Topic 8.4: Fluids and Conservation Laws

What Topic 8.4 requires

Topic 8.4 carries two learning objectives, and the CED names the conservation law in each of them. That naming is the framing to steal, because most other explanations present continuity and Bernoulli as two formulas about pipes.

• 8.4.A asks you to describe the flow of an incompressible fluid through a cross-sectional area by using mass conservation. Under it, 8.4.A.1 states that a difference in pressure between two locations causes a fluid to flow, with two sub-statements: 8.4.A.1.i, the rate at which matter enters a fluid-filled tube open at both ends must equal the rate at which matter exits it, and 8.4.A.1.ii, the rate at which matter flows into a location is proportional to the cross-sectional area of the flow and the speed at which the fluid flows, with the derived equation $V/T = Av$. Then 8.4.A.2 states that the continuity equation for fluid flow describes conservation of mass flow rate in incompressible fluids, with the relevant equation $A_1 v_1 = A_2 v_2$.
• 8.4.B asks you to describe the flow of a fluid as a result of a difference in energy between two locations in a fluid within the fluid-Earth system. 8.4.B.1 states that a difference in gravitational potential energies between two locations in a fluid will result in a difference in kinetic energy and pressure between those two locations, described by conservation laws. 8.4.B.2 states that Bernoulli's equation describes the conservation of mechanical energy in fluid flow, and prints it. 8.4.B.3 states that Torricelli's theorem relates the speed of a fluid exiting an opening to the difference in height between the opening and the top surface of the fluid, and can be derived from conservation of energy principles, with the derived equation $v = \sqrt{2 g \Delta y}$.

Note the CED's own labels. $A_1 v_1 = A_2 v_2$ and Bernoulli's equation are marked relevant equations; $V/T = Av$ and $v = \sqrt{2g\Delta y}$ are marked derived equations. The CED explains the difference: derived equations are printed in the framework for reference or to show the result of a derivation students are expected to be able to perform, and they are not on the equation sheet handed out with the exam. Check that against the AP Physics 1 equation sheet and it holds. The sheet prints seven fluid relations in total: $\rho = m/V$, $P = F_{\perp}/A$, $P = P_0 + \rho g h$, $P_{\text{gauge}} = \rho g h$, $F_b = \rho V g$, $A_1 v_1 = A_2 v_2$, and Bernoulli's equation. Torricelli's theorem is not among them, and neither is $V/T = Av$.

The CED lists five suggested skills for Topic 8.4, the same count it gives Topic 8.1 and one more than it gives Topics 8.2 and 8.3: 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim. Unit 8 is weighted at 10 to 15 percent of the multiple-choice section, with a suggested 12 to 17 class periods.

Continuity is conservation of mass

Say it in those words, because that is what 8.4.A.2 says and it is what a justification question wants to hear. The continuity equation is not a fact about pipes. It is bookkeeping on matter: fluid cannot be created inside a full tube, and it cannot pile up, so whatever flows in per second must flow out per second.

Build it from the volume side. In a time $T$, fluid moving at speed $v$ through a cross-section of area $A$ sweeps out a volume $V = Av T$, so the volume flow rate is

which is 8.4.A.1.ii. That quantity has units of m$^3$/s. For an incompressible fluid the density is fixed, so a constant volume per second is also a constant mass per second, and setting the rate at one location equal to the rate at another gives

The practical form is a ratio: $v_2 = v_1 (A_1/A_2)$. Speed and area are inversely proportional, which is why the exam almost never wants you to compute an area in square meters.

Watch how area scales for a circular pipe, because this is where the arithmetic goes wrong. $A = \pi r^2$, so doubling the radius multiplies the area by 4 and divides the speed by 4. The CED's own sample multiple-choice question for learning objective 8.4.A does exactly this: a fluid moves from a section of radius $R$ into a section of radius $2R$, and the ratio of the speed in the wide section to the speed in the narrow section is $1/4$. That question is aligned to skill 2.C, comparing quantities at different locations in a single scenario, and it needs no calculator at all.

The volume flow rate itself is the thing that does not change, and quoting it is often the fastest way to answer a question about how long a tank takes to fill.

What makes a fluid flow in the first place

Essential knowledge 8.4.A.1 states the cause plainly: a difference in pressure between two locations causes a fluid to flow. That sentence is the hinge between this topic and Topic 8.3, where learning objective 8.3.A asks about the conditions under which a fluid's velocity changes and the answer is a net force.

