A 225 m long train is running at a speed of 30 km/hr. How much time does it take to cross a man running at 3 km/hr in the same direction?
30 seconds
Understanding Train Crossing Problems
This question asks for the time it takes for a train to completely cross a man running in the same direction. When a train crosses a point or a man (considered a point object), the distance covered by the train is equal to its own length.
However, since the man is also moving in the same direction as the train, we need to consider their relative speed. The relative speed is the difference between the speeds of the train and the man because they are moving in the same direction.
Given Information for the Train Crossing Problem
Let's list the details provided in the question:
- Length of the train = 225 meters (m)
- Speed of the train = 30 km/hr
- Speed of the man = 3 km/hr
Calculating Relative Speed in the Same Direction
When two objects move in the same direction, their relative speed is the difference between their individual speeds. This relative speed determines how quickly the distance between them changes.
Relative Speed = Speed of Train - Speed of Man
Relative Speed = \(30 \text{ km/hr} - 3 \text{ km/hr} = 27 \text{ km/hr}\)
Converting Speed to Consistent Units
The train length is given in meters (m), so it's best to convert the speed from kilometers per hour (km/hr) to meters per second (m/s) to ensure consistent units for calculation. The conversion factor is \(\frac{5}{18}\).
Relative Speed in m/s = Relative Speed in km/hr \(\times \frac{5}{18}\)
Relative Speed = \(27 \times \frac{5}{18} \text{ m/s}\)
Relative Speed = \(\frac{3 \times 9 \times 5}{2 \times 9} \text{ m/s}\)
Relative Speed = \(\frac{3 \times 5}{2} \text{ m/s}\)
Relative Speed = \(\frac{15}{2} \text{ m/s} = 7.5 \text{ m/s}\)
Distance and Time Calculation for Crossing
To completely cross the man, the train needs to cover a distance equal to its own length relative to the man.
Distance to be covered = Length of the train = 225 m
Now we can use the standard formula relating distance, speed, and time:
Time = \(\frac{\text{Distance}}{\text{Speed}}\)
Here, the speed is the relative speed calculated above.
Time = \(\frac{\text{Distance}}{\text{Relative Speed}}\)
Time = \(\frac{225 \text{ m}}{7.5 \text{ m/s}}\)
To make the division easier, we can remove the decimal by multiplying both numerator and denominator by 10:
Time = \(\frac{2250}{75} \text{ seconds}\)
Now, perform the division:
\(\frac{2250}{75} = \frac{30 \times 75}{75} = 30\) seconds
So, it takes 30 seconds for the train to cross the man running in the same direction.
| Quantity | Value | Units |
|---|---|---|
| Train Length (Distance) | 225 | m |
| Train Speed | 30 | km/hr |
| Man Speed | 3 | km/hr |
| Relative Speed (km/hr) | 27 | km/hr |
| Relative Speed (m/s) | 7.5 | m/s |
| Time Taken | 30 | seconds |
Revision Table: Train Crossing Concepts
| Scenario | Distance Covered | Speed Used |
|---|---|---|
| Train crossing a point object (pole, man standing still) | Length of the train | Speed of the train |
| Train crossing a man running in the same direction | Length of the train | Relative speed (Train Speed - Man Speed) |
| Train crossing a man running in the opposite direction | Length of the train | Relative speed (Train Speed + Man Speed) |
| Train crossing another train (same direction) | Sum of lengths of both trains | Relative speed (Speed of faster train - Speed of slower train) |
| Train crossing another train (opposite direction) | Sum of lengths of both trains | Relative speed (Speed of Train 1 + Speed of Train 2) |
Additional Information: Speed, Distance, and Time
The relationship between speed, distance, and time is fundamental in solving these types of problems. The core formula is:
Speed = \(\frac{\text{Distance}}{\text{Time}}\)
From this, we can derive:
- Distance = Speed \(\times\) Time
- Time = \(\frac{\text{Distance}}{\text{Speed}}\)
It is crucial to ensure that the units for distance and speed are consistent when performing calculations. If distance is in meters, speed should ideally be in meters per second. If distance is in kilometers, speed should ideally be in kilometers per hour. If not, one of the values must be converted.
A common conversion is from km/hr to m/s, which uses the factor \(\frac{5}{18}\) because:
\(1 \text{ km/hr} = \frac{1000 \text{ meters}}{3600 \text{ seconds}} = \frac{10}{36} \text{ m/s} = \frac{5}{18} \text{ m/s}\)
Conversely, to convert from m/s to km/hr, you multiply by \(\frac{18}{5}\).
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