Worksheet: 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 m2 to 0.025 m2. 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/m3. What is the ratio of the density of the gas where it enters the pipe to its density where it exits the pipe?

  • A3.8
  • B3.3
  • C0.37
  • D2.7
  • E0.3

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 2 . 4 × 1 0 m2. The hose ejects 1.6 kg of water that has a density of 1,025 kg/m3 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 m2 to having a cross-sectional area of 0.15 m2. 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/m3. 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?

  • A 0.50 m2
  • B 0.27 m2
  • C 0.73 m2
  • D 0.45 m2
  • E 0.25 m2

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

  • A 0.69 m/s
  • B 1.3 m/s
  • C 0.75 m/s
  • D 0 m/s
  • E 0.44 m/s

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

  • A 0.75 m/s
  • B 0.69 m/s
  • C 1.3 m/s
  • D 0.50 m/s
  • E 0.19 m/s

Q6:

45 kg of a liquid with a constant density of 1,055 kg/m3 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 2 . 1 2 × 1 0 m2 and leaves the pipe at a point where its cross-sectional area is 1 . 4 1 × 1 0 m2. 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 1 . 2 4 × 1 0 m2 and leaves the pipe at a point where its cross-sectional area is 6 . 3 5 × 1 0 m2. 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 𝑣 = 3 . 1 5 m/s flows smoothly through a cylindrical pipe of radius 𝑟 = 1 . 2 5 m and then flows smoothly through a second, connected cylindrical pipe of radius 𝑟 = 0 . 9 5 1 m, as shown in the diagram. The first pipe has a length 𝐿 = 1 . 4 4 m and the second pipe has a length 𝐿 = 1 . 2 1 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?

  • A1.45
  • B0.486
  • C0.689
  • D1.19
  • E2.06

Q10:

A smoothly flowing liquid flows through a pipe of cross-sectional area 𝐴 at a speed 𝑣 . The liquid then passes into a pipe of cross-sectional area 𝐴 , smoothly flowing at a speed 𝑣 . 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 𝐴 𝐴
  • B ( 𝐴 𝐴 ) ( 𝐴 𝐴 )
  • C 𝐴 𝐴
  • D 𝐴 + 𝐴
  • E ( 𝐴 + 𝐴 ) 2

Q11:

Water with a speed 𝑣 = 1 . 2 5 m/s flows smoothly through a cylindrical pipe of radius 𝑟 = 0 . 3 2 5 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 𝑟 = 0 . 1 1 8 m and length 𝐿 = 0 . 9 7 5 m.

What is the length of the first pipe?

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

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