Two trains are running in opposite directions at the same speed. The length of each train is 486 metres, and they completely cross each other in 27 seconds. What is the speed of each train in km/hr?

Introduction / Context:
This question tests your understanding of relative speed when two trains travel in opposite directions. Both trains have equal length and equal speed. When they cross, the total distance covered relative to each other is the sum of their lengths, and the relative speed is the sum of their individual speeds. From the crossing time, we can find the common speed of each train.

Given Data / Assumptions:
- Length of each train = 486 m.
- Speed of each train = v km/hr (same for both, unknown).
- Trains are moving in opposite directions on parallel tracks.
- Time taken to completely cross each other = 27 seconds.
- Speeds are constant and there is no acceleration.

Concept / Approach:
For two trains moving in opposite directions, relative speed = v1 + v2. Since both trains have the same speed v, relative speed in km/hr is 2v. The distance covered during crossing is the sum of both lengths: 486 m + 486 m. Using distance = speed * time in m/s, we compute the relative speed in m/s, then divide it by 2 and convert to km/hr to get each train's speed.

Step-by-Step Solution:
Step 1: Total distance to be covered during crossing = 486 m + 486 m = 972 m.Step 2: Time taken to cross = 27 seconds.Step 3: Relative speed in m/s = distance / time = 972 / 27 m/s.Step 4: Compute 972 / 27: since 27 * 36 = 972, relative speed = 36 m/s.Step 5: Since both trains have the same speed and are moving in opposite directions, 2 * (speed of each train in m/s) = 36.Step 6: So speed of each train in m/s = 36 / 2 = 18 m/s.Step 7: Convert 18 m/s to km/hr by multiplying by 18 / 5 or equivalently by 3.6.Step 8: Speed in km/hr = 18 * 3.6 = 64.8 km/hr.

Verification / Alternative check:
Using the found speed, convert 64.8 km/hr back to m/s: 64.8 * (5 / 18) = 18 m/s. The relative speed of two trains in opposite directions is 18 + 18 = 36 m/s. At 36 m/s, time to cover 972 m is 972 / 36 = 27 seconds, which exactly matches the given crossing time. This confirms the correctness of 64.8 km/hr.

Why Other Options Are Wrong:
55, 56.4, 73.2, or 60 km/hr all yield different m/s speeds and thus different crossing times when used. For example, 60 km/hr corresponds to 16.67 m/s, so relative speed would be about 33.34 m/s and crossing time would be longer than 27 seconds. Only 64.8 km/hr yields the correct relative speed and crossing time.

Common Pitfalls:
One common error is to forget that both train lengths must be added to get the total crossing distance. Another is to convert km/hr to m/s incorrectly, or to forget to divide the relative speed by 2 when determining each train's speed. Carefully tracking units and using the relative speed concept correctly avoids these mistakes.

Final Answer:
The speed of each train is 64.8 km/hr.