Calculate Relative Speed in Same Direction

When two trains run in the same direction, their relative speed is the difference of their individual speeds: $v_{\text{rel}} = 60 - 40 = 20\text{ km/h}$.

$v_{\text{same}} = 60 - 40 = 20 \text{ km/h}$

Find Total Length of Both Trains ($2L$)

Convert relative speed to meters per second and multiply by the time taken ($50\text{ s}$) to find the combined length of the two trains: $2L = v_{\text{rel}} \times t_{\text{same}}$.

$2L = \left(20 \times \frac{5}{18}\right) \times 50 = \frac{2500}{9} \text{ m}$

Calculate Relative Speed in Opposite Direction

When running in opposite directions, the relative speed is the sum of their speeds: $v_{\text{rel}} = 60 + 40 = 100\text{ km/h}$.

$v_{\text{opp}} = 60 + 40 = 100 \text{ km/h}$

Calculate Time Taken to Cross in Opposite Direction

Divide the total length ($2L$) by the new relative speed in m/s to get the required time.

$t_{\text{opp}} = \frac{2L}{v_{\text{opp}}} = \frac{2500/9}{100 \times 5/18} = \frac{2500}{250} = 10 \text{ sec}$