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 (Figure 1). 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 v², so that the total air resistance force can be expressed by Fair=av² + B/v², where a and 3 are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use a = 0.320 N s²/m² and 3 = 3.59x105 Nm²/s². In steady flight, the engine must provide a forward force that exactly balances the air resistance force. Figure Induced drag Lift Force of air on wings < 1 of 1 > Part A Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. Express your answer in meters per second. IVE ΑΣΦ V = Submit ▾ Part B V = Submit Request Answer Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time). Express your answer in meters per second. 15. ΑΣΦΑ Provide Feedback ? Request Answer m/s ? m/s
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 (Figure 1). 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 v², so that the total air resistance force can be expressed by Fair=av² + B/v², where a and 3 are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use a = 0.320 N s²/m² and 3 = 3.59x105 Nm²/s². In steady flight, the engine must provide a forward force that exactly balances the air resistance force. Figure Induced drag Lift Force of air on wings < 1 of 1 > Part A Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. Express your answer in meters per second. IVE ΑΣΦ V = Submit ▾ Part B V = Submit Request Answer Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time). Express your answer in meters per second. 15. ΑΣΦΑ Provide Feedback ? Request Answer m/s ? m/s
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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![**Educational Content: Understanding Air Resistance and Flight Dynamics**
An airplane in flight is subject to an air resistance force proportional to the square of its speed \(v\). However, another resistive force is present due to the airplane's wings. Air flowing over the wings is pushed down and slightly forward, and Newton's third law tells us the air exerts a force on the wings and airplane that is up and slightly backward (Figure 1). 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:
\[ F_{\text{air}} = \alpha v^2 + \frac{\beta}{v^2} \]
where \(\alpha\) and \(\beta\) are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use:
\[ \alpha = 0.320 \, \text{N} \cdot \text{s}^2/\text{m}^2 \quad \text{and} \quad \beta = 3.59 \times 10^5 \, \text{N} \cdot \text{m}^2/\text{s}^2 \]
In steady flight, the engine must provide a forward force that exactly balances the air resistance force.
**Figure Description:**
The diagram shows an airplane with arrows indicating the forces acting on it. The lift (upward force) and induced drag (backward force) are shown, along with the force of air on the wings.
**Part A**
Calculate the speed at which this airplane will have the maximum **range** (that is, travel the greatest distance) for a given quantity of fuel.
- Express your answer in meters per second.
\[ v = \quad \text{m/s} \]
**Part B**
Calculate the speed for which the airplane will have the maximum **endurance** (that is, will remain in the air the longest time).
- Express your answer in meters per second.
\[ v = \quad \text{m/s} \]
*Submit your answers and check for accuracy.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fb7e309-59c9-4ed8-8b13-a25379b18db0%2F4aa1167c-4be9-43d3-ad6c-5706988a51f8%2Fhhl0nst_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content: Understanding Air Resistance and Flight Dynamics**
An airplane in flight is subject to an air resistance force proportional to the square of its speed \(v\). However, another resistive force is present due to the airplane's wings. Air flowing over the wings is pushed down and slightly forward, and Newton's third law tells us the air exerts a force on the wings and airplane that is up and slightly backward (Figure 1). 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:
\[ F_{\text{air}} = \alpha v^2 + \frac{\beta}{v^2} \]
where \(\alpha\) and \(\beta\) are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use:
\[ \alpha = 0.320 \, \text{N} \cdot \text{s}^2/\text{m}^2 \quad \text{and} \quad \beta = 3.59 \times 10^5 \, \text{N} \cdot \text{m}^2/\text{s}^2 \]
In steady flight, the engine must provide a forward force that exactly balances the air resistance force.
**Figure Description:**
The diagram shows an airplane with arrows indicating the forces acting on it. The lift (upward force) and induced drag (backward force) are shown, along with the force of air on the wings.
**Part A**
Calculate the speed at which this airplane will have the maximum **range** (that is, travel the greatest distance) for a given quantity of fuel.
- Express your answer in meters per second.
\[ v = \quad \text{m/s} \]
**Part B**
Calculate the speed for which the airplane will have the maximum **endurance** (that is, will remain in the air the longest time).
- Express your answer in meters per second.
\[ v = \quad \text{m/s} \]
*Submit your answers and check for accuracy.*
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