Understanding Train Crossing Dynamics

This problem requires us to determine the time it takes for two trains, moving at different speeds, to cross each other when traveling in opposite directions. We are given information about their crossing time when they move in the same direction. The key concepts involved are relative speed and the total distance covered during a crossing maneuver.

Given Information Analysis

Let's break down the information provided in the question:

• The lengths of the two trains are equal. Let's denote the length of each train as \(L\).
• Speed of the first train (\(v_1\)) = 60 km/h.
• Speed of the second train (\(v_2\)) = 40 km/h.
• Time taken for the trains to cross each other when moving in the same direction (\(t_{same}\)) = 50 seconds.
• We need to find the time taken for the trains to cross each other when moving in opposite directions (\(t_{opposite}\)).

Key Concepts: Relative Speed and Distance

When two objects move, their relative speed is the speed at which the distance between them changes. The way we calculate relative speed depends on whether they are moving in the same or opposite directions:

• Same Direction: The relative speed is the difference between their individual speeds. \(v_{rel\_same} = |v_1 - v_2|\)
• Opposite Directions: The relative speed is the sum of their individual speeds. \(v_{rel\_opposite} = v_1 + v_2\)

For two trains of length \(L\) each to completely cross each other, the total distance they need to cover relative to each other is the sum of their lengths, which is \(L + L = 2L\).

Step-by-Step Solution Calculation

Converting Speeds to Consistent Units (m/s)

The speeds are given in kilometers per hour (km/h), but the time is in seconds. To perform calculations correctly, we need to convert the speeds to meters per second (m/s). The conversion factor is: 1 km/h = \(\frac{5}{18}\) m/s.

• Speed of Train 1 (\(v_1\)): \(v_1 = 60 \text{ km/h} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} = \frac{50}{3} \text{ m/s}\).
• Speed of Train 2 (\(v_2\)): \(v_2 = 40 \text{ km/h} = 40 \times \frac{5}{18} \text{ m/s} = \frac{200}{18} \text{ m/s} = \frac{100}{9} \text{ m/s}\).

Calculating the Total Length (\(2L\)) using Same Direction Data

First, let's find the relative speed when the trains are moving in the same direction:

\(v_{rel\_same} = v_1 - v_2 = \frac{50}{3} \text{ m/s} - \frac{100}{9} \text{ m/s}\)

To subtract these fractions, we find a common denominator, which is 9:

\(v_{rel\_same} = \frac{50 \times 3}{3 \times 3} \text{ m/s} - \frac{100}{9} \text{ m/s} = \frac{150}{9} \text{ m/s} - \frac{100}{9} \text{ m/s} = \frac{50}{9} \text{ m/s}\).

Now, we use the relationship Distance = Speed × Time (\(2L = v_{rel\_same} \times t_{same}\)) to find the total length (\(2L\)):

\(2L = \left(\frac{50}{9} \text{ m/s}\right) \times (50 \text{ s})\)

\(2L = \frac{2500}{9} \text{ meters}\).

Calculating Time for Opposite Direction Crossing

Next, we find the relative speed when the trains are moving in opposite directions:

\(v_{rel\_opposite} = v_1 + v_2 = \frac{50}{3} \text{ m/s} + \frac{100}{9} \text{ m/s}\)

Again, using the common denominator 9:

\(v_{rel\_opposite} = \frac{150}{9} \text{ m/s} + \frac{100}{9} \text{ m/s} = \frac{250}{9} \text{ m/s}\).

Finally, we can calculate the time taken to cross in opposite directions using the formula Time = Distance / Speed (\(t_{opposite} = \frac{2L}{v_{rel\_opposite}}\)):

\(t_{opposite} = \frac{2500/9 \text{ meters}}{250/9 \text{ m/s}}\)

To divide the fractions, we multiply by the reciprocal of the denominator:

\(t_{opposite} = \frac{2500}{9} \times \frac{9}{250} \text{ seconds}\)

\(t_{opposite} = \frac{2500}{250} \text{ seconds}\)

\(t_{opposite} = 10 \text{ seconds}\).

Conclusion on Crossing Time

Based on the calculations, the two trains will take 10 seconds to cross each other when they are running in opposite directions.