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AP Physics 2 : Flow Rate

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Example Questions

Example Question #1 : Flow Rate

Water flows through a tube with a diameter of 2m at a rate of $800 \frac{kg}{s}$ . What is the velocity of the water?

$\rho_{water} = 1\frac{kg}{l}$

Possible Answers:

$0.25\frac{m}{s}$

$0.75\frac{m}{s}$

$0.62\frac{m}{s}$

$1.0\frac{m}{s}$

$0.5\frac{m}{s}$

Correct answer:

$0.25\frac{m}{s}$

Explanation:

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

$v = \frac{\dot{V}}{A}$

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

$\dot m = \dot V \cdot \rho_{water}$

Rearrange to solve for volumetric flow rate:

$\dot V = \frac{\dot m}{\rho_{water}} = \frac{800\frac{kg}{s}}{1000\frac{kg}{m^3}} = 0.8 \frac{m^3}{s}$

Next, calculate cross-sectional area:

$A = \pi \frac{d^2}{4} = \pi \frac{(2m)^2}{4} = \pi m^2$

Now we can solve for the velocity:

$v = \frac{0.8\frac{m^3}{s}}{\pi m^2} = 0.25 \frac{m}{s}$

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Example Question #1 : 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 $5\: \frac{m}{s}$ , what is the speed in the second pipe?

Possible Answers:

$5 \frac{m}{s}$

$10 \frac{m}{s}$

$2.5 \frac{m}{s}$

$20 \frac{m}{s}$

Correct answer:

$20 \frac{m}{s}$

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.

vA=f

$v_{1}A_{1}=v_{2}A_{2}$

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

$A=\pi r^{2}$

$A_{1}=\pi r_{1}^{2}=\pi (0.5\m)^{2}=0.25\pi m^{2}$

$A_{2}=\pi r_{2}^{2}=\pi \left ( 0.25 m\right )^{2}=0.0625\pi m^{2}$

Solve the combined equation for v_2 and plug in known values to find the velocity of the water through the second pipe.

$v_{2}=\frac{v_{1}A_{1}}{A_{2}}$

$v_{2}=\frac{(5 \frac{m}{s})(0.25\pi m^{2})}{0.0625\pi m^{2}}=20 \frac{m}{s}$

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Example Question #2 : Flow Rate

A diameter garden hose with a diameter of 3cm sprays water travels through a hose at $1\frac{m}{s}$ . At the end of the garden hose, the diameter reduces to 2cm. What is the speed of the water coming out at the end?

Possible Answers:

$2 \frac{m}{s}$

$1.5 \frac{m}{s}$

$2.25 \frac{m}{s}$

$2.75 \frac{m}{s}$

$0.67 \frac{m}{s}$

Correct answer:

$2.25 \frac{m}{s}$

Explanation:

Use the continuity equation for incompressible fluids.

A_1v_1 =A_2v_2

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.

$\left(\frac{\pi d_1^2}{4}\right)v_1 =\left(\frac{\pi d_2^2}{4}\right)v_2$

$v_2 = \frac{v_1 d_1^2}{d_2^2}= \frac{1\frac{{m}}{{s}}(3^2)}{2^2}= \left(\frac{3}{2}\right)^2=2.25 \frac{{m}}{{s}}$

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Example Question #1 : Flow Rate

An civil engineer is designing the outflow of a pond. The pond has a radius of r=200m , and the maximum sustained rainfall rate is $v=2.1*10^{-5}\frac{m}{s}$ , about 3 inches per hour. If the engineer makes the outflow with a cross-sectional area of A=0.5 m^2 , what maximum velocity will the outflow of water have during a heavy rainstorm if the surface level of the pond does not change?

Possible Answers:

$v=4.2*10^{-3}\frac{m}{s}$

$v=8.4*10^{-3}\frac{m}{s}$

$v=5.56\frac{m}{s}$

$v=8.8\frac{m}{s}$

$v=16\frac{m}{s}$

Correct answer:

$v=5.56\frac{m}{s}$

Explanation:

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

$A_{1}* v_{1}=A_{2}* v_{2}$

Find the surface area of the pond:

$A_{1}=\pi r^2 = 3.14* 200^2 =1.3* 10^5m^2$

Substitute into the flow rate equation:

$1.3*10^5 * 2.1*10^{-5}=0.5* v_2$

$v_2=5.56\frac{m}{s}$

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Example Question #1 : Flow Rate

An incompressible fluid flows through a pipe. At location 1 along the pipe, the volume flow rate is $10 \frac{m^3}{s}$ . At location 2 along the pipe, the area halves. What is the volume flow rate at location 2?

