Train Speed Calculation Opposite Direction

This problem requires calculating the speed of a train based on the time it takes to pass two individuals moving in opposite directions at different speeds. The core concept is using relative speed and the relationship between distance, speed, and time.

Understanding Relative Speed

When two objects move towards each other (in opposite directions), their relative speed is the sum of their individual speeds. The distance the train covers to "pass" a person is equal to the length of the train itself.

Setting Up the Equations

Let the speed of the train be $S \text{ m/s}$. Let the length of the train be $L$ meters.

• Scenario 1: The train passes a person walking at $4 \text{ m/s}$. The relative speed is $(S + 4) \text{ m/s}$. The time taken is $10$ seconds. Using Distance = Speed × Time, we get: $L = (S + 4) \times 10$ $L = 10S + 40$
• Scenario 2: The train passes a person walking at $10 \text{ m/s}$. The relative speed is $(S + 10) \text{ m/s}$. The time taken is $8$ seconds. Using Distance = Speed × Time, we get: $L = (S + 10) \times 8$ $L = 8S + 80$

Solving for Train Speed

Since the length of the train ($L$) is the same in both scenarios, we equate the two expressions for $L$: $10S + 40 = 8S + 80$

Now, we solve this equation for $S$:

• Rearrange the terms to group speeds: $10S - 8S = 80 - 40$
• Simplify: $2S = 40$
• Calculate the speed: $S = \frac{40}{2}$ $S = 20$

The speed of the train is $20 \text{ m/s}$.