Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

In the design of certain small turbo-prop aircraft, the landing speed \(V\) (in \(\mathrm{ft} / \mathrm{sec})\) is determined by the formula \(W=0.00334 V^{2} S\), where \(W\) is the gross weight (in pounds) of the aircraft and \(S\) is the surface area (in \(\mathrm{ft}^{2}\) ) of the wings. If the gross weight of the aircraft is between 7500 pounds and 10,000 pounds and \(S=210 \mathrm{ft}^{2}\), determine the range of the landing speeds in miles per hour.

Step-by-Step Solution

Answer

The landing speed ranges from 222 mph to 256 mph.

Step 1: Understanding the Problem

The given formula relates the gross weight \( W \), landing speed \( V \), and wing surface area \( S \). We need to determine the range of \( V \) when \( W \) varies from 7500 to 10000 pounds and \( S = 210 \, \mathrm{ft}^2 \). The original formula is written: \[ W = 0.00334 \, V^2 \, S \] We'll solve this equation for \( V \) and then convert the speed from feet per second to miles per hour.

Step 2: Rearranging the Formula for V

Given the formula \[ W = 0.00334 \, V^2 \, S \] we rearrange it to solve for \( V \): \[ V^2 = \frac{W}{0.00334 \, S} \] Thus, \[ V = \sqrt{\frac{W}{0.00334 \, S}} \] We'll substitute \( S = 210 \, \mathrm{ft}^2 \) to find \( V \) in \( \mathrm{ft/s} \) for the specified range of \( W \).

Step 3: Calculating Landing Speed with Minimum Weight

Substitute \( W = 7500 \) and \( S = 210 \) in the formula \[ V = \sqrt{\frac{7500}{0.00334 \, \times \, 210}} \] Calculate \( V \): \[ V \approx \sqrt{106110} \approx 326 \text{ ft/s}\]

Step 4: Calculating Landing Speed with Maximum Weight

Substitute \( W = 10000 \) and \( S = 210 \) in the formula \[ V = \sqrt{\frac{10000}{0.00334 \, \times \, 210}} \] Calculate \( V \): \[ V \approx \sqrt{141480} \approx 376 \text{ ft/s}\]

Step 5: Converting Speed to Miles per Hour

To convert from feet per second to miles per hour, use the conversion: \[ 1 \text{ ft/s} = 0.6818 \text{ mph} \]Thus, convert \( 326 \text{ ft/s} \) and \( 376 \text{ ft/s} \): \[ 326 \text{ ft/s} \approx 326 \times 0.6818 \approx 222 \text{ mph} \] \[ 376 \text{ ft/s} \approx 376 \times 0.6818 \approx 256 \text{ mph} \]

Step 6: Conclusion for Landing Speeds

The landing speed \( V \) varies from approximately \( 222 \text{ mph} \) to \( 256 \text{ mph} \), corresponding to the aircraft gross weight between 7500 and 10000 pounds.

Key Concepts

Turbo-Prop Aircraft

Turbo-prop aircraft are a type of aircraft that uses a turbo-prop engine, which is essentially a jet engine driving a propeller. These aircraft are known for their efficiency and versatility over short to medium distances.
Unlike traditional jet engines, turbo-prop engines are optimized to deliver power to the propeller, making them especially fuel-efficient at lower flight speeds and altitudes.
This efficiency makes them particularly suitable for regional flights, where the cost of fuel and operational range can significantly impact the economics of flight operations. Understanding the particular dynamics of turbo-prop aircraft not only provides insights into their operational characteristics but also aids in applications such as calculating landing speeds under different conditions.

Gross Weight

The gross weight of an aircraft includes the total weight with all its elements combined. It's the sum total of the aircraft itself, including the structure, fuel, passengers, cargo, and other equipment onboard. Monitoring and managing this weight is essential because it directly influences an aircraft’s performance characteristics, such as lift, fuel efficiency, and landing speed.
In this exercise, the gross weight is a variable ranging from 7500 to 10000 pounds, which significantly affects the calculated landing speed.

• Higher gross weights result in a greater gravitational pull requiring more lift, influencing speed and altitude stability.
• It's crucial to not exceed the maximum allowable gross weight as it leads to safety risks and potential structural damage.

Wing Surface Area

Wing surface area, often denoted as 'S', is crucial in determining the lift that an aircraft can generate. Larger wing areas produce more lift, allowing the aircraft to carry more weight. This principle is fundamental in aviation, impacting everything from takeoff, in-flight stability, to landing mechanisms.
In our scenario, the wing surface area is constant at 210 square feet. This constancy is used to isolate the effects of varying gross weight on landing speed.

• A fixed wing surface area means calculations can focus specifically on other factors like weight, simplifying the analysis.
• Understanding the relationship between wing area and aerodynamic forces is vital for designing efficient, safe aircraft capable of slower landing speeds.

Unit Conversion

Unit conversion is a fundamental skill in science and engineering and is crucial in this exercise as it allows the transition from one measurement scale to another, ensuring meaningful interpretations in varying contexts. In this exercise, we convert landing speeds from feet per second (ft/s) to miles per hour (mph).
The conversion factor used is:

• 1 ft/s = 0.6818 mph

Applying this conversion factor helps interpret speed measurements more commonly understood in real-world contexts. For example:

• 326 ft/s converts to approximately 222 mph.
• 376 ft/s converts to approximately 256 mph.

These conversions are crucial for comprehending aircraft performance parameters in practical scenarios, like determining safe landing speeds at different weights.