Rate of flow, A * v, must remain constant. Use the continuity equation, .

Solving the initial cross-sectional area yields: . The initial radius is 5cm.

Then find the final area of the pipe: . The final radius is 3cm.

Using these values in the continuity equation allows us to solve the final velocity.

Just like the volume flow rate equation for fluids, the flow rate of blood through the body is equal to area times velocity.

The flow rate is a constant, so depending on the area that the blood is travelling through, the velocity is constantly changing; therefore the volume flow rate though the aorta is equal to the volume flow rate in the capillaries.

Because we can determine the area of the aorta and area of the capillaries, knowing the velocity through the aorta can give the velocity through the capillaries.

The equation for volumetric flow rate is , where is the cross-sectional area of the tube and is the flow velocity. We can re-write this equation to solve for the velocity and include the tube radius.

Halving the radius will reduce the tube area by a factor of four.

Volumetric flow rate is constant, thus, any reduction in area will cause a corresponding increase in velocity.

Halving the radius will thus quadruple the velocity, resulting in the greatest increase of the given options.

Using the continuity equation we know that . The question tells us that the cross-sectional area at point 2 is nine times greater that at point 1 ().

Using the continuity equation we can make A₁= 1 and A₂ = 9.

Flow speed at point 1 is nine times that at point 2.

This problem covers the concept of continuity. As the tube diameter changes, the volumetric flow of water stays constant. Therefore, we can calculate the volumetric flow at the diameter of 0.5m, and use that to find the velocity of water at 1m.

Here,  is the cross-sectional area of the pipe

Apply this flow rate to a tube diameter of 1m to find the velocity:

Alternatively, this question can be solved by setting up a proportion.

Rearranging for :

Plugging in our values:

The continuity equation states that:

In other words, the volumetric flow rate stays constant throughout a pipe of varying diameter. If the diameter decreases (constricts), then the velocity must increase.

To establish the change in cross-sectional area, we need to find the area in terms of the diameter:

If the diameter is halved, the area is quartered.

To keep the volumetric flow constant, the velocity would have to increase by a factor of 4.

Flow rate in a pipe must be constant in order to create linear flow. This flow rate is given by the product of the cross-sectional area and the velocity of the fluid.

The cross-sectional areas of the pipe and nozzle can be found using their radii. Note that you were given dimensions in terms of diameter, so be sure to divide by 2 to get the radius.

Use these areas and the initial velocity to calculate the final velocity in the nozzle.

Use Bernoulli's equation: 

Plug in our given values: 

Rearrange to isolate : 

Bernoulli's equation states that:

Essentially, this equation notes that pressure, velocity, and height of a fluid during flow can be related by a constant term. Very similarly, kinetic energy and potential energy sum to a constant mechanical energy for static compounds.

Neglecting the term for pressure in Bernoulli's equation, there are direct correlations between kinematic and gravitational terms. Each term has an equivalent in both equations.

Kinematic terms:

Gravitational terms:

In Bernoulli's equation, kinetic energy is the kinematic equivalent and potential energy is the gravitational equivalent.

Bernoulli’s principle is an equation that states the relationship between the pressure and velocity of a fluid flowing through a pipe.

It operates on three main assumptions. First, the fluid has to have laminar flow. There are two types of flow: laminar and turbulent. Laminar flow is characterized by uniform and ordered flow, whereas turbulent flow is characterized by haphazard and irregular flow. For the Bernoulli equation, the fluid must flow uniformly (laminar). Second, the fluid must be incompressible. This means that the fluid must not change volume (should not be compressed) when an outside pressure is applied. Third, the fluid must experience no friction during flow. Recall that during fluid flow friction usually occurs between layers of fluid because the molecules from each layer interact with each other; however, for the Bernoulli equation to be valid this friction between layers of fluid must be negligible (must be frictionless).