Problem 87 - Applied Statistics and Probability for Engineers

Question

An aircraft is flying at a constant altitude with velocity magnitude \(r_{1}\) (relative to the air) and angle \(\theta_{1}\) (in a twodimensional coordinate system). The magnitude and direction of the wind are \(r_{2}\) and \(\theta_{2},\) respectively. Suppose that the wind angle is uniformly distributed between 10 and 20 degrees and all other parameters are constant. Determine the probability density function of the magnitude of the resultant vector \(r=\left[r_{1}^{2}+r_{2}^{2}+r_{1} r_{2}\left(\cos \theta_{1}-\cos \theta_{2}\right)\right]^{0.5}\)

Step-by-Step Solution

Answer

The PDF of the resultant velocity magnitude \( r \) is influenced uniformly due to the balanced effect of \( \theta_2 \) over 10 degrees.

Step 1: Identify Variables and Relationship

Given parameters include the aircraft velocity magnitude \( r_1 \), its direction \( \theta_1 \), wind velocity magnitude \( r_2 \), and its direction \( \theta_2 \). We need to find the probability density function (PDF) of the resultant velocity magnitude \( r \). The angle \( \theta_2 \) has given bounds of 10 to 20 degrees and is assumed to be uniformly distributed. The resultant vector's magnitude is calculated using:\[r = \left[r_{1}^{2} + r_{2}^{2} + r_{1}r_{2}(\cos(\theta_1) - \cos(\theta_2))\right]^{0.5}\]

Step 2: Define Problem Constraints

Since \( \theta_2 \) is uniformly distributed between 10 and 20 degrees, we first need to focus on this parameter. The uniform distribution implies that the probability density function (PDF) for \( \theta_2 \) is constant: \( f(\theta_2) = \frac{1}{10} \) for \( \theta_2 \) within the 10 to 20-degree range. With this information, \( \theta_2 \) must be converted from degrees to radians for calculations.

Step 3: Substitute and Simplify

Substitute \( \theta_2 \) from the uniform distribution into the magnitude equation and analyze:\[r = \left[r_{1}^{2} + r_{2}^{2} + r_{1}r_{2}\left(\cos(\theta_1) - \cos(\theta_2)\right)\right]^{0.5}\]Focus on the component \( \cos(\theta_2) \) as it is the variable dependent on \( \theta_2 \). The function \( \cos(\theta_2) \) will vary as \( \theta_2 \) changes, affecting the resulting \( r \).

Step 4: Analyze Effect on Resultant Magnitude r

The resultant magnitude \( r \) is affected by the cosine difference in the formula. As \( \theta_2 \) spans 10 to 20 degrees, \( \cos(\theta_2) \) spans a certain range. Given the cos function's slowly changing nature over the small angle difference, \( r \) will have minimal variation.

Step 5: Determine Probability Density Function for r

Given the uniform spread of \( \theta_2 \), one must integrate the effect of \( \theta_2 \) on \( r \). However, since the change in cosine value over a 10-degree range is minimal, and no specific distribution is heavily defined for \( r \) besides the randomness of \( \theta_2 \), a basic resultant is computed. This suggests a continuous variation of \( r \) with possibly minuscule differences in frequency across small \( r \) intervals.

Step 6: Conclusion and Formulation

Because \( \theta_2 \) influences the system minimally over its 10-degree range, and under the uniform distribution condition, the density of \( r \) defined by \( f(r) \) remains affected by the form, not a distinct value distribution. This results in a general linear profile attribute, indicating equidistribution in its bounded segment.

Key Concepts

Uniform Distribution

In statistics, a uniform distribution is where every outcome in a given interval is equally likely. When applied to the angle \( \theta_2 \) in the context of the aircraft navigation problem, it suggests that all angles between 10 and 20 degrees have the same probability of occurrence. This equates to a constant probability density function (PDF) within that interval. For the uniform distribution, the probability density function for any angle \( \theta_2 \) within the range is \( f(\theta_2) = \frac{1}{b-a} \) where \( a \) and \( b \) define the interval. Here, \( a = 10 \) degrees and \( b = 20 \) degrees, so the PDF is \( f(\theta_2) = \frac{1}{10} \). This implies a consistent probability for each degree, ensuring that \( \theta_2 \) affects the system evenly across its range.

Resultant Vector Magnitude

The resultant vector magnitude, \( r \), is a critical component in determining the final velocity of the aircraft as it navigates through the wind. It's derived by combining the aircraft's velocity vector with the wind's velocity vector. Using the formula: \[ r = \sqrt{r_1^2 + r_2^2 + 2r_1r_2 \cos(\theta_1 - \theta_2)} \] This calculation finds the combined effect of the two velocities taking into account their directionality. Yet, since \( \theta_2 \) is uniformly distributed and the change is mild over the given interval, the impact on \( r \) remains subtle. This keeps the overall resultant almost constant, notwithstanding the minor fluctuations in angle due to wind.

Trigonometric Functions

Trigonometric functions play a pivotal role in calculating vector magnitudes, most notably with the cosine function affecting the resultant vector's magnitude. In this exercise, the relationship \( r = \sqrt{r_1^2 + r_2^2 + 2r_1r_2 \cos(\theta_1 - \theta_2)} \) is used where \( \cos(\theta_1 - \theta_2) \) adjusts the magnitude depending on the angle between the velocity vector of the aircraft and the wind. As \( \theta_2 \) variances over its uniform distribution, \( \cos(\theta_2) \) slightly shifts while having only a minimal impact due to trigonometric properties over small angle changes. Understanding these properties is crucial for anticipating the behavior of the resultant vector in real-world applications like aircraft navigation.

Aircraft Navigation

Aircraft navigation involves finding the most efficient path considering external factors like wind. By navigating with a clear understanding of both vectors and their combined magnitude, pilots can strategize routes for optimized flight paths. Through the exercise problem, we see how the aircraft's velocity integrates with wind velocity to produce the resultant vector that guides navigation. The uniform distribution of \( \theta_2 \) highlights the variability of wind direction without excessive complexity, representative of actual flight conditions. This model aids in illustrating the dynamics airplanes encounter, ensuring a balance between natural forces and navigational tactics.