A 150 metre long train crosses another 210 metre long train running in the opposite direction in 10.8 seconds. If the shorter train crosses a pole in 12 seconds, what is the speed of the longer train in km/h?

Introduction / Context:
This problem involves two trains moving in opposite directions, which means their relative speed is the sum of their individual speeds. We are given the time they take to cross each other and additional information about how long the shorter train takes to cross a pole. Using these pieces of data, we can find the speed of the longer train in km/h.

Given Data / Assumptions:

• Length of the shorter train = 150 m.
• Length of the longer train = 210 m.
• Time to cross each other when moving in opposite directions = 10.8 s.
• The shorter train crosses a pole in 12 s.
• Speeds are constant and motion is along straight tracks.

Concept / Approach:
When a train crosses a pole, the distance is equal to its own length. We first find the speed of the shorter train by using its length and the time taken to cross the pole. Then we use the combined distance of both trains when they cross each other and the time taken to find their relative speed. The speed of the longer train is then obtained by subtracting the speed of the shorter train from the relative speed.

Step-by-Step Solution:
Step 1: Speed of the shorter train = 150 m / 12 s = 12.5 m/s. Step 2: Total distance covered when trains cross each other = 150 + 210 = 360 m. Step 3: Relative speed when moving in opposite directions = 360 / 10.8 m/s. Step 4: Compute relative speed: 360 / 10.8 = 33.333... m/s. Step 5: Speed of longer train in m/s = relative speed - speed of shorter train = 33.333... - 12.5 = 20.833... m/s. Step 6: Convert to km/h: 20.833... * 18/5 ≈ 75 km/h.

Verification / Alternative check:
Take longer train speed as 75 km/h, which is 75 * 5/18 = 20.833... m/s. Sum of speeds in opposite directions = 20.833... + 12.5 = 33.333... m/s. Time to cross a combined distance of 360 m is 360 / 33.333... = 10.8 s, matching the given data exactly, which verifies the result.

Why Other Options Are Wrong:
Speeds such as 80 km/h or 70 km/h would change the relative speed and thus alter the crossing time away from 10.8 s. Options 54 km/h and 45 km/h are too low and would yield relative speeds that are inconsistent with the required crossing time for the given combined length of 360 m.

Common Pitfalls:
Learners sometimes forget to add the lengths of both trains when they cross each other. Another common mistake is using the difference instead of the sum of speeds for opposite directions. Errors also occur when converting between m/s and km/h or when misreading which train is shorter or longer.

Final Answer:
The speed of the longer train is 75 kmph.