The two descriptions are the same physics at two altitudes. A parcel of fluid with higher pressure behind it than in front of it has a net force on it, so it accelerates: that is Newton's second law, Topic 8.3. Track the same parcel down the pipe and add up the work done on it, and you get Bernoulli's equation: that is conservation of energy, Topic 8.4.

This also fixes the order of work inside a problem. Continuity first, because it needs only geometry and gives you the speeds. Bernoulli second, because it needs the speeds and gives you the pressures.

Bernoulli is conservation of energy per unit volume

Essential knowledge 8.4.B.2 says Bernoulli's equation describes the conservation of mechanical energy in fluid flow. Here is the printed form:

Every term is an energy divided by a volume, which is why they can all be measured in pascals. Take the conservation of energy statement you already know and divide it by $V$:

The check that convinces people: a pascal is a newton per square meter, and multiplying top and bottom by a meter turns that into a newton meter per cubic meter, which is a joule per cubic meter. Pressure is an energy density. Once you see that, the equation reads as a sentence rather than a formula: the total mechanical energy per cubic meter of fluid is the same at location 1 and location 2.

Two consequences carry most of the multiple-choice questions.

• Along a horizontal flow, $y_1 = y_2$, the height terms cancel, and pressure and speed trade off directly. Faster fluid means lower pressure. This is not because moving fluid "has less pressure" by nature; it is because the energy that went into $\tfrac{1}{2}\rho v^2$ had to come out of $P$.
• In a fluid at rest, all the $v$ terms are zero and Bernoulli collapses to $P_1 + \rho g y_1 = P_2 + \rho g y_2$, which rearranges into $P = P_0 + \rho g h$ from Topic 8.2. The static pressure relation is the special case of Bernoulli with nothing moving, which is worth knowing if only because it means one fewer equation to trust.

One bookkeeping rule: pick a reference height for $y$, and a pressure convention, gauge or absolute, before you substitute, then use the same choice on both sides. Only differences in $y$ appear, so the zero can go anywhere, but it cannot move partway through a problem.

Torricelli's theorem, and what a derived equation means

Essential knowledge 8.4.B.3 gives Torricelli's theorem, $v = \sqrt{2g\Delta y}$, for the speed of fluid leaving an opening a height $\Delta y$ below the top surface, and states that it can be derived from conservation of energy principles. Since it is a derived equation rather than a relevant one, it is not printed on the equation sheet, so the derivation is the thing to own.

Put point 1 at the top surface and point 2 at the opening. Both are open to the atmosphere, so $P_1 = P_2$ and the pressure terms cancel. If the tank is wide compared with the hole, continuity makes the surface's speed negligible next to the jet's, so $v_1 \approx 0$. What is left is

The density cancels from every term, leaving $v_2 = \sqrt{2g(y_1 - y_2)} = \sqrt{2g\Delta y}$.

That result is worth a second look: it is the same speed an object would reach after falling freely through the same height, and it does not depend on the fluid's density at all. Mercury and water leave a hole at the same depth at the same speed. Students expect the heavier fluid to shoot out faster, and the cancellation on line 3 is the answer to why it does not.

Two cautions. The approximation $v_1 \approx 0$ is an approximation, and it is only good while the hole is small compared with the tank. And $\Delta y$ is measured from the fluid surface down to the hole, not from the floor, not from the top of the container.

The boundary statement, and the exception it carries

Topic 8.4 is the only topic in Unit 8 with a boundary statement, and it is one sentence: all fluids will be assumed to be ideal, and all pipes are assumed to be completely filled by the fluid, unless otherwise stated.

The last three words are part of the rule. "Unless otherwise stated" means a question is free to hand you a fluid that is not ideal, and if it does, the assumptions below stop applying. Do not read the boundary statement as a promise that non-ideal fluids never appear; read it as the default when a question is silent.

What ideal means is fixed elsewhere in the unit. Essential knowledge 8.1.A.4 defines an ideal fluid as incompressible and having no viscosity, and 8.2.A.3 adds that the volume and density of a given amount of an incompressible fluid are constant regardless of the pressure exerted on it. Put together, the default gives you three things:

• No viscosity. No internal friction, so no energy is lost to the fluid rubbing against itself or the pipe walls, which is what lets Bernoulli's equation be an equality rather than an inequality.
• Incompressible. The density is the same at every point, which is what lets the single symbol $\rho$ appear on both sides of Bernoulli's equation and what turns a constant volume flow rate into a constant mass flow rate.
• Pipes completely filled. No air gaps, so the cross-sectional area in $A_1 v_1 = A_2 v_2$ really is the pipe's area.