Possible Answers:

$4 \frac{m^3}{s}$

$5 \frac{m^3}{s}$

$20 \frac{m^3}{s}$

$10 \frac{m^3}{s}$

Correct answer:

$10 \frac{m^3}{s}$

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.

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Example Question #6 : Flow Rate

What is the volumetric flow rate of oil in a 12m diameter pipe? The velocity of the oil is $5\frac{m}{s}$ .

Possible Answers:

$Q=60\frac{m^3}{s}$

$Q=124.7\frac{m^3}{s}$

$Q=226.5\frac{m^3}{s}$

$Q=565.2\frac{m^3}{s}$

Correct answer:

$Q=565.2\frac{m^3}{s}$

Explanation:

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

Q=v*a

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:

$a=\pi*r^{2}=\pi*\left ( \frac{12}{2} \right )^{2}\approx113.04m^2$

The volumetric flow rate is:

$Q=v*a=5*113.04=565.2\frac{m^3}{s}$

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Example Question #13 : Fluid Dynamics

What is the volumetric flow rate of ethanol flowing through a square pipe with sidelength 4m? The velocity of the ethanol is $7\frac{m}{s}$ .

Possible Answers:

$Q=112\frac{m^3}{s}$

$Q=213\frac{m^3}{s}$

$Q=68\frac{m^3}{s}$

$Q=28\frac{m^3}{s}$

Correct answer:

$Q=112\frac{m^3}{s}$

Explanation:

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

Q=v*a

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:

a=4*4=16m^2

The volumetric flow rate is:

$Q=v*a=7*16=112\frac{m^3}{s}$

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Example Question #3 : Flow Rate

A pipe narrows from a 30m diameter to a 15m diameter. What is the velocity of the fluid when it exits the pipe (at the 15m end) if it entered the pipe at $15\frac{m}{s}$ ?

Possible Answers:

$v_{out}=59.97\frac{m}{s}$

$v_{out}=15\frac{m}{s}$

$v_{out}=100\frac{m}{s}$

$v_{out}=30.15\frac{m}{s}$

Correct answer:

$v_{out}=59.97\frac{m}{s}$

Explanation:

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

Q=v*a

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 $Q_{in}=Q_{out}$ , therefore

$v_{in}*a_{in}=v_{out}*a_{out}$

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

$a_{in}=\pi*r^{2}=\pi*\left ( \frac{30}{2} \right )^{2}\approx706.5m^2$

The area of the exit cross-section is:

$a_{out}=\pi*r^{2}=\pi*\left ( \frac{15}{2} \right )^{2}\approx176.71m^2$

Plug in these variables into the continuity equation and solve:

$15*706.5=v_{out}*176.71$

$v_{out}=59.97\frac{m}{s}$

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Example Question #4 : Flow Rate

Fluid enters a pipe at $24\frac{m}{s}$ and exits the pipe at $48\frac{m}{s}$ . What is the diameter of the pipe exit if the entrance has a 26m diameter?

Possible Answers:

$d_{out}=20.11m$

$d_{out}=9.19m$

$d_{out}=18.38m$

$d_{out}=10.55m$

Correct answer:

$d_{out}=18.38m$

Explanation:

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

Q=v*a

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 $Q_{in}=Q_{out}$ , therefore:

$v_{in}*a_{in}=v_{out}*a_{out}$

In this problem,

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

$a_{in}=\pi*r^{2}=\pi*\left ( \frac{26}{2} \right )^{2}\approx530.93m^2$

Plugging our variables into the continuity equation gives us

$24*530.93=48*a_{out}$

$a_{out}=265.46m^2$

$r_{out}=\sqrt{\frac{a}{\pi}}=\sqrt{\frac{265.46}{\pi}}=9.19m$

$d_{out}=2*r_{out}=2*9.19=18.38m$

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Example Question #1 : Flow Rate

Fluid flows through a pipe whose diameter goes from 20m to 12m . How are flow in the two parts of the pipe related?

Possible Answers:

The fluid in both sections of the pipe are not related

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

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

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

Correct answer:

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

Explanation:

Based on the continuity equation, both parts of the flow must have the same volumetric flow rate.

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AP Physics 2 : Flow Rate

Study concepts, example questions & explanations for AP Physics 2

All AP Physics 2 Resources

Example Questions

Example Question #1 : Flow Rate

Example Question #1 : Flow Rate

Example Question #2 : Flow Rate

Example Question #1 : Flow Rate

Example Question #1 : Flow Rate

Example Question #6 : Flow Rate

Example Question #13 : Fluid Dynamics

Example Question #3 : Flow Rate

Example Question #4 : Flow Rate

Example Question #1 : Flow Rate

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