Question 15: Man vs. Train: Time to Cross in Different Scenarios

Objective:

Learn how to calculate the time taken by a train to cross a stationary object (a man, pole, or similar) using the train’s length and speed.

Key Concepts:

1. Speed, Distance, and Time Relationship:
   The fundamental formula is:

   Time=DistanceSpeed\text{Time} = \frac{\text{Distance}}{\text{Speed}}Time=SpeedDistance​

2. Distance Covered by the Train:
   When a train crosses a stationary object (like a man), the distance covered is equal to the length of the train.

3. Convert Speed to m/s:
   Since the train’s length is in meters and time is calculated in seconds, convert speed from km/h to m/s using:

   Speed (m/s)=Speed (km/h)×518\text{Speed (m/s)} = \text{Speed (km/h)} \times \frac{5}{18}Speed (m/s)=Speed (km/h)×185​

Given Information:

1. Length of the train: 75 meters
2. Speed of the train: 20 km/h

We need to calculate the time taken to cross a man standing on the platform.

Step-by-Step Solution:

Step 1: Convert Speed to m/s

Convert the train’s speed from km/h to m/s:

Speed (m/s)=20×518=10018=5.56 m/s (approx)\text{Speed (m/s)} = 20 \times \frac{5}{18} = \frac{100}{18} = 5.56 \, \text{m/s} \, (\text{approx})Speed (m/s)=20×185​=18100​=5.56m/s(approx)

Step 2: Use the Time Formula

The time taken by the train to cross the man is calculated as:

Time=DistanceSpeed\text{Time} = \frac{\text{Distance}}{\text{Speed}}Time=SpeedDistance​

Here, the distance is equal to the length of the train:

Time=755.56≈13.5 seconds\text{Time} = \frac{75}{5.56} \approx 13.5 \, \text{seconds}Time=5.5675​≈13.5seconds

Final Answer:

The train will take approximately 13.5 seconds to cross the man.