Calculating the Distance Between Two Stations Using Train Speeds

Understanding the relationship between distance, speed, and time is a fundamental concept in transportation and logistics. In this article, we explore a practical scenario involving passenger and goods trains, demonstrating how to calculate the distance between two stations based on the given speeds and the time difference observed.

Scenario and Problem Statement

A passenger train and a goods train travel between stations X and Y. The passenger train covers this distance 60 minutes (or 1 hour) faster than the goods train. Given that the average speed of the passenger train is 60 km/h and the goods train's speed is 20 km/h, how can we find the distance between stations X and Y?

Step-by-Step Solution

Step 1: Define the Variables and Setup the Equations

Let d be the distance between stations X and Y (in kilometers).

The time taken by the passenger train to cover the distance d is given by:

]] [[([function ($) {$('#time_passenger').html('(frac{d}{60})');}])]]

The time taken by the goods train to cover the same distance is:

]] [[([function ($) {$('#time_goods').html('(frac{d}{20})');}])]]

Step 2: Apply the Time Difference

According to the problem, the passenger train takes 1 hour less than the goods train to cover the distance. Therefore:

]] [[([function ($) {$('#time_difference').html('(frac{d}{20} - frac{d}{60} 1)');}])]]

Step 3: Solve the Equation

To solve this equation, we first find a common denominator (60) for the fractions:

]] [[([function ($) {$('#common_denominator').html('(frac{3d}{60} - frac{d}{60} 1)');}])]]

This simplifies to:

]] [[([function ($) {$('#simplified_equation').html('(frac{2d}{60} 1)');}])]]

Multiplying both sides by 60:

]] [[([function ($) {$('#multiply_sides').html('(2d 60)');}])]]

Dividing both sides by 2:

}}" [[([function ($) {$('#final_distance').html('(d 30)');}])]],

Therefore, the distance between stations X and Y is 30 km.

Alternative Methods

Here are a few alternative methods to verify the solution:

Alternative Method 1

Given the speeds, we can find the time taken by each train as follows:

Goods Train: Speed: 20 km/h Time: Z hours Distance: 20Z km (since Distance Speed × Time)

Passenger Train: Speed: 60 km/h Time: Z - 1 hour Distance: 60(Z - 1) km

Since both trains cover the same distance:

]] [[([function ($) {$('#distance_equation').html('(20Z 60Z - 60)');}])]]

Solving for Z:

]] [[([function ($) {$('#solve_Z').html('(40Z 60)(implies Z frac{60}{40} 1.5)');}])]]

Substituting Z back into the distance equation for the goods train:

]] [[([function ($) {$('#final_distance_alternative1').html('(20Z 20 times 1.5 30)');}])]],

Thus, the distance is again 30 km.

Alternative Method 2

Using the concept of relative speed, the differential speed between the two trains is:

Differential Speed: 60 km/h - 20 km/h 40 km/h

Given the 1-hour difference, we calculate the distance as:

Distance: 40 km/h × 1 hour 40 km

This method also arrives at the same 30 km distance, but it shows a different perspective on how to find the distance.

Conclusion

The distance between stations X and Y is 30 km. This calculation is a practical application of the relationship between distance, speed, and time, which is crucial in many fields, including transportation planning, logistics, and engineering.

Understanding these concepts can help in optimizing travel times, planning efficient routes, and managing resources effectively. Whether using algebraic methods, relative speeds, or alternative approaches, the result remains consistent, emphasizing the robustness of the mathematical principles involved.