A 150 m long train overtakes a man moving at a speed of 6 km/hr (in same…

Concept

When two bodies move in the SAME direction, the speed at which one gains on (overtakes) the other is the DIFFERENCE of their speeds; when they move in OPPOSITE directions, the speed at which they approach or separate is the SUM of their speeds. In every crossing problem, the time taken for one moving body to completely clear another equals the total distance still to be covered (the sum of both lengths when both are extended bodies, or just the moving object's own length when passing a point such as a person) divided by this relative speed.

Application

1. The 150 m train overtaking the man covers a relative distance equal to its own length (150 m) in 36 s, since the man can be treated as a point. So the relative speed = 150 m / 36 s = 25/6 m/s.

2. Convert this relative speed to km/hr: (25/6) x (18/5) = 15 km/hr. Since both move in the same direction, this relative speed equals (train's speed - man's speed), so the train's own speed = 15 + 6 = 21 km/hr.

3. The train now crosses a second, 500 m long train moving in the opposite direction at 31 km/hr. Because the directions are opposite, the relative speed for this crossing = 21 + 31 = 52 km/hr.

4. Convert 52 km/hr to m/s: 52 x 5/18 = 130/9 m/s (approximately 14.44 m/s).

5. To completely cross another train, the total distance to be covered is the sum of both trains' lengths: 150 + 500 = 650 m.

6. Time = distance / relative speed = 650 / (130/9) = 650 x 9/130 = 45 seconds.

Cross-check

Cross-check entirely in km/hr and hours: the combined length 650 m = 0.65 km, and the combined closing speed is 52 km/hr, so time = 0.65/52 hours = 0.0125 hours = 0.0125 x 3600 = 45 seconds - the same result, confirming the working.

So the train takes 45 seconds to completely cross the second train.