49 km/h
The objective is to determine the train's speed. This involves calculating the time taken to traverse a platform and the time taken to pass a man moving in the opposite direction.
- Train length: 150 meters.
- Platform traversal: The train covers its own length plus the platform's length. The platform's length will be derived from the time and speed.
- Man traversal (opposite direction): The train covers its own length plus the distance the man travels during that time.
- Unit consistency: All speeds must be standardized to meters per second (m/s).
- Relative speed (opposite direction): The sum of individual speeds.
- Train length, \( L = 150 \) meters.
- Time to pass platform, \( t_1 = 30 \) seconds.
- Time to pass man, \( t_2 = 10 \) seconds.
- Man's speed, \( v_m = 5 \) km/h.
Convert the man's speed from km/h to m/s:
\[v_m = 5 \times \frac{1000}{3600} = \frac{5000}{3600} \approx 1.39 \text{ m/s}\]
Let the train's speed be denoted as \( v \) m/s.
Scenario: Passing the man
Due to the opposite direction of movement, the relative speed is \( v + v_m \).
The train covers its 150-meter length in 10 seconds while passing the man:
\[(v + v_m) \times 10 = 150\]
\[v + v_m = \frac{150}{10} = 15 \text{ m/s}\]
\[v = 15 - v_m = 15 - 1.39 = 13.61 \text{ m/s}\]
The platform is passed in 30 seconds. This involves the train covering its length plus the platform's length. We can calculate the platform's length:
Distance covered during platform traversal = speed × time = \( v \times 30 = 13.61 \times 30 = 408.3 \) meters.
Platform length = Total distance covered - Train length = \( 408.3 - 150 = 258.3 \) meters.
Convert the train's speed from m/s to km/h:
\[v = 13.61 \times \frac{3600}{1000} = 13.61 \times 3.6 = 49 \text{ km/h (approximately)}\]
The approximate speed of the train is 49 km/h.