Understanding the Train Crossing Platform Problem

This problem involves a train moving at a constant speed and crossing a stationary platform. A key concept here is understanding the total distance the train needs to cover to completely cross the platform.

Defining the Scenario and Variables

We are given the following information:

• The length of the train is equal to the length of the platform. Let's denote this length by L (in metres).
• The speed of the train is 108 km/hr.
• The time taken for the train to cross the platform is one minute.

We need to find the length of the train, L, in metres.

Calculating Total Distance Covered

When a train crosses a platform, the total distance it travels from the moment the front of the train enters the platform until the moment the rear of the train leaves the platform is equal to the length of the platform plus the length of the train.

Total distance = Length of train + Length of platform

Since the length of the train and the platform are equal (both are L), the total distance covered is:

Total distance \( = L + L = 2L \)

Converting Units for Calculation

The speed is given in kilometres per hour (km/hr) and the time is in minutes. To use the standard speed-distance-time formula, we need to convert these units to metres per second (m/s) and seconds (s), respectively.

Converting Speed from km/hr to m/s

To convert speed from km/hr to m/s, we multiply by the conversion factor \( \frac{5}{18} \).

Speed \( = 108 \text{ km/hr} \)

Speed in m/s \( = 108 \times \frac{5}{18} \text{ m/s} \)

Speed in m/s \( = \frac{108}{18} \times 5 \text{ m/s} \)

Speed in m/s \( = 6 \times 5 \text{ m/s} \)

Speed \( = 30 \text{ m/s} \)

Converting Time from Minutes to Seconds

There are 60 seconds in one minute.

Time \( = 1 \text{ minute} \)

Time in seconds \( = 1 \times 60 \text{ seconds} \)

Time \( = 60 \text{ seconds} \)

Applying the Speed, Distance, Time Formula

The relationship between speed, distance, and time is:

Distance \( = \text{Speed} \times \text{Time} \)

We know the total distance covered is \( 2L \), the speed is \( 30 \text{ m/s} \), and the time is \( 60 \text{ seconds} \).

Substitute these values into the formula:

\( 2L = 30 \text{ m/s} \times 60 \text{ s} \)

\( 2L = 1800 \text{ metres} \)

Solving for the Length of the Train (L)

Now we have an equation to find \( L \):

\( 2L = 1800 \)

To find \( L \), divide both sides of the equation by 2:

\( L = \frac{1800}{2} \)

\( L = 900 \text{ metres} \)

Thus, the length of the train (and the platform) is 900 metres.

Revision Table: Train Crossing Concepts

Here's a quick summary of the key points when a train crosses a platform:

Additional Information: Related Problems

Understanding how to solve train crossing problems is essential for many exams. Related problems might involve:

• A train crossing a pole or a person (where the distance covered is just the length of the train).
• Two trains crossing each other (either moving in the same direction or opposite directions). This requires calculating the relative speed of the trains and the total distance covered (sum of both train lengths).
• A train crossing a bridge or tunnel (similar to crossing a platform, distance is train length + bridge/tunnel length).

Always pay close attention to the units given in the problem and ensure consistency (e.g., using metres and seconds) before applying the speed-distance-time formula.