Problem 104 - University Physics with Modern Physics

Question

CALC An airplane in flight is subject to an air resistance force proportional to the square of its speed \(v .\) But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.104). The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to \(v^{2},\) so that the total air resistance force can be expressed by \(F_{\text { air }}=\alpha v^{2}+\beta / v^{2},\) where \(\alpha\) and \(\beta\) are positive constants that depend on the shape and size of the airplane and the density of the air. For a Cessna \(150,\) a small single-engine airplane, \(\alpha=0.30 \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2}\) and \(\beta=3.5 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{s}^{2} .\) In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed (in \(\mathrm{km} / \mathrm{h} )\) at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. (b) Calculate the speed (in \(\mathrm{km} / \mathrm{h} )\) for which the airplane will have the maximum endurance (that is, remain in the air the longest time).

Step-by-Step Solution

Answer

Maximum range speed: 205.74 km/h. Maximum endurance speed: 159.05 km/h.

Step 1: Identify the Objective

We need to find the speed at which the Cessna 150 will have both the maximum range and the maximum endurance. These speeds are determined by balancing different aspects of forces acting on the airplane.

Step 2: Understand Maximum Range Condition

For the maximum range, the speed should be such that the fuel consumption per unit distance is minimized. This corresponds to minimizing the air resistance force per unit speed, which can be written as \( F_{\text{air}}/v = \alpha v + \beta/v^3 \).

Step 3: Set Derivative for Minimum Fuel Consumption (Range)

To find the speed that minimizes fuel consumption per distance, take the derivative of \( \alpha v + \beta/v^3 \) with respect to \( v \), and set it to zero: \( \frac{d}{dv}(\alpha v + \beta/v^3) = \alpha - \frac{3\beta}{v^4} = 0 \).

Step 4: Solve for Speed (Range)

Solving \( \alpha - \frac{3\beta}{v^4} = 0 \), we get \( v^4 = \frac{3\beta}{\alpha} \). Therefore, \( v = \sqrt[4]{\frac{3\beta}{\alpha}} \). Substitute \( \alpha = 0.30 \) and \( \beta = 3.5 \times 10^5 \) to find \( v \).

Step 5: Calculate Maximum Range Speed

Calculate \( v = \sqrt[4]{\frac{3 \times 3.5 \times 10^5}{0.30}} = \sqrt[4]{3.5 \times 10^6} \). After calculation, this speed is approximately 57.15 m/s. Convert this to km/h: \( 57.15 \times 3.6 \approx 205.74 \text{ km/h} \).

Step 6: Understand Maximum Endurance Condition

For maximum endurance, the airplane should minimize fuel consumption per unit time. This corresponds to minimizing \( F_{\text{air}} = \alpha v^2 + \beta/v^2 \).

Step 7: Set Derivative for Minimum Fuel Consumption (Endurance)

Take the derivative of \( \alpha v^2 + \beta/v^2 \) with respect to \( v \), and set it to zero: \( 2\alpha v - \frac{2\beta}{v^3} = 0 \).

Step 8: Solve for Speed (Endurance)

Solving \( 2\alpha v = \frac{2\beta}{v^3} \) leads to \( v^4 = \frac{\beta}{\alpha} \). Hence, \( v = \sqrt[4]{\frac{\beta}{\alpha}} \). Substitute \( \alpha = 0.30 \) and \( \beta = 3.5 \times 10^5 \) to find \( v \).

Step 9: Calculate Maximum Endurance Speed

Calculate \( v = \sqrt[4]{\frac{3.5 \times 10^5}{0.30}} = \sqrt[4]{1.1667 \times 10^6} \). This speed is approximately 44.18 m/s. Convert this to km/h: \( 44.18 \times 3.6 \approx 159.05 \text{ km/h} \).

Key Concepts

Induced Drag

Induced drag is a critical concept for understanding how airplanes operate in flight. This type of drag occurs due to the generation of lift by an aircraft's wings. As the air moves over the wings, it is deflected downwards and slightly forward, which according to Newton's Third Law, results in a force exerted by the air in the opposite direction. The air pushes back and slightly down on the wings, creating a component of force known as induced drag.

This drag is essential because it affects how much power the engine needs to maintain flight. At lower speeds, induced drag is higher because the wings need to generate more lift to keep the plane aloft, leading to a larger angle of attack. Interestingly, induced drag is inversely proportional to the square of the plane's speed ( 1/v^2 ). This means that as a plane goes faster, the induced drag decreases, which is a crucial factor when considering the efficiency of an aircraft in flight.

Maximum Range

Maximum range in aviation is all about how far an aircraft can travel on a given amount of fuel. Achieving this involves finding the optimal speed where fuel consumption per unit distance is minimized. At this point, the total drag force, represented by \(F_{\text{air}} = \alpha v^2 + \beta / v^2\), is balanced efficiently.

To determine the speed for maximum range, we need to minimize the air resistance force per unit of velocity, \(\alpha v + \beta/v^3\). By setting the derivative of this function with respect to speed \(v\) to zero, the resulting equation allows us to calculate the optimal speed. For the Cessna 150, this speed is found to be approximately 205.74 km/h. At this speed, the plane can travel the furthest distance possible before needing to refuel.

Maximum Endurance

Maximum endurance refers to the duration an aircraft can remain airborne. It focuses on minimizing fuel consumption over time rather than distance. This means flying at a speed that allows the aircraft to stay in the air the longest while using the least amount of fuel each second.

• The factors involved in determining maximum endurance include managing thrust and drag to ensure fuel efficiency over time.
• The derivative of the air resistance force \(\alpha v^2 + \beta/v^2\) with respect to speed (v) helps identify this critical speed.

For the Cessna 150, this optimal speed is determined to be about 159.05 km/h. At this speed, the aircraft uses fuel most efficiently to prolong its time in the air, making it ideal for endurance flights such as long surveillance missions or holding patterns in aviation.

Newton's Third Law

Newton's Third Law of Motion is fundamental in understanding how forces act on an airplane in flight. The law states that for every action, there is an equal and opposite reaction. This principle explains how the wings of an aircraft generate lift and, consequently, induced drag.

• When the airplane wings push down on the air to generate lift, the air reacts by exerting an equal force upwards, which is the lift that keeps the airplane aloft.
• Similarly, the backward component of this reaction force is what creates induced drag.

In aviation, understanding Newton's Third Law helps engineers and pilots anticipate how changes in speed and angle affect lift and drag forces. It is a vital concept when optimizing performance and fuel efficiency, as it underpins the very mechanics of flight and maneuverability.