Question

A train 150 meters long passes a platform in 30 seconds and passes a man running at 5 km/h in the opposite direction in 10 seconds. What is the speed of the train?

• A. 49 km/h
• B. 60 km/h
• C. 66 km/h
• D. 72 km/h

Hint:

Convert units consistently (km/h to m/s using \(\frac{5}{18}\)). Use relative speed for opposite directions and verify with the second condition.

Answer 1

The Correct Option is A

To solve the problem, we need to find the speed of the train given the time it takes to pass a platform and a man running in the opposite direction.

1. Understanding the Concepts:

- The train length = 150 meters.
- When the train passes a platform, it covers the length of the train plus the platform length (platform length is unknown, so we use the time and speed to find the platform length).
- When the train passes a man running in the opposite direction, it covers just the length of the train plus the distance the man moves during that time.
- Speeds must be converted to consistent units (m/s).
- The relative speed when moving in opposite directions is the sum of their speeds.

2. Given Values:

- Length of train, \( L = 150 \) meters.
- Time to pass platform, \( t_1 = 30 \) seconds.
- Time to pass man, \( t_2 = 10 \) seconds.
- Speed of man, \( v_m = 5 \) km/h.

3. Converting Units:

Convert speed of man to meters per second (m/s):

\[ v_m = 5 \times \frac{1000}{3600} = \frac{5000}{3600} \approx 1.39 \text{ m/s} \]

4. Calculating the speed of the train:

Let the speed of the train be \( v \) m/s.

Passing the man:

Since the man is running in the opposite direction, relative speed = \( v + v_m \).

The train covers its own length (150 m) in 10 seconds while passing the man:

\[ (v + v_m) \times 10 = 150 \]

\[ v + v_m = \frac{150}{10} = 15 \text{ m/s} \]

\[ v = 15 - v_m = 15 - 1.39 = 13.61 \text{ m/s} \]

5. Checking with platform passing time:

Time to pass platform is 30 seconds. When passing the platform, the train covers its own length plus the platform length. But since the platform length is unknown, we can find it:

Distance covered while passing platform = speed × time = \( v \times 30 = 13.61 \times 30 = 408.3 \) meters.

Platform length = distance covered - length of train = \( 408.3 - 150 = 258.3 \) meters.

6. Convert speed of train to km/h:

\[ v = 13.61 \times \frac{3600}{1000} = 13.61 \times 3.6 = 49 \text{ km/h (approximately)} \]

Final Answer:

The speed of the train is approximately 49 km/h.