Bernoulli: track pressure, speed and height together
You will be able to: Apply Bernoulli’s equation along a streamline and state the conditions behind pressure comparisons.
When can faster flow have lower pressure?
Water speeds up through a horizontal narrowing in an ideal pipe. Its kinetic energy per volume increases. With no pump or losses between the chosen points, a drop in pressure supplies that increase. If the pipe also climbs, gravitational energy matters too.
A useful starting point: A narrower pipe means faster flow at the same rate →
Words and symbols before equations
- Streamline
- A curve tangent to the local flow direction.
- P
- Pressure term in Pa, representing pressure-work transfer per fluid volume.
- ½ρv²
- Kinetic energy per volume, in J/m³, equivalent to Pa.
- ρgy
- Gravitational potential energy per volume relative to a chosen height zero.
- Bernoulli assumptions
- Steady, incompressible, nonviscous flow along one streamline, without an intervening pump or dissipative loss.
What this picture assumes
Same steady ideal streamline, no pump or losses between points. Water density 1000 kg/m³, g=10 m/s², inlet absolute P=160 kPa, v=2 m/s, y=0. Continuity gives v₂=2(A₁/A₂). Chosen ranges keep calculated absolute pressures positive; energy-per-volume terms are displayed in kPa.
Connect the picture to the physics
Pressure forces do work as fluid moves, while speed and height determine kinetic and gravitational energy. Under the stated conditions, P+½ρv²+ρgy is the same at two points along a streamline. Each term has the same units; the pressure term should not be mistaken for a separate stored potential energy of the fluid.
In a horizontal section, height terms cancel. Then increasing speed requires decreasing pressure. With a height change, include ρgΔy: a rising stream may use both pressure and kinetic energy to gain gravitational energy. “Faster always means lower pressure” is not a universal rule across unrelated flows or different energy inputs.
Use continuity first if the second speed is unknown. Keep pressure references consistent and choose one height zero. If a pump adds energy or viscosity removes mechanical energy between the points, the simple constant-sum form needs additional terms. Check that a proposed pressure remains physically compatible with the liquid model.
A worked example, step by step
Water at ρ=1000 kg/m³ flows horizontally from v_1=2 m/s and P_1=120 kPa to v_2=4 m/s. Find P_2, neglecting losses.
- At equal heights, P_2=P_1+½ρ(v_1²−v_2²).
- The kinetic term increase is ½(1000)(16−4)=6000 Pa.
- P_2=120,000−6000=114,000 Pa=114 kPa.
- The pressure drop supplies the 6 kJ/m³ kinetic-energy increase. No pump or loss was included.
Use the same streamline, valid assumptions and consistent pressure reference. Speed alone does not determine pressure.
If speed stays constant while a pipe rises, what happens to pressure in the ideal no-pump model?
Compare with an explanation
It falls by ρgΔy to supply the gravitational-energy increase.
Predict. Change one thing. Explain.
Start with a horizontal pipe and vary the area ratio, using continuity for outlet speed. Then raise the outlet while keeping the speeds fixed. Compare the pressure, kinetic and gravitational terms at the two points.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
v₂=4 m/s; y₂=0 m. P₂=154 kPa absolute. Point 1: 160+2+0=162 kPa. Point 2: 154+8+0=162 kPa. The shared zero of height is at point 1.
Same steady ideal streamline, no pump or losses between points. Water density 1000 kg/m³, g=10 m/s², inlet absolute P=160 kPa, v=2 m/s, y=0. Continuity gives v₂=2(A₁/A₂). Chosen ranges keep calculated absolute pressures positive; energy-per-volume terms are displayed in kPa.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionWater with ρ=1000 kg/m³ and g=10 m/s² flows at equal speed at two points. Point 2 is 3 m higher; P_1=150 kPa. (a) Identify which terms cancel. (b) Find the change in ρgy. (c) Find P_2. (d) State a required assumption.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The kinetic terms cancel because speeds match.
- 1 point: ρgΔy=30,000 Pa.
- 1 point: P_2=120 kPa.
- 1 point: Steady incompressible nonviscous flow along the same streamline with no intervening pump or loss; a clearly stated relevant assumption earns the point.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does Bernoulli express?
Mechanical-energy conservation under its stated flow assumptions.
RECALL 2When can faster be linked directly to lower pressure?
Along the same ideal streamline at equal height and equal total mechanical-energy-per-volume sum.
RECALL 3Why include height?
Gravity changes the energy balance.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Bernoulli: track pressure, speed and height together
- P_1+½ρv_1²+ρgy_1=P_2+½ρv_2²+ρgy_2.
- All terms are energy per volume, in Pa=J/m³.
Remember: Use the same streamline, valid assumptions and consistent pressure reference. Speed alone does not determine pressure.
Conditions: Same steady ideal streamline, no pump or losses between points. Water density 1000 kg/m³, g=10 m/s², inlet absolute P=160 kPa, v=2 m/s, y=0. Continuity gives v₂=2(A₁/A₂). Chosen ranges keep calculated absolute pressures positive; energy-per-volume terms are displayed in kPa.
Refresh Kid · Unit 8 · Objectives 8.4.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.4, objectives 8.4.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
Want to work through this with a tutor?
Bring your question about Bernoulli: track pressure, speed and height together. Your explanation and answers remain free to access.