Be careful about one thing this page cannot tell you: the AP Physics 1 CED does not use the word turbulence anywhere, so do not write that the course "excludes turbulent flow" as though quoting a rule. What the CED excludes by default is viscosity and compressibility, in the words above.

Graphs, experiments, and where skill 1.B lands

Two of Topic 8.4's five suggested skills point straight at the laboratory: 1.B, create quantitative graphs with appropriate scales and units, including plotting data, and 3.A, create experimental procedures appropriate for a given scientific question. Topic 8.1 is the only other topic in Unit 8 that lists either of them. Take that as a warning that this topic is a natural home for an experimental free-response question.

It is not a hypothetical warning. The sample free-response set in the CED includes an Experimental Design and Analysis question aligned to learning objectives 1.5.B, 8.2.B, and 8.4.B. Students are given a cylinder of liquid, meter sticks, and a pressure sensor but no scale, and asked to design a linear-graph method for the liquid's density; then a second cylinder of water has a small hole punched at various heights, the exit speed is measured at each, and the students are asked to plot $v^2$ against a quantity of their own choosing so the graph comes out linear, and to get a value for $g$ from it.

That second part is the graph to be able to build cold. Square Torricelli's theorem:

Plot $v^2$ on the vertical axis and $\Delta y$ on the horizontal axis and you get a straight line through the origin whose slope is $2g$, so $g$ is half the slope. Notice what the linearization bought: the raw $v$ against $\Delta y$ plot is a curve, and you cannot read a physical constant off a curve by eye. Choosing which quantity to square so that a graph comes out linear is the recurring move in AP Physics 1 experimental questions, and the third worked example below runs the numbers.

The CED also suggests a Bernoulli bar-chart activity here: draw bars for $P$, $\rho g y$, and $\tfrac{1}{2}\rho v^2$ at two or more points in a flow, then defend a claim about the pressure using the chart as evidence. Bar charts make the conservation statement visible, because the three bars must add to the same total at every point.

Traps that cost points

• Using continuity on a compressible flow. $A_1 v_1 = A_2 v_2$ is stated in 8.4.A.2 for incompressible fluids. What is conserved in general is the mass flow rate; the areas-and-speeds form is the incompressible special case.
• Adding an area to a Bernoulli equation. Bernoulli has no $A$ in it. Areas enter only through continuity, which supplies the speeds you then substitute.
• Mixing gauge and absolute pressures across the equal sign. Both are fine, but a $1.0 \times 10^5$ Pa discrepancy is the classic result of using one on each side.
• Assuming faster always means lower pressure. That shortcut is only true when the height terms cancel. In a pipe that both narrows and climbs, the speed term and the height term can push the pressure in opposite directions, and only the arithmetic settles it.
• **Forgetting that $v$ in Torricelli's theorem is the exit speed, not a final speed.** Fluid leaving a hole horizontally is a projectile from that moment on, and how far it travels is a separate projectile motion question.
• Writing the density into a Torricelli answer. It cancels. If $\rho$ survives to your final expression for exit speed, an algebra step went wrong.

How Topic 8.4 is tested, and where it leads

The recurring question shapes, mapped to the CED's suggested skills for this topic:

1. Given a change in a pipe's radius or area, give the factor by which the speed changes, with no numbers (skill 2.C).
2. Run continuity and then Bernoulli through a pipe that changes both width and height, and find a pressure (skills 2.A and 2.C).
3. Derive Torricelli's theorem from Bernoulli's equation, stating which terms you cancelled and why (skill 2.A).
4. Choose axes that linearize fluid data, plot them, take a slope, and turn it into a physical quantity (skill 1.B).
5. Design a procedure to measure a fluid property with the equipment listed, then say what you would graph (skill 3.A).
6. Justify a claim about pressure at two points in a flow, using conservation of mass and conservation of energy by name (skill 3.B).

The sixth is the one that separates answers. "The pressure is lower because the fluid is faster" earns less than "the flow speeds up because the cross-sectional area decreased and mass is conserved, and the pressure must drop because the total mechanical energy per unit volume is conserved."

Topic 8.4 closes both Unit 8 and the course, and the CED calls Unit 8 the culminating unit, tying together threads woven through the whole year. Every piece here is borrowed: conservation of energy from Unit 3, and projectile kinematics from Unit 1 whenever a jet leaves a hole. The fluids guide has the standard routines, Topic 8.3 has the force side of the same physics, and the Unit 8 page shows how the four topics fit together.