Introduction / Context:
This question links two related scenarios: trains running in the same direction and in opposite directions. The trains have equal lengths and we know the time they take to cross each other in one case. Using that, we can deduce the train length and then find the time to cross in the other case. It illustrates how relative speed changes with direction.
Given Data / Assumptions:
- Speed of faster train = 60 km/h.
- Speed of slower train = 40 km/h.
- Trains have equal lengths.
- Time to cross each other in same direction = 50 s.
- We need time to cross each other in opposite directions.
Concept / Approach:
When trains move in the same direction, relative speed is the difference between their speeds. The distance for complete crossing is the sum of both train lengths. When they move in opposite directions, the distance is the same, but the relative speed is the sum of their speeds. Using the time in the same direction case, we can compute the combined length and then use it with the new relative speed to get the required time.
Step-by-Step Solution:
Relative speed in same direction = 60 − 40 = 20 km/h.
Convert to m/s: 20 * (5 / 18) = 50 / 9 m/s.
Let length of each train be L m, so total length = 2L.
Time to cross in same direction = 50 s, so 2L = (50 / 9) * 50 = 2500 / 9 m.
Thus, 2L = 2500 / 9 ⇒ L = 1250 / 9 m (not needed explicitly later).
Now in opposite directions, relative speed = 60 + 40 = 100 km/h = 100 * (5 / 18) = 250 / 9 m/s.
Time to cross in opposite directions = total length / new relative speed = (2500 / 9) / (250 / 9) = 10 s.
Verification / Alternative check:
We can also reason that time in opposite directions must be smaller by a factor equal to the ratio of relative speeds. Relative speed same direction is 20 km/h; opposite direction is 100 km/h. Factor = 20/100 = 1/5. So time opposite direction = 50 * (1/5) = 10 s. This matches our detailed calculation.
Why Other Options Are Wrong:
Values 11 s, 12 s, or 8 s do not follow from the ratio of relative speeds or the distance found from the first case. Only 10 s gives the correct proportional reduction in time when relative speed increases by a factor of 5 from 20 km/h to 100 km/h.
Common Pitfalls:
Some students incorrectly keep using the same relative speed in both cases, or forget to convert km/h to m/s consistently. Others may mistakenly treat the train lengths as changing. Remember that only the relative speed changes when the direction arrangement changes; the train lengths and total crossing distance remain the same.
Final Answer:
They will take
10 sec to cross each other when running in opposite directions.
Discussion & Comments