Understanding Angular and Linear Displacement in Airplane Movements

An airplane's movement can be quite complex when considering the angles and distances involved. Understanding displacement in various directions is crucial for both theoretical calculations and practical applications. In this article, we’ll explore how to calculate the net displacement of an airplane that moves 400 meters due north, 300 meters south, and 1200 meters upward.

Breaking Down the Movements: North-South and Upward Directions

The airplane’s journey can be broken down into distinct movements along the north-south and vertical axes:

Movement due North: 400 meters

Movement due South: 300 meters

Movement upward: 1200 meters

Step 1: Calculating Net North-South Displacement

The first step in calculating the net displacement is to find the net displacement in the North-South direction. Since the airplane first moves north and then south, we can subtract the southward movement from the northward movement:

Net North-South Displacement: 400 meters - 300 meters 100 meters North

Step 2: Calculating Net Vertical Displacement

The vertical displacement is straightforward since it only involves upward motion:

Vertical Displacement: 1200 meters upward

Step 3: Combining Displacements with a Right Triangle

We can visualize the net displacement as a right triangle where the legs are the net north-south displacement and the vertical displacement:

One leg: 100 meters (Net North-South) The other leg: 1200 meters (Vertical)

Step 4: Calculating the Magnitude of the Resultant Displacement

To find the overall magnitude of the displacement, we use the Pythagorean theorem:

Magnitude of Resultant Displacement: (sqrt{100^2 1200^2}) (sqrt{10000 1440000}) (sqrt{1450000}) ≈ 1204.16 meters

Step 5: Determining the Direction of Net Displacement

To find the direction of the net displacement, we use the arctangent function:

Angle (theta): (tan^{-1}left(frac{1200}{100}right) tan^{-1}(12) approx 85^{circ})

This angle is measured from north towards the upward direction.

Conclusion

The net displacement of the airplane is approximately 1204.16 meters at an angle of 85.0 degrees from the north towards the upward direction.

Practical Considerations

While the above calculations provide a precise vector displacement, in practical terms, the net displacement can vary based on starting position:

If the plane starts near the equator, the net north-south displacement is minimal (100 meters). If the plane starts near the North Pole, the net displacement can be much greater due to the curvature of the Earth.

Therefore, the final displacement depends significantly on the initial latitude of the airplane’s starting point.

Additional Insights

From a vector perspective, each movement can be represented as a vector in the i (north-south) and j (vertical) directions:

North Movement: 400i South Movement: -300i Upward Movement: 1200j

The net displacement vector can be represented as:

Net Displacement Vector: (100i 1200j)

The magnitude of this vector is calculated as:

Magnitude of Net Displacement Vector: (sqrt{100^2 1200^2}) 1204.16 meters

Thus, using vector analysis, we can accurately describe the airplane's net movement in both linear and angular terms.