A train overtakes two persons who are walking at 15 m/s and 35 m/s, respectively, in the same direction as that of the train in 20 seconds and 40 seconds, respectively. The length of the train is:

Correct option is C

Given
 Speed of the first person = 15 m/s
 Speed of the second person = 35 m/s
 Time to overtake the first person = 20 seconds
 Time to overtake the second person = 40 seconds

Formula Used

$$\text{Distance} = \text{Relative speed} \times \text{Time}$$

Solution

$$\text{Relative speed with respect to the first person:} \\ \ \\\text{Since the train and the first person are moving in the same direction, the relative speed is:} \\ \ \\\text{Relative speed} = V_t - 15 \, \text{m/s}\\ \ \\\text{Relative speed with respect to the second person:} \\ \ \\\text{Relative speed} = V_t - 35 \, \text{m/s}\\ \ \\\text{The distance traveled by the train to overtake the first person is the length of the train, and this distance is covered in 20 seconds :} \\ \ \\\text{Length of the train} = (V_t - 15) \times 20\\ \ \\\text{Similarly, the distance traveled by the train to overtake the second person is also the length of the train, covered in 40 seconds. So:} \\ \ \\\text{Length of the train} = (V_t - 35) \times 40\\ \ \\\text{Since both expressions represent the length of the train} \\ \ \\(V_t - 15) \times 20 = (V_t - 35) \times 40\\ \ \\20V_t - 300 = 40V_t - 1400 \\ \ \\20V_t = 1100 \\ \ \\V_t = 55 \, \text{m/s}\\ \ \\\text{Calculate the length of the train:} \\ \ \\\text{Substitute } V_t = 55 \, \text{m/s} \text{ into one of the equations for the length of the train} \\ \ \\\text{Length of the train} = (55 - 15) \times 20 = 40 \times 20 = 800\ m$$​

​