The Question is from neet-physics mechanical-properties-of-fluids , and it was created by the author NAGA SAI

Question

A bird with a mass of 0.5 kg is soaring at a constant speed.  Its wings have a combined surface area of 0.1 m².  Assuming the air density is 1.2 kg/m³ and the pressure difference between the top and bottom surfaces of the wings is 40 Pa, what is the average downward velocity of air deflected by the bird's wings? (Assume all lift is due to downward air deflection).

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Newton's Laws of Motion (specifically 3rd Law and 2nd Law) and Fluid Dynamics (pressure and force)
	**Related Topics**: Lift, Drag, Bernoulli's Principle, Momentum
	**Prerequisites**: Understanding of force, pressure, momentum, and Newton's laws.
## SOLUTION PATHWAY
**Given Information**:
	- Mass of bird ($m$): 0.5 kg
	- Wing surface area ($A$): 0.1 m²
	- Air density ($ρ$): 1.2 kg/m³
	- Pressure difference ($ΔP$): 40 Pa
	- Constant speed (implies net force on bird is zero)
**Method & Steps**:
	1. **Calculate the lift force:**
		Reasoning: The lift force must equal the bird's weight since it's soaring at a constant speed.
		Formula/Rule: Lift Force ($F_L$) = Weight = $mg$
		Working: $F_L = (0.5 	ext{ kg}) * (9.8 	ext{ m/s²}) = 4.9 	ext{ N}$
	2. **Relate lift force to the rate of change of momentum of the air:**
		Reasoning: The lift force is a consequence of the bird's wings deflecting air downwards. This downward deflection of air creates a change in momentum of the air, and by Newton's third law, an upward force on the wings.
		Formula/Rule: $F_L = ΔP * A = \dot{m}v$, where $\dot{m}$ is the mass flow rate of the deflected air and $v$ is the downward velocity of the air.
		Working: $4.9 	ext{ N} = 40 	ext{ Pa} * 0.1 	ext{ m²}$ (Verification of Lift Force calculation using pressure difference)
	3. **Calculate the mass flow rate:**
		Reasoning: Mass flow rate is the product of density, area, and velocity.
		Formula/Rule: $\dot{m} = ρAv$
	4. **Solve for the downward velocity ($v$):**
		Working: $F_L = ρAv²$, so $v = \sqrt{F_L / (ρA)} = \sqrt{(4.9 	ext{ N}) / (1.2 	ext{ kg/m³} * 0.1 	ext{ m²})} = \sqrt{40.83} ≈ 5.77 	ext{ m/s}$
		Result: $v ≈ 5.77$ m/s
		Verification: The calculated downward velocity of the air, when used in the momentum equation, produces a lift force equal to the bird's weight.
## OPTION ANALYSIS
[A] :x: Incorrect. This might result from an error in calculating the lift force or misapplying the momentum principle.
[B] ✓: Correct. This is the correct application of the lift force and momentum change relationship.
[C] :x: Incorrect. This could be due to mathematical errors or incorrect use of the formulas.
[D] :x: Incorrect.  Likely a result of a calculation error or misunderstanding of the relationship between force, momentum, and velocity.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrectly applying Newton's third law. Remember that the force on the bird is equal and opposite to the force on the air.
		2. Confusing pressure with force. Ensure proper use of the area in calculations.
	**Quick Tips**:
		1. Visualize the air deflection to understand the momentum change.
		2. Double-check units to ensure consistency throughout calculations.

NAGA SAI · Answer

2.89 m/s

NAGA SAI · Answer

The correct Option is "5.77 m/s"

NAGA SAI · Answer

8.17 m/s

NAGA SAI · Answer

11.09 m/s

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