Two Planes Passing Each Other: A Real-World Application of Speed and Distance

Introduction

In the realm of aviation, the concept of speed and distance is fundamental. This article explores a real-world scenario where two airplanes are traveling towards each other, starting a specified distance apart. We'll apply mathematical principles to determine the individual speeds of the airplanes, providing a detailed and understandable explanation of how to solve such problems.

Problem Statement

To make the problem more relatable, let's frame it as follows:

Two planes are 520 miles apart. One plane is traveling 100 mph faster than the other. The planes pass each other in 0.5 hours. The question asks for the speed of each plane.

Solution Approach

Let's denote the speed of the slower plane as x mph. Then, the speed of the faster plane is x 100 mph.

Step 1: Combined Speed

When traveling towards each other, the combined speed of the two planes is the sum of their individual speeds. Therefore, the combined speed is:

x (x 100) 2x 100 mph

Step 2: Distance Covered

Given that the planes pass each other in 0.5 hours, we can use the distance formula:

Distance Speed x Time

Substituting the known values:

520 (2x 100) x 0.5

Step 3: Solving for x

Multiplying both sides by 2 to eliminate the fraction:

1040 2x 100

Subtract 100 from both sides:

940 2x

Divide by 2:

x 470 mph

Thus, the speed of the slower plane is 470 mph. To find the speed of the faster plane:

x 100 470 100 570 mph

The summary of the speeds is:

Slower plane: 470 mph Faster plane: 570 mph

Verification and Other Examples

To verify our solution, we can use an alternative method:

Distance Rate x Time Plane 1 distance 0.5x Plane 2 distance 0.5(x 100) 0.5x 0.5(x 100) 520 0.5x 0.5x 50 520 1x 50 520 1x 470 x 470 mph

Therefore, the speeds remain the same: 470 mph for the slower plane and 570 mph for the faster plane.

Conclusion

The application of speed and distance formulas is a crucial skill in various fields, including aviation. This problem demonstrates how to solve for individual speeds when the relative speed and distance are known. By following these steps, you can tackle similar problems with ease.