Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Vertical wind shear occurs when wind speed varies at different heights above the ground. Wind shear is of great importance to pilots during takeoffs and landings. If the wind speed is \(v_{1}\) at height \(h_{1}\) and \(v_{2}\) at height \(h_{2}\), then the average wind shear \(s\) is given by the slope formula $$ s=\frac{v_{2}-v_{1}}{h_{2}-h_{1}} . $$ If the wind speed at ground level is \(22 \mathrm{mi} / \mathrm{hr}\) and \(s\) has been determined to be \(0.07\), find the wind speed 185 feet above the ground.

Step-by-Step Solution

Answer

The wind speed 185 feet above the ground is 34.95 mi/hr.

Step 1: Understand the Given Information

We are given the wind speed at ground level, \(v_1 = 22 \text{ mi/hr}\), and the average wind shear, \(s = 0.07\). We want to find the wind speed \(v_2\) at a height \(h_2 = 185 \text{ feet}\) above ground level, remembering that \(h_1 = 0\) because it is at ground level.

Step 2: Identify the Formula to Use

The formula for average wind shear is given by:\[s = \frac{v_2 - v_1}{h_2 - h_1}\]Plug the known values into this formula to solve for \(v_2\).

Step 3: Substitute the Known Values

Substitute \(s = 0.07\), \(v_1 = 22\), \(h_1 = 0\), and \(h_2 = 185\) into the formula:\[0.07 = \frac{v_2 - 22}{185 - 0}\]

Step 4: Solve for Wind Speed \(v_2\)

To isolate \(v_2\), first multiply both sides of the equation by 185:\[0.07 \times 185 = v_2 - 22\]Calculate \(0.07 \times 185 = 12.95\). So,\[12.95 = v_2 - 22\]Next, add 22 to both sides of the equation to solve for \(v_2\):\[v_2 = 12.95 + 22 = 34.95 \]Thus, the wind speed at 185 feet above ground is \(34.95 \text{ mi/hr}\).

Key Concepts

Slope Formula

The slope formula is a mathematical tool used to determine the rate of change between two points. In the context of wind shear, it helps us calculate the change in wind speed with respect to change in height. This concept is crucial for understanding how the wind behaves at varying altitudes, especially significant for aviation purposes.

The formula is:

• \[ s = \frac{v_2 - v_1}{h_2 - h_1} \]

Where:

• \( s \) is the average wind shear, representing the change in wind speed per unit height.
• \( v_1 \) is the wind speed at height \( h_1 \). In many cases, this might be the ground level speed.
• \( v_2 \) is the wind speed at height \( h_2 \), the height at which we want to calculate the wind speed.

This equation essentially measures how quickly wind speed increases or decreases with altitude, which is vital for safe takeoff and landing of aircraft.

Wind Speed Calculation

Calculating wind speed at different heights involves understanding the relationship between wind speed and altitude through the slope formula. This method helps predict potential challenges aircraft might face during ascent or descent.

Let's illustrate this with an example:

• We know the wind speed at ground level is \( v_1 = 22 \text{ mi/hr} \), and the average wind shear is \( s = 0.07 \).
• The height we are interested in is \( h_2 = 185 \text{ feet} \), and since the ground is at \( h_1 = 0 \text{ feet} \), let's substitute these values into the formula.

Substituting the knowns into the slope formula:

• \( 0.07 = \frac{v_2 - 22}{185 - 0} \)

To find the unknown wind speed \( v_2 \), we perform the following calculations:

• Multiply both sides by 185 to solve for \( v_2 \): \( 0.07 \times 185 = v_2 - 22 \).
• Calculate \( 0.07 \times 185 = 12.95 \).
• Then, add 22 to both sides: \( v_2 = 12.95 + 22 \).
• Thus, \( v_2 = 34.95 \text{ mi/hr} \).

This calculation shows the wind speed at 185 feet is approximately 34.95 miles per hour.

Average Wind Shear

Average wind shear is a measure of how wind speed changes with height, expressed as a rate of speed change per unit of height. It is a significant factor in aviation, influencing how pilots approach takeoff and landing procedures.

Wind shear can impact aircraft performance because:

• Swift changes in wind speed and direction can lead to a loss of control if adjustments are not made.
• Pilots need to be aware of potential shifts to manage better fuel efficiency and flight safety.

To calculate average wind shear:

• Use the slope formula, which provides a straightforward calculation to determine this rate.
• In our example, the average wind shear \( s \) is given, enabling us to work backward to find the unknown wind speed at a certain altitude.

Therefore, understanding and accurately calculating average wind shear is crucial for predicting how conditions might shift as an aircraft changes elevation. It allows for adjustments in flight plans to mitigate risks associated with sudden wind changes.