Two trains are moving in opposite directions at 60 km/h and 90 km/h respectively. Their lengths are 1.10 km and 0.90 km. What time, in seconds, does the slower train take to completely cross the faster train?

Introduction / Context:
This question extends standard train crossing problems by giving lengths in kilometres instead of metres. Two trains move in opposite directions, so their relative speed is the sum of their speeds. To find the crossing time, we consider the total distance that must be covered (sum of their lengths) and divide it by their relative speed, taking care to keep units consistent.

Given Data / Assumptions:

• Speed of the slower train = 60 km/h.
• Speed of the faster train = 90 km/h.
• Length of the slower train = 1.10 km.
• Length of the faster train = 0.90 km.
• The trains move in opposite directions at constant speeds.

Concept / Approach:
When two objects move towards each other on a straight line, their relative speed equals the sum of their speeds. The total distance that needs to be covered for them to completely cross each other is the sum of their lengths. Here, all data can be kept in kilometres and hours initially, and the final time can then be converted to seconds. The core formula is time = distance / speed.

Step-by-Step Solution:
Step 1: Total distance to be covered when they cross each other = 1.10 + 0.90 = 2.00 km. Step 2: Relative speed in opposite directions = 60 + 90 = 150 km/h. Step 3: Time in hours = distance / speed = 2.00 / 150 hours. Step 4: Simplify: 2 / 150 = 1 / 75 hours. Step 5: Convert time into seconds: (1 / 75) * 3600 seconds. Step 6: Compute: 3600 / 75 = 48 seconds.

Verification / Alternative check:
We can verify by converting speeds to m/s and lengths to metres. Total distance = 2.00 km = 2000 m. Relative speed = 150 km/h = 150 * 5/18 = 41.666... m/s. Time = 2000 / 41.666... ≈ 48 seconds. This agrees exactly with the previous calculation, confirming the result.

Why Other Options Are Wrong:
Times like 42, 44, 46, or 50 seconds correspond to different effective relative speeds when recalculated against the fixed distance of 2.00 km. None of them match the given speeds of 60 km/h and 90 km/h when checked rigorously. Only 48 seconds fits the precise relationship between distance and relative speed.

Common Pitfalls:
Common errors include forgetting to convert kilometres to metres or miscalculating the relative speed by subtracting instead of adding for opposite directions. Some learners may also misuse unit conversion between hours and seconds. Carefully keeping track of units and remembering that relative speed is the sum for opposite directions prevents these mistakes.

Final Answer:
The slower train takes 48 sec to completely cross the faster train.