# Worksheet: The Continuity Equation for Fluids

In this worksheet, we will practice calculating the rate of transfer of smoothly flowing fluids through channels with varying cross-sections.

**Q1: **

A gas flows smoothly through a pipe. The pipe’s cross-sectional area
contracts from 0.075 m^{2} to
0.025 m^{2}.
The gas enters the pipe moving at
1.8 m/s and
leaves the pipe
moving at 2.0 m/s.
The density of the gas as it enters the pipe is
1.4 kg/m^{3}.
What is the ratio of the density of the gas where it enters the pipe to its density
where
it exits the pipe?

**Q2: **

Water flows smoothly through a pipe at a speed of 1.66 m/s. The water then flows smoothly from the first pipe into and through a second pipe, at a speed of 1.22 m/s. What is the ratio of the cross-sectional area of the first pipe to that of the second pipe?

**Q3: **

A fire hose has a length of
15 m and a cross-sectional area
of
m^{2}.
The hose ejects 1.6 kg of water that
has a density of
1,025 kg/m^{3} per second. What is the time interval
between the water entering one end of the hose and leaving the opposite end?

**Q4: **

A gas flows smoothly through a pipe that expands from having a
cross-sectional area of 0.10 m^{2} to having a cross-sectional area of
0.15 m^{2}.
The gas enters the pipe moving at 1.4 m/s and leaves it moving at
1.2 m/s. The density of the gas as it enters the pipe is
0.98 kg/m^{3}. How
much does the gas decrease in density between entering the pipe and exiting it?

**Q5: **

Water flows smoothly into and through a primary pipe that divides into two secondary pipes. The secondary pipes change in thickness along their lengths, as shown in the diagram. The cross-sectional area of the pipe where it divides is identical to its cross-sectional area where water flows into it. Water flows at 0.25 m/s out of the secondary pipe that has the greater cross-sectional exit area. Water flows at 1.0 m/s out of the secondary pipe that has the smaller cross-sectional exit area.

What is the difference in the cross-sectional areas of the secondary pipes where water enters them?

- A0.27 m
^{2} - B0.50 m
^{2} - C0.25 m
^{2} - D0.73 m
^{2} - E0.45 m
^{2}

What is the difference in the speed of water flow where water enters the two secondary pipes?

- A0.44 m/s
- B0 m/s
- C0.69 m/s
- D1.3 m/s
- E0.75 m/s

What is the speed of water flow where water enters the primary pipe?

- A0.69 m/s
- B0.19 m/s
- C0.75 m/s
- D1.3 m/s
- E0.50 m/s

**Q6: **

45 kg
of a liquid with a constant density of 1,055 kg/m^{3}
flows smoothly through a 2.5 m
long pipe each second. What is the cross-sectional
area of the pipe?

**Q7: **

Water flows smoothly through a pipe. The water enters the
pipe at a point where its cross-sectional area is
m^{2}
and leaves the pipe at a point where its cross-sectional area is
m^{2}.
What is the ratio of the water’s speed where it enters the
pipe to its speed where it exits the pipe?

**Q8: **

Water flows smoothly through a pipe that changes
thickness along its length. The water enters the pipe at a point where its cross-sectional
area is m^{2} and leaves the pipe at a point
where its cross-sectional area is m^{2}. The
water increases in speed by 2.50 m/s between entering and exiting the pipe.
At what speed does the water exit the pipe?

**Q9: **

Water with a speed m/s flows smoothly through a cylindrical pipe of radius m and then flows smoothly through a second, connected cylindrical pipe of radius m, as shown in the diagram. The first pipe has a length m and the second pipe has a length m. What is the ratio of the time the water takes to pass through the first pipe to the time it takes to pass through the second pipe?

**Q10: **

A smoothly flowing liquid flows through a pipe of cross-sectional area at a speed . The liquid then passes into a pipe of cross-sectional area , smoothly flowing at a speed . Which of the following formulas correctly represents the ratio of the liquid’s speed in the first pipe to its speed in the second pipe?

- A
- B
- C
- D
- E

**Q11: **

Water with a speed m/s flows smoothly through a cylindrical pipe of radius m, as shown in the diagram. The water passes through the pipe in 0.955 s before smoothly flowing through a second cylindrical pipe of radius m and length m.

What is the length of the first pipe?

What is the time interval between the water entering and leaving the second pipe?