Problem 38 - Algebra and Trigonometry

Question

Aerodynamic Lift The lift \(L\) on an airplane wing at take-off varies jointly as the square of the speed \(s\) of the plane and the area \(A\) of its wings. A plane with a wing area of 500 \(\mathrm{ft}^{2}\) traveling at 50 \(\mathrm{mi} / \mathrm{h}\) experiences a lift of 1700 \(\mathrm{Jb}\) . How much lift would a plane with a wing area of 600 \(\mathrm{ft}^{2}\) traveling at 40 \(\mathrm{mi} / \mathrm{h}\) experience?

Step-by-Step Solution

Answer

The lift on the new plane is 1305.6 lb.

Step 1: Understanding the Formula

The lift \( L \) varies jointly as the square of the speed \( s \) and the wing area \( A \). This can be expressed as \( L = k \cdot s^2 \cdot A \), where \( k \) is a constant of proportionality that we need to solve for.

Step 2: Solving for the Constant

Substitute the known values into the formula to find \( k \). For the plane traveling at 50 mph with a wing area of 500 ft² and experiencing a lift of 1700 lb, use \( 1700 = k \cdot 50^2 \cdot 500 \).

Step 3: Calculating the Constant

Solve for \( k \):\[ 1700 = k \cdot 2500 \cdot 500 \]\[ 1700 = k \cdot 1,250,000 \]\[ k = \frac{1700}{1,250,000} = 0.00136 \]

Step 4: Substitute New Values

Use the constant \( k \) to find the lift for the new plane. Substitute \( s = 40 \) and \( A = 600 \) into the formula: \( L = 0.00136 \cdot 40^2 \cdot 600 \).

Step 5: Calculating the New Lift

Compute the values:\[ L = 0.00136 \cdot 1600 \cdot 600 \]\[ L = 0.00136 \cdot 960,000 \]\[ L = 1305.6 \]

Step 6: Conclusion

The lift on a plane with a wing area of 600 ft² traveling at 40 mph is approximately 1305.6 lb.

Key Concepts

aerodynamic lift

In the world of aviation, the concept of aerodynamic lift is fundamental to understanding how airplanes manage to glide through the sky. Aerodynamic lift refers to the upward force that allows a plane to take off, remain airborne, and facilitate smooth turns during flight. This force opposes the weight of the airplane and is crucial for flight stability and control.
\
Lift is generated by the airplane's wings and is influenced by several factors, including the wing's shape, tilt, and motion through the air. The mathematics behind lift can be expressed by the equation \[L = k \cdot s^2 \cdot A\]where \(L\) is the lift, \( s \) is the speed of the airplane, \( A \) is the wing area, and \( k \) is a constant of proportionality. Understanding this equation helps in optimizing the plane's design and performance, ensuring that the right amount of lift is generated during flight.

proportionality constant

The proportionality constant, often denoted as \( k \), is an important factor in understanding how different variables interact in a joint variation scenario like that of the aerodynamic lift. In the lift equation \[L = k \cdot s^2 \cdot A,\]\( k \) represents the effect of all other physical factors that are not explicitly mentioned, such as air density and wing efficiency.
\
To calculate \( k \), we must first have a set of known values for lift, speed, and wing area. By substituting these into our formula, we can solve for \( k \), allowing us to predict lift under different conditions confidently.
\
For example, by knowing that a plane with 500 ft² of wing area traveling at 50 mph experiences a lift of 1700 lb, we substituted these values to find \( k = 0.00136 \). This constant is essential for calculating and understanding lift for other scenarios, showing how proportionality constants are crucial for scientific and engineering calculations.

speed and wing area relationship

In aviation, the relationship between speed, wing area, and lift is intricate and crucial for ensuring safe and efficient flights. This relationship can be seen in how the equation of lift makes use of the square of the speed \((s^2)\) and wing area \((A)\) to determine lift:\[L = k \cdot s^2 \cdot A\]

• Speed: The speed \((s)\) of an aircraft greatly influences lift, as increasing speed results in exponentially more lift. This is because lift is proportional to the square of the speed, meaning even small increases in speed can significantly impact lift.
• Wing Area: The wing area \((A)\) also plays a critical role. Larger wing areas can generate more lift because they provide a greater surface for the air to flow over, thus enhancing the lifting force.

By analyzing how these two factors work together, aviation engineers can tweak design features and flight strategies to ensure planes fly safely and efficiently. An increase in either speed or wing area will directly increase the lift, permitting various adjustments depending on the flight conditions.