# AP Physics 2 : Flow Rate

## Example Questions

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### Example Question #1 : Flow Rate

Water flows through a tube with a diameter of 2m at a rate of . What is the velocity of the water?

Explanation:

The velocity of the water can be determined from the following formula:

We need to calculate the volumetric flow rate and the cross-sectional area. For the flow rate:

Rearrange to solve for volumetric flow rate:

Next, calculate cross-sectional area:

Now we can solve for the velocity:

### Example Question #2 : Flow Rate

Suppose that water flows from a pipe with a diameter of 1m into another pipe of diameter 0.5m. If the speed of water in the first pipe is , what is the speed in the second pipe?

Explanation:

To find the answer to this question, we'll need to use the continuity equation to determine the flow rate, which will be the same in both pipes.

We'll also need to calculate the area of the pipe using the equation:

Solve the combined equation for  and plug in known values to find the velocity of the water through the second pipe.

### Example Question #3 : Flow Rate

A diameter garden hose with a diameter of 3cm sprays water travels through a hose at . At the end of the garden hose, the diameter reduces to 2cm. What is the speed of the water coming out at the end?

Explanation:

Use the continuity equation for incompressible fluids.

The cross sectional area of the garden hose at both ends are circular regions. Rewrite the equation replacing areas with the formula for an area of a circle and solve for the velocity at the second point.

### Example Question #4 : Flow Rate

An civil engineer is designing the outflow of a pond. The pond has a radius of , and the maximum sustained rainfall rate is , about 3 inches per hour. If the engineer makes the outflow with a cross-sectional area of , what maximum velocity will the outflow of water have during a heavy rainstorm if the surface level of the pond does not change?

Explanation:

This is a volume flow rate problem. Because the water is an incompressible fluid, we can apply the flow rate equation:

Find the surface area of the pond:

Substitute into the flow rate equation:

### Example Question #5 : Flow Rate

An incompressible fluid flows through a pipe. At location 1 along the pipe, the volume flow rate is . At location 2 along the pipe, the area halves. What is the volume flow rate at location 2?

Explanation:

When the area halves, the velocity of the fluid will double. However, the volume flow rate (the product of these two quantities) will remain the same. In other words, the volume of water flowing through location 1 per second is the same as the volume of water flowing through location 2 per second.

### Example Question #6 : Flow Rate

What is the volumetric flow rate of oil in a  diameter pipe?  The velocity of the oil is .

Explanation:

The volumetric flow rate of fluid is found using the equation:

Where  is the velocity of the fluid and  is the cross-sectional area of the space through which the fluid is flowing.

In this problem the cross-section of the pipe is a circle. The area of the cross-section is:

The volumetric flow rate is:

### Example Question #7 : Flow Rate

What is the volumetric flow rate of ethanol flowing through a square pipe with sidelength 4m? The velocity of the ethanol is .

Explanation:

The volumetric flow rate of fluid is found using the equation:

Where  is the velocity of the fluid and  is the cross-sectional area of the space through which the fluid is flowing.

In this problem the cross-section of the pipe is a square. The area of the cross-section is:

The volumetric flow rate is:

### Example Question #8 : Flow Rate

A pipe narrows from a  diameter to a  diameter. What is the velocity of the fluid when it exits the pipe (at the  end) if it entered the pipe at ?

Explanation:

The volumetric flow rate of fluid is found using the equation:

Where  is the velocity of the fluid and  is the cross-sectional area of the space through which the fluid is flowing. Use the continuity equation, we see that , therefore

In this problem, the cross-section of the pipe is a circle, which is

The area of the exit cross-section is:

Plug in these variables into the continuity equation and solve:

### Example Question #9 : Flow Rate

Fluid enters a pipe at  and exits the pipe at . What is the diameter of the pipe exit if the entrance has a  diameter?

Explanation:

The volumetric flow rate of fluid is found using the equation:

Where  is the velocity of the fluid and  is the cross-sectional area of the space through which the fluid is flowing.

Use the continuity equation, we see that , therefore:

In this problem,

The cross-section of the pipe is a circle, which is:

Plugging our variables into the continuity equation gives us

### Example Question #10 : Flow Rate

Fluid flows through a pipe whose diameter goes from  to . How are flow in the two parts of the pipe related?

The fluid in both sections of the pipe have the same velocity

The fluid in both sections of the pipe have the same cross-sectional area

The fluid in both sections of the pipe are not related

The fluid in both sections of the pipe have the same volumetric flow rate