Solving the Train Crossing the Bridge Problem

This problem asks us to find the time it takes for a train of a specific length, moving at a certain speed, to completely cross a bridge of another specific length. To solve this, we need to understand the total distance the train must travel and ensure all units are consistent.

Understanding the Total Distance

When a train crosses a bridge, the train must travel a distance equal to its own length plus the length of the bridge. Imagine the front of the train just reaching one end of the bridge. For the train to completely cross, the rear of the train must reach the other end of the bridge. This means the train's front has traveled the length of the bridge, and then the entire train length must also pass that point. Thus, the total distance covered is the sum of the length of the train and the length of the bridge.

• Length of the train = 600 m
• Length of the bridge = 200 m
• Total distance the train must travel = Length of train + Length of bridge
• Total distance = $600 \text{ m} + 200 \text{ m} = 800 \text{ m}$

Converting Speed Units

The speed of the train is given in kilometers per hour (km/h), but the distances are in meters. To calculate the time in seconds, we need to convert the speed from km/h to meters per second (m/s). The conversion factor is $\frac{5}{18}$, because $1 \text{ km} = 1000 \text{ m}$ and $1 \text{ hour} = 3600 \text{ seconds}$. So, $\frac{1000 \text{ m}}{3600 \text{ s}} = \frac{10}{36} \text{ m/s} = \frac{5}{18} \text{ m/s}$.

• Given speed of the train = 72 km/h
• Conversion factor from km/h to m/s = $\frac{5}{18}$
• Speed in m/s = $72 \times \frac{5}{18}$ m/s
• Speed in m/s = $\frac{72}{18} \times 5$ m/s
• Speed in m/s = $4 \times 5$ m/s
• Speed in m/s = 20 m/s

Calculating the Time Taken

Now that we have the total distance in meters and the speed in meters per second, we can calculate the time taken using the formula: Time = Distance / Speed.

• Total distance = 800 m
• Speed = 20 m/s
• Time taken = $\frac{\text{Total Distance}}{\text{Speed}}$
• Time taken = $\frac{800 \text{ m}}{20 \text{ m/s}}$
• Time taken = $\frac{800}{20}$ seconds
• Time taken = 40 seconds

Therefore, the train will take 40 seconds to cross the 200 m long bridge.

Comparing with Options

Let's look at the given options:

Our calculated time is 40 seconds, which matches Option 3.

Revision Table: Train Crossing Time Calculation

Additional Information on Train Problems

Problems involving trains crossing objects often depend on the nature of the object being crossed.

• Crossing a point object (like a pole, a person, or a signal post): In this case, the distance covered by the train is simply the length of the train itself, as the object has negligible length compared to the train.
• Crossing a platform, bridge, or tunnel: As seen in this problem, the distance covered is the sum of the length of the train and the length of the object being crossed.
• Crossing another train:
  • If trains are moving in the same direction, the relative speed is the difference between their speeds. The distance covered is the sum of their lengths.
  • If trains are moving in opposite directions, the relative speed is the sum of their speeds. The distance covered is the sum of their lengths.

Always ensure units are consistent (meters and seconds, or kilometers and hours) before applying the time, distance, speed formula.