This question tests the quadratic relationship between velocity and dynamic pressure in Bernoulli's equation. Dynamic pressure is defined as ½ρv², which depends on velocity squared. If velocity increases by a factor of 4 (v₂ = 4v₁), then dynamic pressure increases by a factor of 4² = 16, since ½ρv₂² = ½ρ(4v₁)² = 16(½ρv₁²). Choice A incorrectly uses the velocity ratio directly without squaring, while choice B might result from confusing this with a volume relationship. This quadratic relationship is crucial for understanding why small velocity changes can cause large pressure changes in fluid systems, explaining why high-speed flows can generate significant forces.

This question tests understanding of the continuity equation for incompressible fluid flow. The continuity equation states that for steady flow of an incompressible fluid, the product of cross-sectional area and velocity must remain constant: A₁v₁ = A₂v₂. In this catheter scenario, we have A₁ = 4.0 mm², v₁ = 0.50 m/s, and A₂ = 1.0 mm². Solving for v₂: v₂ = (A₁/A₂)v₁ = (4.0/1.0) × 0.50 = 2.0 m/s. Choice A incorrectly divides velocities instead of multiplying, a common error when students confuse the inverse relationship between area and velocity. To verify continuity problems, always check that flow rate Q = Av remains constant at both points.

This question tests application of Bernoulli's principle to blood flow through a stenosis. Bernoulli's equation for horizontal flow states that P + ½ρv² = constant, meaning that as velocity increases, pressure must decrease to maintain energy conservation. Since the stenosis reduces area to half (A₂ = 0.50A₁), the continuity equation tells us velocity doubles (v₂ = 2v₁). According to Bernoulli's principle, this higher velocity at the stenosis corresponds to lower pressure. Choice C incorrectly assumes pressure remains constant in horizontal flow, confusing the fact that elevation change is zero with pressure change being zero. Remember: in constrictions, velocity increases and pressure decreases, which explains why stenoses can be dangerous in arteries.

This question tests continuity equation application in bifurcating vessels. The continuity equation requires that flow rate entering equals flow rate exiting: Q₀ = Q₁ + Q₂, or A₀v₀ = A₁v₁ + A₂v₂. Given A₀ = 6 mm², v₀ = 0.40 m/s, A₁ = 2 mm², A₂ = 4 mm², and v₁ = v₂ = v, we solve: 6(0.40) = 2v + 4v = 6v, giving v = 0.40 m/s. Choice C incorrectly assumes flow splits proportionally to area ratios without considering the constraint of equal velocities. When branches have equal velocities, the parent velocity equals the branch velocity only when total branch area equals parent area, which is true here (2 + 4 = 6 mm²).

This question tests conceptual understanding of Bernoulli's principle in respiratory flow. Bernoulli's equation shows that total mechanical energy per unit volume (P + ½ρv²) remains constant in ideal flow. When velocity increases from 2.0 m/s to 5.0 m/s in the narrowed region, the kinetic energy density (½ρv²) increases significantly. To maintain constant total energy, the static pressure P must decrease correspondingly. Choice A represents a common misconception that higher velocity means higher pressure, confusing dynamic pressure (½ρv²) with static pressure (P). Remember the trade-off: in steady horizontal flow, regions of high velocity have low static pressure, which explains phenomena like the Venturi effect.

This question tests calculation of flow velocity from volumetric flow rate using the continuity equation. The relationship between volumetric flow rate Q and average velocity v is Q = Av, where A is cross-sectional area. Given Q = 2.0 mL/s = 2.0 cm³/s and r = 0.50 mm = 0.050 cm, the area is A = πr² = π(0.050)² = 0.00785 cm². Therefore, v = Q/A = 2.0/0.00785 = 255 cm/s = 2.55 m/s ≈ 2.5 m/s. Choice B incorrectly uses diameter instead of radius or makes a unit conversion error, a common mistake when working with small medical tubing. Always verify units: when Q is in cm³/s and A in cm², velocity comes out in cm/s, which must be converted to m/s.

This question tests the inverse relationship between area and velocity in the continuity equation. For incompressible flow, A₁v₁ = A₂v₂, which means velocity is inversely proportional to area. When area increases by a factor of 3 (A₂ = 3A₁), velocity must decrease by the same factor: v₂ = (A₁/A₂)v₁ = (1/3)v₁. Choice A incorrectly assumes velocity increases with area, reversing the relationship—a common conceptual error. Choice D squares the area ratio, suggesting confusion with pressure relationships. Remember: in steady flow, wider pipes have slower flow, narrower pipes have faster flow, maintaining constant volumetric flow rate Q = Av.

This question tests the qualitative relationship between velocity and pressure in Bernoulli's equation. For horizontal flow of an ideal fluid, Bernoulli's equation states P + ½ρv² = constant. Since segment 2 has higher velocity (v₂ > v₁) due to its smaller diameter, the kinetic energy term ½ρv₂² is larger. To maintain constant total energy, the pressure P₂ must be lower than P₁, making P₂ - P₁ < 0. Choice A represents the common misconception that faster flow means higher pressure, confusing the colloquial use of "pressure" with static pressure in fluid dynamics. Choice D incorrectly suggests density changes, but the problem states blood is incompressible. This pressure drop in constrictions is clinically relevant for understanding stenosis effects.

This question tests Bernoulli's equation with gravitational potential energy changes. The complete Bernoulli equation is P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. Since the pipe has constant diameter, v₁ = v₂, so the kinetic energy terms cancel. This gives P₂ - P₁ = ρg(h₁ - h₂) = -ρgΔh = -1000(9.8)(0.50) = -4900 Pa = -4.9 kPa. The negative sign indicates pressure decreases with height, as expected since the fluid must overcome gravity. Choice C might result from using incorrect units or forgetting the density term. Remember: when fluid flows upward against gravity at constant velocity, pressure must decrease by ρgh to provide the driving force.

This question tests combined application of continuity and Bernoulli's equations to predict pressure changes. First, continuity gives us v₂: since A₂ = 0.5A₁, we have v₂ = 2v₁ = 0.60 m/s. Then, applying Bernoulli's equation for horizontal flow: P₁ + ½ρv₁² = P₂ + ½ρv₂². Rearranging: P₂ - P₁ = ½ρ(v₁² - v₂²) = ½(1060)(0.30² - 0.60²) = ½(1060)(-0.27) = -143 Pa. Since this is negative, P₂ < P₁, making choice C correct. Choice A incorrectly suggests pressure must rise to accelerate fluid, misunderstanding that it's the pressure gradient that causes acceleration, not the reverse. When solving Bernoulli problems, always calculate the kinetic energy change first, then determine the corresponding pressure change.