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How do conservation of mass and energy govern a flowing fluid, and why does a fluid speed up where a pipe narrows?

Topic 8.4 Fluids and Conservation Laws: apply the continuity equation and Bernoulli's equation to ideal fluid flow.

A focused answer to AP Physics 1 Topic 8.4, covering the continuity equation A1 v1 = A2 v2 from conservation of mass, Bernoulli's equation from conservation of energy, the inverse speed-area and pressure-speed relationships, and applications to flowing fluids, with full worked examples.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

For an ideal (incompressible) fluid, conservation of mass gives the continuity equation $A_1 v_1 = A_2 v_2$ : the same volume passes every cross-section each second, so the fluid speeds up where the pipe narrows and slows where it widens (speed is inversely proportional to area). Conservation of energy gives Bernoulli's equation, $P + \tfrac{1}{2}\rho v^2 + \rho g y = \text{constant}$ along a streamline. On a horizontal pipe this means the pressure is lower where the fluid moves faster: in the narrow, fast section the pressure drops. These two relations explain why a thumb on a hose makes the water shoot out faster, and why pressure falls in a constriction (the Venturi effect), and they connect fluid flow back to the conservation laws of mechanics.

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What this topic is asking
Conservation of mass: the continuity equation
Conservation of energy: Bernoulli's equation
Putting the two laws together
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What this topic is asking

The College Board (Topic 8.4) wants you to apply the two great conservation laws to ideal fluid flow: conservation of mass, expressed as the continuity equation $A_1 v_1 = A_2 v_2$ , and conservation of energy, expressed as Bernoulli's equation. Together they predict how the speed and pressure of a flowing fluid change as a pipe widens or narrows.

Conservation of mass: the continuity equation

The continuity equation is conservation of mass in disguise. Because the fluid is incompressible, what flows in must flow out, so the volume per second ( $Av$ , the volume flow rate) is the same everywhere along the pipe. This forces an inverse relationship between speed and area: where the pipe is narrow, the fluid must move faster to carry the same volume through the smaller opening. This is why a river speeds up through a narrow gorge and why pinching a hose makes the water jet out.

Conservation of energy: Bernoulli's equation

Bernoulli's equation is conservation of energy written for a unit volume of flowing fluid: it balances pressure energy, kinetic energy and gravitational potential energy, exactly the energy bookkeeping of Unit 3 adapted to fluids. The most-tested consequence is the speed-pressure trade-off on a horizontal pipe: combining it with continuity, the narrow section has the higher speed and therefore the lower pressure. This counterintuitive result, low pressure where the fluid rushes fastest, is the Venturi effect, and it underlies how aeroplane wings, carburettors and atomisers work.

Putting the two laws together

The power of this topic comes from using the two equations in sequence. Continuity first fixes the speeds from the geometry: a known speed in a wide section gives the speed in a narrow section through $A_1 v_1 = A_2 v_2$ . Bernoulli then converts those speeds into pressures: knowing the speeds and one pressure, you solve for the other. This two-step routine, mass conservation for speeds, then energy conservation for pressures, handles the standard exam problem of a pipe that changes cross-section. For a tank draining through a hole, the same energy reasoning gives Torricelli's result, that the efflux speed equals $\sqrt{2gh}$ , the speed an object would reach falling the same height, because the fluid's pressure energy at depth converts to kinetic energy at the opening. The strategic insight is that fluids are not a separate world: the continuity equation is conservation of mass, and Bernoulli's equation is conservation of energy, the same pillars that ran through Units 3 and 4. Recognizing which law to apply, mass for "how fast" and energy for "what pressure", is exactly what the conservation-law free-response question rewards, and it closes the fluids unit by uniting it with the rest of AP Physics 1.

