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:
Step-by-step Solution:
1. Define Variables and Relative Speed - Let the length of the train be L (in meters) and its speed be S (in kmph).
- When a train overtakes someone moving in the same direction, we use their relative speed, which is the difference between their speeds.
- The key formula is: Length = Relative Speed × Time. We'll need to convert speeds from kmph to m/s by multiplying by 5/18.
2. Set Up Equations for Both Scenarios - Scenario 1 (Person at 2 kmph): The relative speed is (S - 2) kmph.
L = (S - 2) × (5/18) × 9 seconds => L = (S - 2) × 2.5
- Scenario 2 (Person at 4 kmph): The relative speed is (S - 4) kmph.
L = (S - 4) × (5/18) × 10 seconds => L = (S - 4) × 25/9
3. Solve for the Train's Speed (S) - Since L is the same in both equations, we can set them equal to each other:
- (S - 2) × 2.5 = (S - 4) × 25/9
- (S - 2) × 5/2 = (S - 4) × 25/9
- 9 × (S - 2) = 10 × (S - 4) (after cross-multiplying and simplifying)
- 9S - 18 = 10S - 40
- S = 22 kmph.
4. Find the Length of the Train (L) - Now substitute S = 22 back into the first (simpler) equation:
- L = (22 - 2) × 2.5
- L = 20 × 2.5 = 50 meters.
Final Answer: The length of the train is 50 m.