A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is :

### Concept & Equations of Motion When a train overtakes people walking in the same direction, the relative speed is the train's speed minus the person's speed. The length of the train is constant, so we can equate the distance formulas for both scenarios. $$ \text{Length} = (\text{Speed}_{\text{train}} - \text{Speed}_{\text{person}}) \times \text{Time} $$ ### Step-by-Step Solution * Let the speed of the train be $v$ kmph and its length be $L$ meters. * For the first person: Speed is $2$ kmph, time is $9$ seconds. * Relative speed = $(v - 2)$ kmph. Length $L = (v - 2) \times \frac{5}{18} \times 9$. * For the second person: Speed is $4$ kmph, time is $10$ seconds. * Relative speed = $(v - 4)$ kmph. Length $L = (v - 4) \times \frac{5}{18} \times 10$. * Equate both expressions for $L$: $(v - 2) \times 9 = (v - 4) \times 10$. (The $\frac{5}{18}$ factor cancels out). * Solve for $v$: $9v - 18 = 10v - 40 \Rightarrow v = 22$ kmph. * Substitute $v$ back to find $L$: $L = (22 - 2) \times \frac{5}{18} \times 9 = 20 \times \frac{5}{2} = 50$ meters. ### Exam Strategy & Shortcut Use the cross-multiplication formula to find the train's speed directly: $v = \frac{(v_2 \times t_2) - (v_1 \times t_1)}{t_2 - t_1}$. Here, $v = \frac{(4 \times 10) - (2 \times 9)}{10 - 9} = \frac{40 - 18}{1} = 22$ kmph. Then, calculate length: $L = (22 - 2) \times \frac{5}{18} \times 9 = 50$ m. ### Common Pitfall Forgetting to convert the kmph speed to m/s when calculating the final length $L$. The conversion $\frac{5}{18}$ cancels out while finding speed, but is strictly required for the length calculation. ### Final Answer Therefore, the correct answer is **50 m**.