Two trains, 130 and 110 metres long, are going in the same direction. The faster train takes one minute to pass the other completely. If they are moving in opposite directions, they pass each other completely in 3 seconds. Find the speed of the faster train.

### Concept & Logic The total distance covered when trains pass each other is the sum of their lengths, regardless of direction. In the same direction, relative speed = $v_1 - v_2$. In opposite directions, relative speed = $v_1 + v_2$. $$ \text{Relative Speed} = \frac{\text{Total Distance}}{\text{Time}} $$ ### Step-by-Step Solution 1. **Find Total Distance:** - Length of first train = $130$ m. - Length of second train = $110$ m. - Total Distance = $130 + 110 = 240$ m. 2. **Calculate Relative Speeds:** - Same direction (Time = $60$ s): $v_1 - v_2 = \frac{240}{60} = 4$ m/s. - Opposite directions (Time = $3$ s): $v_1 + v_2 = \frac{240}{3} = 80$ m/s. 3. **Solve for Faster Train Speed ($v_1$):** - Add the two equations: - $(v_1 - v_2) + (v_1 + v_2) = 4 + 80$. - $2v_1 = 84$. - $v_1 = 42$ m/s. ### Exam Strategy & Shortcut Once you have the relative speeds (4 and 80), the speed of the faster entity is simply the average of the two relative speeds: $\frac{80 + 4}{2} = 42$ m/s. The speed of the slower entity would be $\frac{80 - 4}{2} = 38$ m/s. ### Common Pitfall Forgetting to convert "one minute" into $60$ seconds, leading to drastically incorrect values for the relative speed in the same direction. ### Final Answer Therefore, the correct answer is **42 m/sec**.