Speed and pressure in a narrowing pipe

Water (density $1000$ kg/m cubed) flows in a horizontal pipe at $2.0$ m/s where the area is $0.030$ m squared and the pressure is $1.5 \times 10^5$ Pa. The pipe narrows to $0.010$ m squared. Find the speed and pressure in the narrow section.

step 1 Speed from continuity

$A_1 v_1 = A_2 v_2$ : $v_2 = \dfrac{A_1 v_1}{A_2} = \dfrac{(0.030)(2.0)}{0.010} = 6.0$ m/s. The fluid speeds up threefold as the area shrinks to one third.

step 2 Set up Bernoulli on a horizontal pipe

With $y$ constant, $P_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \tfrac{1}{2}\rho v_2^2$ .

step 3 Solve for the narrow-section pressure

$P_2 = P_1 + \tfrac{1}{2}\rho(v_1^2 - v_2^2) = 1.5 \times 10^5 + \tfrac{1}{2}(1000)(2.0^2 - 6.0^2) = 1.5 \times 10^5 + \tfrac{1}{2}(1000)(4 - 36)$ .

step 4 Evaluate and interpret

$P_2 = 1.5 \times 10^5 + \tfrac{1}{2}(1000)(-32) = 1.5 \times 10^5 - 1.6 \times 10^4 = 1.34 \times 10^5$ Pa. The pressure dropped in the narrow section, even though the fluid sped up, exactly as Bernoulli predicts: fast flow means low pressure.

Try this

Q1. Water flows at $3.0$ m/s through a pipe of area $0.040$ m squared into a section of area $0.010$ m squared. Calculate the new speed. [2 points]

Cue. $A_1 v_1 = A_2 v_2$ : $v_2 = (0.040)(3.0)/0.010 = 12$ m/s.

Q2. On a horizontal pipe, state how the pressure changes where the fluid flows faster. [1 point]

Cue. The pressure decreases, by Bernoulli's equation.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2024 (style)7 marksSection II (long FRQ). Water (density

1000

kg/m cubed) flows through a horizontal pipe that narrows from cross-sectional area

0.020

m squared to

0.0050

m squared. In the wide section the speed is

1.5

m/s and the pressure is

1.8 \times 10^5

Pa. (a) Calculate the speed of the water in the narrow section. (b) State which conservation law you used and why. (c) Calculate the pressure in the narrow section using Bernoulli's equation.

Show worked answer →

A 7-point FRQ on continuity and Bernoulli's equation.

(a) Speed in narrow section (2 points): continuity gives $A_1 v_1 = A_2 v_2$ , so $v_2 = \dfrac{A_1 v_1}{A_2} = \dfrac{(0.020)(1.5)}{0.0050} = 6.0$ m/s.
(b) Law (2 points): conservation of mass, expressed as the continuity equation, because an incompressible fluid carries the same volume per second through every cross-section; a smaller area requires a higher speed.
(c) Pressure (3 points): for a horizontal pipe, Bernoulli gives $P_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \tfrac{1}{2}\rho v_2^2$ . So $P_2 = P_1 + \tfrac{1}{2}\rho(v_1^2 - v_2^2) = 1.8 \times 10^5 + \tfrac{1}{2}(1000)(1.5^2 - 6.0^2) = 1.8 \times 10^5 + \tfrac{1}{2}(1000)(2.25 - 36) = 1.8 \times 10^5 - 1.69 \times 10^4 = 1.63 \times 10^5$ Pa.

Markers reward continuity for the speed, naming conservation of mass, and applying Bernoulli's equation to find the lower pressure in the faster, narrower section.

AP 2023 (style)1 marksSection I (multiple choice). An ideal fluid flows through a horizontal pipe that narrows. As the fluid moves into the narrower section, what happens to its speed and pressure? (A) both increase (B) both decrease (C) speed increases, pressure decreases (D) speed decreases, pressure increases. Justify your reasoning.

Show worked answer →

A 1-point MCQ on the continuity and Bernoulli relations. The answer is (C).

Continuity ( $A_1 v_1 = A_2 v_2$ ) requires the speed to increase where the area shrinks. Bernoulli's equation then requires the pressure to fall where the speed rises (on a horizontal pipe), since the energy per unit volume is conserved. The trap is (A): pressure drops, not rises, in the fast-moving narrow section.

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Sources & how we know this

AP Physics 1: Algebra-Based Course and Exam Description — College Board (2024)