Ten sense-making tasks on continuity and Bernoulli's equation. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Water flows steadily from left to right through a horizontal pipe that completely fills with water. The cross-sectional area of each section is shown. Ignore viscosity.
If any are equal, show the tie. If a ranking cannot be determined, say so. Explain your reasoning for both rankings.
Back to topWater flows steadily through a horizontal pipe that narrows and then widens again. Three students discuss what happens in the narrow section.
Ava: "The water gets squeezed in the narrow part, so the pressure there is higher. That's why water sprays harder when you put your thumb over a hose."
Ben: "The water speeds up in the narrow part, and the pressure there is lower than in the wide parts."
Cam: "It's the same water flowing through the whole pipe, so its speed can't change. Only the pressure changes."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement.
Back to topWater flows steadily to the right through a horizontal pipe. Three thin vertical tubes, open to the air at the top, are attached to the pipe. The height of the water in each tube shows the pressure in the pipe below it. A student predicts the water heights shown:
Student's explanation: "The water is forced into a smaller space in the middle, so the pressure there is highest and the water rises highest in tube 2."
What, if anything, is wrong with the student's prediction? If something is wrong, explain the error and describe the correct water heights. If nothing is wrong, explain why it is correct.
Back to topWater flows steadily through a pipe that is always completely full. Ignore viscosity. For each situation, choose what happens to the quantity listed as the water moves from the first section to the second:
A. increases B. decreases C. stays the same D. cannot be determined
| Situation | Choice |
|---|---|
| (a) Horizontal pipe, wide to narrow. Speed of the water: | |
| (b) Horizontal pipe, wide to narrow. Volume flow rate (m³/s): | |
| (c) Horizontal pipe, wide to narrow. Pressure: | |
| (d) Horizontal pipe, wide to narrow. Mass of water passing each second: | |
| (e) Same-width pipe that goes uphill. Speed: | |
| (f) Same-width pipe that goes uphill. Pressure: | |
| (g) Pipe that goes uphill and gets wider. Pressure: |
Explain the rule you used. For (g), what extra information would you need?
Back to topA large open tank sits on the floor, filled with water to a height of 1.0 m. Two small holes of equal area, $1.0 \times 10^{-4}\ \text{m}^2$, are opened in its side. Hole A is 0.20 m below the water's surface; hole B is 0.80 m below the surface. The tank is large enough that the water level drops very slowly.
For each quantity, state whether it is greater for A, greater for B, or the same for both. If it's different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Speed of the water as it leaves the hole | |
| (b) Volume flow rate out of the hole | |
| (c) Time for the water to fall to the floor | |
| (d) Horizontal distance from the tank wall to where the stream hits the floor |
Explain each answer. Predict (d) before you calculate, then check it with numbers.
Back to topWater flows smoothly out of a faucet. Right at the faucet, the stream has a cross-sectional area of $1.0 \times 10^{-4}\ \text{m}^2$ and the water moves downward at 1.0 m/s. Ignore air resistance.
The equation below is Bernoulli's equation applied to water flowing through a pipe in some physical situation:
$$2.5 \times 10^5\ \text{Pa} + \tfrac{1}{2}(1000\ \text{kg/m}^3)(2.0\ \text{m/s})^2 = P_2 + \tfrac{1}{2}(1000\ \text{kg/m}^3)(4.0\ \text{m/s})^2 + (1000\ \text{kg/m}^3)(10\ \text{m/s}^2)(5.0\ \text{m})$$
Water flows steadily to the right through the horizontal pipe shown. The wide sections have area $0.020\ \text{m}^2$ and the narrow section has area $0.010\ \text{m}^2$. In the wide section on the left, the water moves at 2.0 m/s and the pressure is $1.60 \times 10^5\ \text{Pa}$.
Water flows at 1.0 m/s through a horizontal pipe with a diameter of 4.0 cm, where the pressure is $2.0 \times 10^5\ \text{Pa}$. The pipe narrows to a diameter of 2.0 cm. Find the speed and pressure in the narrow section.
Student solution:
"The diameter is cut in half, so the speed doubles."
$v_2 = 2(1.0\ \text{m/s}) = 2.0\ \text{m/s}$
$P_2 = P_1 + \tfrac{1}{2}\rho(v_1^2 - v_2^2) = 2.0 \times 10^5\ \text{Pa} + \tfrac{1}{2}(1000\ \text{kg/m}^3)\big[(1.0\ \text{m/s})^2 - (2.0\ \text{m/s})^2\big] = 1.985 \times 10^5\ \text{Pa}$
Identify the error in the solution. Explain the physics of why it's an error, then find the correct speed and pressure.
Back to topA large open tank holds water 1.8 m deep. A valve at the bottom of the tank is closed.
Jamie writes: "Bernoulli: $\rho g h = \tfrac{1}{2}\rho v^2$, so $v = \sqrt{2gh} = \sqrt{2(10\ \text{m/s}^2)(1.8\ \text{m})} = 6.0\ \text{m/s}$. The water at the bottom of the tank is moving at 6.0 m/s."