Two trains are coming from opposite directions with speeds of 75 km/hr and 100 km/hr on two parallel tracks. At some moment the distance between them is 100 km. After T hours, distance between them is again 100 km. T is equal to (I.A.M., 2007)

### Concept & Relative Speed When two bodies move in opposite directions, their relative speed is the sum of their individual speeds. To be 100 km apart again, the trains must first meet (covering the initial 100 km separation) and then separate by another 100 km. $$Relative\ Speed = S_1 + S_2$$ $$Total\ Distance = D_{initial} + D_{final}$$ ### Step-by-Step Solution 1. **Given:** Speeds are 75 km/hr and 100 km/hr. Initial distance = 100 km, Final required distance = 100 km. 2. Calculate the relative speed since they are moving in opposite directions: $75 + 100 = 175$ km/hr. 3. Determine the total relative distance they need to cover. They must cover the 100 km between them to cross paths, plus another 100 km to move apart again. Total distance = $100 + 100 = 200$ km. 4. Calculate the time ($T$) taken to cover this relative distance: $$Time = \frac{Distance}{Relative\ Speed}$$ $$T = \frac{200}{175} = \frac{8}{7} = 1 \frac{1}{7}\ hours$$ ### Exam Strategy & Shortcut Instead of modeling the exact meeting point, always use relative speed for these types of questions. Add the initial and final separation distances, and divide by the sum of speeds. $200 / 175 = 8/7$. ### Common Pitfall A common mistake is assuming the trains only need to cover 100 km total, forgetting that they must first cross the initial 100 km gap before creating a new 100 km gap. ### Final Answer Therefore, the correct answer is **$1 \frac{1}{7}$ hr**.