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Fluids and
Conservation Laws

Master pressure, Bernoulli's equation, and the continuity equation to understand how fluids behave in pipes, tanks, and everyday systems.

AP Physics 1

What Is a Fluid?

A fluid is any substance that flows and takes the shape of its container. Both liquids and gases qualify. Unlike solids, fluids cannot resist shear stress: they deform continuously under an applied force.

Density \(\rho\)
Mass per unit volume: \(\rho = \frac{m}{V}\), measured in \(\text{kg/m}^3\).
Pressure \(P\)
Force per unit area: \(P = \frac{F}{A}\), measured in Pascals (Pa).
Flow Rate \(Q\)
Volume passing a point per second: \(Q = Av\), measured in \(\text{m}^3/\text{s}\).

Pressure in Fluids at Rest

In a static fluid, pressure increases with depth. At any depth \(h\) below the surface, the absolute pressure is the atmospheric pressure plus the weight of the fluid column above.

surface
\(h\)
\(P\)
\(P_0\) (atmospheric)
\(P = P_0 + \rho g h\)

Pascal's Principle

A change in pressure applied to an enclosed, incompressible fluid is transmitted equally to every point in the fluid and to the walls of the container. This is the basis for hydraulic systems.

\(F_1\)
 small
\(F_2\)
 large
\(A_1\)
\(A_2\)
 Pressure transmitted equally

The Continuity Equation

For an incompressible fluid flowing through a pipe, mass is conserved. What flows in must flow out. If the pipe narrows, the fluid speeds up.

\(v_1\) (slow)
\(v_2\) (fast)
\(A_1\)
\(A_2\)
\(A_1 v_1 = A_2 v_2\)
 Conservation of Mass for Fluids

Bernoulli's Equation

Bernoulli's equation applies conservation of energy to a flowing fluid. Along a streamline, the total of pressure energy, kinetic energy per volume, and gravitational potential energy per volume is constant.

Pressure Energy
\(P\): static pressure at a point in the fluid.
Kinetic Energy
\(\frac{1}{2}\rho v^2\): energy from fluid motion per unit volume.
Potential Energy
\(\rho g y\): energy from the fluid's height above a reference.

Visualizing Bernoulli's Principle

reference level (y = 0) Point 1
\(y_1\)
 Point 2
\(y_2\)
\(v_1\) (slow)
\(v_2\) (fast)
High \(P\), low \(v\)
Low \(P\), high \(v\)

Continuity vs Bernoulli

Continuity Equation Bernoulli's Equation
Conservation of Mass Energy
Equation \(A_1 v_1 = A_2 v_2\) \(P + \frac{1}{2}\rho v^2 + \rho g y = \text{const}\)
Tells you How speed changes with pipe area How pressure, speed, and height trade off
Conditions Incompressible, steady flow Incompressible, steady, non-viscous, along a streamline
Number of terms 2 3 (per side)

How to Solve Fluid Problems

1
Identify the two points. Label them 1 and 2 along the streamline. Note which quantities are known and unknown.
2
Apply continuity first. If areas or speeds are involved, use \(A_1 v_1 = A_2 v_2\) to express one speed in terms of the other.
3
Apply Bernoulli's equation. Write out all six terms. Cancel any terms that are zero or equal on both sides (e.g., same height means \(\rho g y\) cancels).
4
Substitute and solve. Plug in known values and the expression from continuity. Solve for the unknown.
5
Check units and signs. Pressure should be positive. Speed should be positive. Does the answer make physical sense?

Example 1: Speed in a Pipe

Water flows through a horizontal pipe that narrows from a cross-sectional area of \(0.04\,\text{m}^2\) to \(0.01\,\text{m}^2\). The speed in the wide section is \(2\,\text{m/s}\). Find the speed in the narrow section.

Example 2: Pressure Drop

Using the same pipe from Example 1, the pressure in the wide section is \(200{,}000\,\text{Pa}\). Find the pressure in the narrow section. (Water density: \(\rho = 1000\,\text{kg/m}^3\), same height.)

Water through a Venturi

Transparent water-filled pipe with a narrow central throat and wide inlet and outlet.For steady horizontal water flow with negligible losses, narrowing the diameter from 8.0 cm to 4.0 cm raises speed from 1.5 to 6.0 m/s. With P₁ = 250 kPa, P₂ = 233.125 kPa.

