A train overtakes two persons who are walking in the same direction at 2 km/h and 4 km/h for 9 seconds and 10 seconds respectively. What is the length of the train in meters?

Steps to Solve

1. Understanding relative speed
   To find the length of the train, we need to understand the relative speed of the train with respect to the persons. The relative speed is calculated as the speed of the train minus the speed of the person.

2. Calculating relative speeds
   Assuming the speed of the train is $V_t$ km/h, the relative speeds with respect to each person are:

   • For the first person (2 km/h):
     $$ V_{rel1} = V_t - 2 $$
   • For the second person (4 km/h):
     $$ V_{rel2} = V_t - 4 $$

3. Finding the length of the train using time
   Given the time taken to completely overtake each person, the length of the train can be expressed as:

   • Length = Relative Speed × Time
   • For the first person (9 seconds):
     $$ L = (V_t - 2) \times \frac{9}{3600} $$
   • For the second person (10 seconds):
     $$ L = (V_t - 4) \times \frac{10}{3600} $$

4. Setting up the equations
   Since the lengths calculated for both individuals must be equal, we set the equations equal to each other:
   $$ (V_t - 2) \times 9 = (V_t - 4) \times 10 $$

5. Solving for the speed of the train
   Distributing and rearranging gives:
   $$ 9V_t - 18 = 10V_t - 40 $$ Rearranging yields:
   $$ V_t = 22 \text{ km/h} $$

6. Calculating the length of the train
   Choose either formula to find the length. Using the first person's formula:
   $$ L = (22 - 2) \times \frac{9}{3600} \text{ km} $$
   Converting km to meters:
   $$ L = 20 \times \frac{9}{3600} \times 1000 $$
   Calculate L:
   $$ L = 50 \text{ m} $$