THE PRACTICE STUDIO
Go slowly. Ask why. Make it yours.
PRACTICE / Bernoulli equation
Pressure pays for a speed increase
Water accelerates from 2 m/s to 6 m/s in a horizontal ideal pipe. Its density is 1000 kg/m³.
Find p₁ − p₂.
Organize the givens & assumptions
- v₁ = 2 m/s
- v₂ = 6 m/s
- ρ = 1000 kg/m³
Steady incompressible inviscid flow on a horizontal streamline; no pumps or losses; water density 1000 kg/m³.
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Problem: Pressure pays for a speed increase
Water accelerates from 2 m/s to 6 m/s in a horizontal ideal pipe. Its density is 1000 kg/m³.
Question: Find p₁ − p₂.
Givens:
- v₁ = 2 m/s
- v₂ = 6 m/s
- ρ = 1000 kg/m³
Assumptions: Steady incompressible inviscid flow on a horizontal streamline; no pumps or losses; water density 1000 kg/m³.
Choices (LaTeX):
A. -16000\ \mathrm{Pa}
B. 32000\ \mathrm{Pa}
C. 16000\ \mathrm{Pa}
D. 2000\ \mathrm{Pa}
Ask for my approach first. Help me distinguish a physical misconception from an algebra error without revealing which choice is correct.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.Start with the smallest hint.
Work through it with me.
One decision at a time. Pause, predict, then unpack the algebra.
Compare squared speeds because the pressure difference supplies a kinetic energy density increase.
Choose a physical relation
With no height change, an increase in flow kinetic energy per volume requires a decrease in pressure.
Isolate what is asked
Compare squared speeds because the pressure difference supplies a kinetic energy density increase.
Why is this step valid?
Subtract the squares of the speeds, not the square of their difference.
Substitute and interpret
Pressure falls by 16000 Pa as speed rises; the pressure work supplies the kinetic-energy increase.
Continuity requires the slower section to have three times the area of the faster section. For each cubic metre, the 16 kPa pressure drop supplies 16 kJ of kinetic energy. This restates the energy balance per volume; the separate continuity check also confirms that acceleration occurs in the narrower section.
TRY THE IDEA AGAIN
Would an extra uphill height change allow the same pressure-drop formula alone?
YOUR LEARNING NOTE
How does this idea feel now?
Confidence is a reflection, not proof of mastery.
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