Question

An airplane has a mass of 2×10^6 kg and air flows past the power surface of the wings at 100ms¯¹. If the wings have a surface area of 1200m², how fast must the air flow over the upper surface of the wing if the plane is to stay in the air? Consider only the Bernoulli's effect. ​

Answer 1

To keep the plane in the air, the air flowing over the upper surface of the wings must be faster than the air flowing past the lower surface. This is due to Bernoulli's principle, which states that as the speed of a fluid increases, its pressure decreases. We can use the equation v₂ = sqrt(v₁² + 2(P₁ - P₂)/ρ) to calculate the speed of the air over the upper surface of the wing.

Step-by-step explanation:

To keep the plane in the air, the air flowing over the upper surface of the wings must be faster than the air flowing past the lower surface. This is due to Bernoulli's principle, which states that as the speed of a fluid increases, its pressure decreases. The difference in pressure between the upper and lower surfaces of the wing creates lift.

To calculate the speed of the air over the upper surface of the wing, we can use the equation:

P₁ + ½ρv₁² = P₂ + ½ρv₂²

P₁ is the pressure below the wing, P₂ is the pressure above the wing, ρ is the density of the air, v₁ is the speed of the air below the wing, and v₂ is the speed of the air above the wing.

We can rearrange the equation to solve for v₂:

v₂ = sqrt(v₁² + 2(P₁ - P₂)/ρ)

Plugging in the given values, we get:

v₂ = sqrt(100² + 2(0 - P₂)/(1.2))

Since we don't have the specific values for P₁ and P₂, we cannot calculate the exact speed of the air over the upper surface of the wing. However, we can determine that it must be greater than 100 m/s in order for the plane to stay in the air.

Learn more about Bernoulli's principle

Answer 2

190 m/s

Step-by-step explanation:

For the plane to stay in the air, the lift force must equal the weight.

The lift force is also equal to the pressure difference across the wings (pressure at the bottom minus pressure at the top) times the area of the wings.

Therefore:

mg = (P₂ − P₁) A

P₂ − P₁ = mg / A

Using Bernoulli equation:

P₁ + ½ ρ v₁² + ρgh₁ = P₂ + ½ ρ v₂² + ρgh₂

P₁ + ½ ρ v₁² = P₂ + ½ ρ v₂²

½ ρ (v₁² − v₂²) = P₂ − P₁

½ ρ (v₁² − v₂²) = mg / A

v₁² − v₂² = 2mg / (Aρ)

v₁² = v₂² + 2mg / (Aρ)

Substituting values (assuming air density of 1.225 kg/m³):

v₁² = (100 m/s)² + 2 (2×10⁶ kg) (9.8 m/s²) / (1200 m² × 1.225 kg/m³)

v₁² = 36,666.67 m²/s²

v₁ = 191 m/s

Rounding to two significant figures, the air must move at 190 m/s over the top of the wing.