Problem 86 - Fundamentals of Physics

Question

A radar station detects an airplane approaching directly from the east. At first observation, the airplane is at distance \(d_{1}=360 \mathrm{~m}\) from the station and at angle \(\theta_{1}=40^{\circ}\) above the horizon (Fig. \(4-49\) ). The airplane is tracked through an angular change \(\Delta \theta=123^{\circ}\) in the vertical east-west plane; its distance is then \(d_{2}=790 \mathrm{~m}\). Find the (a) magnitude and (b) direction of the airplane's displacement during this period.

Step-by-Step Solution

Answer

Magnitude is \(~823.3 \mathrm{~m}\), direction is \(~77.5^{\circ}\) above the horizon.

Step 1: Understand the Problem

We need to find both the magnitude and direction of the airplane's displacement as it changes position from \( (d_1, \theta_1) \) to \( (d_2, \theta_2) \). This involves trigonometry and vector analysis as the problem gives two angles and distances relative to the radar station.

Step 2: Calculate the Initial Position

Convert the initial radar coordinates of the airplane into Cartesian coordinates using trigonometry. The airplane's initial height \( h_1 \) is given by the vertical component \( h_1 = d_1 \sin(\theta_1) \). The horizontal distance \( x_1 \) is \( x_1 = d_1 \cos(\theta_1) \).

Step 3: Calculate the Final Position

Following the same process, calculate the final position: \( \theta_2 = \theta_1 + \Delta \theta \).Compute the final height \( h_2 = d_2 \sin(\theta_2) \). Compute the final horizontal distance \( x_2 = d_2 \cos(\theta_2) \).

Step 4: Determine Displacement Components

Find the change in each component: \( \Delta x = x_2 - x_1 \) and \( \Delta h = h_2 - h_1 \). These represent the horizontal and vertical components of displacement.

Step 5: Calculate Displacement Magnitude

Use the Pythagorean theorem to calculate the magnitude of the displacement \( \Delta d \) using its components: \( \Delta d = \sqrt{(\Delta x)^2 + (\Delta h)^2} \).

Step 6: Determine Displacement Direction

Calculate the angle \( \theta \) of displacement relative to the horizontal axis using \( \theta = \tan^{-1}(\Delta h / \Delta x) \). This gives the direction of displacement in degrees above the horizon.

Key Concepts

Vector Analysis

Vector analysis helps us understand situations involving direction and magnitude, like the airplane tracked by a radar station. In this case, the position of the airplane changes from an initial to a final point. This can be described in terms of vectors. A vector has two key properties: its length (magnitude) and where it points (direction).
Vectors can be added, subtracted, and multiplied by scalars. Here, the initial and final positions are vectors. Calculating the displacement means finding the vector that shows the change in position from start to finish.
By converting the airplane's radar coordinates into Cartesian coordinates, we can treat the positions as vectors and use arithmetic operations to get the displacement vector. The displacement is a new vector resulting from subtracting the initial position vector from the final position vector. This gives both the magnitude and direction of the airplane's movement during the tracked period.

Pythagorean Theorem

The Pythagorean Theorem is crucial in finding the magnitude of the displacement vector. This theorem applies to right-angled triangles and states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In our case, after finding the horizontal component (\(\Delta x\)) and the vertical component (\(\Delta h\)) of the airplane's displacement vector, we can think of these as the two sides of a right triangle. The total displacement (\(\Delta d\)) is like the hypotenuse.
Using the Pythagorean theorem, the magnitude of displacement is calculated as follows:

• The formula is \(\Delta d = \sqrt{(\Delta x)^2 + (\Delta h)^2}\).
• This yields a single number representing how far the airplane has moved, regardless of the path it took.

Understanding this theorem is key to solving many problems that involve distances and coordinates, helping us find how far an object traveled in linear space.

Cartesian Coordinates

Cartesian coordinates are a system that uses two numbers to pinpoint a position on a plane, much like a map. These numbers are usually denoted as \(x\) and \(y\), representing horizontal and vertical positions respectively.
For the radar problem, the airplane's position is first described using polar coordinates: distance from the radar and angle above the horizon. We convert these to Cartesian coordinates to simplify the vector calculation process.

• To find the initial position: - The vertical position (height \(h_1\)) is calculated with \(h_1 = d_1 \sin(\theta_1)\). - The horizontal position (\(x_1\)) is obtained by \(x_1 = d_1 \cos(\theta_1)\).
• Similarly, the final position is calculated using the same method with different distance and angle values.

Understanding how to switch between polar and Cartesian coordinates helps in tackling various problems in physics, engineering, and mathematics, allowing for easier calculations and visualizations of movement and change.