Calculating Train Time to Cross a Bridge

This problem involves understanding the relationship between speed, distance, and time, and how to apply these concepts when a train crosses an object like a pole or a bridge.

Understanding the Problem Components

We are given the train's speed, the time it takes to cross a pole, and the length of a bridge. We need to find the time it takes for the train to cross the bridge.

• Train Speed: 36 km/h
• Time to cross a pole: 25 seconds
• Bridge length: 250 m

Step 1: Convert Speed to Consistent Units

The speed is given in kilometers per hour (km/h), but distances are in meters (m) and time is in seconds (s). We need to convert the speed to meters per second (m/s).

The conversion factor from km/h to m/s is $\frac{5}{18}$.

Speed in m/s = $36 \times \frac{5}{18}$ m/s

Speed in m/s = $2 \times 5$ m/s

Speed in m/s = $10$ m/s

So, the train's speed is 10 m/s.

Step 2: Calculate the Train's Length

When a train crosses a pole, the distance covered by the train is equal to its own length. We know the speed and the time taken to cross the pole, so we can calculate the distance (train length) using the formula: Distance = Speed $\times$ Time.

Train Length = Speed $\times$ Time to cross pole

Train Length = $10 \text{ m/s} \times 25 \text{ s}$

Train Length = $250$ meters

The length of the train is 250 meters.

Step 3: Calculate the Total Distance to Cross the Bridge

When a train crosses a bridge, the total distance the train's front end must travel from the moment it reaches the bridge's start until the train's rear end leaves the bridge's end is the sum of the train's length and the bridge's length.

Total Distance = Train Length + Bridge Length

Total Distance = $250 \text{ m} + 250 \text{ m}$

Total Distance = $500$ meters

Step 4: Calculate the Time to Cross the Bridge

Now that we have the total distance to be covered and the train's speed, we can calculate the time taken to cross the bridge using the formula: Time = Distance / Speed.

Time to cross bridge = Total Distance / Speed

Time to cross bridge = $\frac{500 \text{ m}}{10 \text{ m/s}}$

Time to cross bridge = $50$ seconds

Therefore, it will take 50 seconds for the train to cross the bridge.

Revision Table: Train Speed, Distance, and Time

Here is a quick summary of the key concepts used in this problem.

Additional Information: Relative Speed Concepts

While this problem didn't require relative speed, it's a related concept in train problems. Relative speed is important when considering two moving objects, such as two trains or a train and a person running.

• If two objects move in the same direction, their relative speed is the difference between their speeds.
• If two objects move in opposite directions, their relative speed is the sum of their speeds.

In problems involving a train crossing a stationary object (like a pole, bridge, or platform), we only consider the train's speed because the object is not moving relative to the ground.