{relative}}=v_{\text{faster}}-v_{\text{slower}}=18\text{km/h}$

Step 2: Convert the Relative Speed to Meters per Second (m/s)

To convert speed from kilometers per hour (km/h) to meters per second (m/s), multiply by $\frac{5}{18}$:

$v_{\text{relative}}=18\times \frac{5}{18}=5\text{m/s}$

Step 3: Calculate the Length of the Faster Train

From the perspective of a passenger sitting in the slower train, their window acts as a fixed point. The time taken for the faster train to pass completely by this window depends only on the length of the faster train (L) and the relative speed between the two trains.

The formula relating distance, speed, and time is:

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

Here, the distance covered is equal to the length of the faster train (L), the speed is the relative speed (5 m/s), and the time taken is 30 seconds.

$L=5\text{m/s}\times 30\text{s}$

$L=150\text{meters}$

Therefore, the length of the faster train is 150 meters.