Problem 94 - University Physics with Modern Physics

Question

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 (\(\textbf{Fig. P6.94}\)). 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 the total air resistance force can be expressed by \(F_air = \alpha v^{2} + \beta /v{^2}\), 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, \(\alpha = 0.30 \, \mathrm{N} \cdot \mathrm{s^{2}/m^{2}}\) and \(\beta = 3.5 \times 10^5 \, \mathrm{N} \cdot \mathrm{m^2/s^2}\). In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed (in km/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 km/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: 166.8 km/h; maximum endurance speed: 132.6 km/h.

Step 1: Understand the Problem

We need to determine the speed at which the airplane will maximize its range and endurance. The total air resistance force is given by \( F_{air} = \alpha v^2 + \frac{\beta}{v^2} \), where \( \alpha = 0.30 \; \mathrm{N \cdot s^2/m^2} \) and \( \beta = 3.5 \times 10^5 \; \mathrm{N \cdot m^2/s^2} \).

Step 2: Formulate for Maximum Range

For maximum range, the product of velocity \( v \) and the air resistance force \( F_{air} \) should be minimized. The expression for this product is \( P = v(\alpha v^2 + \frac{\beta}{v^2}) = \alpha v^3 + \frac{\beta}{v} \).

Step 3: Differentiate and Find Critical Points for Range

To find the speed for maximum range, take the derivative of \( P \) with respect to \( v \) and set it to zero: \[\frac{d}{dv}(\alpha v^3 + \frac{\beta}{v}) = 3\alpha v^2 - \frac{\beta}{v^2} = 0.\] Solve for \( v \).

Step 4: Solve for Speed for Maximum Range

Rearrange the equation to find \( v^4 = \frac{\beta}{3\alpha} \). Thus, \[ v = \left(\frac{\beta}{3\alpha}\right)^{1/4}. \] Substitute \( \alpha \) and \( \beta \) to calculate \( v \) in m/s, then convert to km/h.

Step 5: Formulate for Maximum Endurance

For maximum endurance, the ratio \( \frac{F_{air}}{v} \) should be minimized, given by \( E = \frac{\alpha v^2 + \frac{\beta}{v^2}}{v} = \alpha v + \frac{\beta}{v^3} \).

Step 6: Differentiate and Find Critical Points for Endurance

Take the derivative of \( E \) with respect to \( v \) and set it to zero: \[\frac{d}{dv}(\alpha v + \frac{\beta}{v^3}) = \alpha - \frac{3\beta}{v^4} = 0.\] Solve for \( v \).

Step 7: Solve for Speed for Maximum Endurance

Rearrange the equation to find \( v^4 = \frac{3\beta}{\alpha} \). Thus, \[ v = \left(\frac{3\beta}{\alpha}\right)^{1/4}. \] Substitute \( \alpha \) and \( \beta \) to calculate \( v \) in m/s, then convert to km/h.

Key Concepts

Induced Drag

Induced drag is a type of aerodynamic resistance that occurs due to the generation of lift. As an airplane's wings work to lift the plane off the ground, it disrupts the airflow around it. This disruption creates vortices at the wingtips, leading to a downward and backward deflection of the air.

When air is pushed downwards, the airplane experiences a counteracting force upwards, which is lift, and a backward force, known as induced drag.

• Induced drag is significant at lower speeds because the aircraft requires a higher angle of attack to generate the needed lift, which subsequently increases vortex strength.
• Unlike parasitic drag, induced drag reduces as speed increases because the lift-to-drag ratio improves at higher velocities.

Therefore, to minimize induced drag, pilots often aim to maintain an optimal speed that balances these forces, especially during critical phases of flight such as takeoff and landing.

Lift Force

Lift force is a fundamental aspect of aviation, crucial for an airplane to become airborne. Lift is generated by the difference in pressure between the upper and lower surfaces of an aircraft's wings when air passes over them.

Bernoulli's principle states that as the speed of a fluid increases, the pressure decreases. Thus, when air moves faster over the curved upper part of the wing than the bottom, the pressure above the wing decreases, creating lift.

• Lift must always counteract the aircraft's weight for it to fly.
• It depends on factors such as air density, wing area, and the square of the aircraft's velocity.

Without sufficient lift, the airplane cannot take off or maintain altitude. Adjusting wing flaps and angle of attack helps manage lift during various flight conditions.

Newton's Third Law

Newton's third law of motion plays a key role in understanding flight dynamics, stating that for every action, there is an equal and opposite reaction. This concept is evident in the interaction between the airplane and the air.

When the wings push the air downward, the air exerts an upward reaction force on the wings, creating lift. At the same time, this interaction results in induced drag, which acts opposite to the direction of flight.

• This principle is essential for explaining how lift and drag forces emerge during flight.
• It demonstrates the balance required to maintain steady flight, balancing lift and drag with the airplane's weight and propulsion.

This law is fundamental in designing aircraft that effectively manipulate aerodynamic forces for controlled flight.

Air Resistance Force

Air resistance, also known as drag, is the force that opposes an aircraft's motion through the air. It is composed of both parasitic drag and induced drag. In the context of flight, air resistance is a crucial factor that affects an airplane's speed, fuel efficiency, and range.

Parasitic drag increases with the square of speed and includes skin friction, form drag, and interference drag.

• At higher speeds, parasitic drag dominates because it is directly proportional to the square of the velocity.
• Reducing drag is vital for fuel efficiency and maximizing flight range.

Engineers design sleek, aerodynamic shapes to minimize air resistance, allowing airplanes to move more smoothly and efficiently through the air.