A second pipe example: 8 cm to 4 cm

Horizontal pipe diameters 8 and 4 centimeters, speeds 1.5 and 6 meters per second, and pressures 250 and 233.125 kilopascals.For steady horizontal water flow with negligible losses, narrowing the diameter from 8.0 cm to 4.0 cm raises speed from 1.5 to 6.0 m/s. With P₁ = 250 kPa, P₂ = 233.125 kPa.Open full diagram

Example 3: Elevated Tank

A large open tank is filled with water to a height of \(h = 5\,\text{m}\) above a small spout at the bottom. How fast does water exit the spout? (Assume the tank surface barely moves.)

\(P_0, v \approx 0\)
\(h\)
\(v_2\)
\(P_0\) (open to air)

Ideal outlet flow

This separate outlet example uses a 1 m depth. Both the free surface and outlet are at atmospheric pressure, and losses are neglected; these assumptions give the indicated speed.

An ideal water outlet 1 m below the free surface has speed √(2gh) = 4.43 m/s. Surface and outlet are at atmospheric pressure; losses are neglected.
An ideal water outlet 1 m below the free surface has speed √(2gh) = 4.43 m/s. Surface and outlet are at atmospheric pressure; losses are neglected.

Tank-drain geometry and free-jet trajectory

Water exits a small side hole of an open elevated tank and falls toward the ground.The head driving the outlet speed is the vertical difference between the water surface and the opening. The opening’s height above the ground separately determines the jet’s flight time.

Outlet speed and jet range

Outlet speed and jet range. With a 4 m water head and g = 10 m/s², exit speed is about 8.94 m/s. A separate 2 m drop from outlet to ground gives flight time about 0.632 s and range about 5.66 m.With a 4 m water head and g = 10 m/s², exit speed is about 8.94 m/s. A separate 2 m drop from outlet to ground gives flight time about 0.632 s and range about 5.66 m.

Buoyancy and Archimedes' Principle

Any object submerged (fully or partially) in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.

Floats
\(\rho_{\text{object}} < \rho_{\text{fluid}}\): buoyant force can equal weight before fully submerging.
Sinks
\(\rho_{\text{object}} > \rho_{\text{fluid}}\): even fully submerged, buoyant force is less than weight.

The Energy Perspective

Bernoulli's equation is really just conservation of energy per unit volume. Each term represents a form of energy density:

1
\(P\): pressure energy density (work done by the fluid on its surroundings per unit volume).
2
\(\frac{1}{2}\rho v^2\): kinetic energy density, analogous to \(\frac{1}{2}mv^2\) scaled by volume.
3
\(\rho g y\): gravitational potential energy density, analogous to \(mgy\) scaled by volume.

Common Mistakes

✕ Mixing up \(P\) and \(F\)
Pressure is force per unit area. Using \(P = F\) instead of \(P = F/A\) gives wrong units and values.
✕ Forgetting height terms
When the pipe changes elevation, you must include \(\rho g y\) terms. Setting them to zero only works for horizontal flow.
✕ Higher speed = higher pressure
Bernoulli says the opposite: faster flow means lower pressure. This counterintuitive result is a frequent exam trap.

Summary of Fluid Laws

Law Equation Conserves Key Insight
Hydrostatic \(P = P_0 + \rho g h\) Equilibrium Deeper = more pressure
Pascal's \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\) Pressure Force multiplied by area ratio
Continuity \(A_1 v_1 = A_2 v_2\) Mass Narrower = faster
Bernoulli \(P + \frac{1}{2}\rho v^2 + \rho g y = \text{const}\) Energy Faster = lower pressure
Archimedes \(F_b = \rho_f V_d g\) Force balance Buoyancy = displaced weight

Question 1 of 5

Water flows through a horizontal pipe. The pipe narrows so that \(A_2 = \frac{1}{3}A_1\). If the speed in the wide section is \(v_1 = 3\,\text{m/s}\), what is \(v_2\)?

Question 2 of 5

According to Bernoulli's Principle, when an ideal fluid flows through a region where its speed increases, the pressure in that region:

Question 3 of 5

A large water tank has an open top. A small hole is at a depth of \(h = 3.2\,\text{m}\) below the water surface. Using Torricelli's theorem, the exit speed of the water is approximately:

Question 4 of 5

A hydraulic press has pistons with areas \(A_1 = 0.005\,\text{m}^2\) and \(A_2 = 0.5\,\text{m}^2\). A force of 100 N is applied to the small piston. What force is exerted by the large piston?

Question 5 of 5

A solid cube of density \(600\,\text{kg/m}^3\) is placed in water (\(\rho_w = 1000\,\text{kg/m}^3\)). What fraction of the cube's volume is submerged when it floats?