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AP Physics C: Fluids Investigation

Narrative Inquiry: Can the Field Station Fill Its Emergency Water Tank?

Pressure, density, continuity, volume flow rate, and Bernoulli's equation in one real-world decision.

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Submission Progress: 0 of 8 required sections complete

Answer checks unlock only after all required written responses and calculation fields have been completed.

Teacher Overview

Estimated Time

80 minutes total: 10 minutes reading and planning, 55 minutes solving, 10 minutes online research and refinement, 5 minutes final submission.

Learning Goal

Students will model a practical fluid system by extracting data from context, researching missing constants, making assumptions, and defending whether the system can meet a design requirement.

Format

Students complete a multi-step AP Physics C style free-response problem and receive automated feedback only after full submission.

Academic honesty note: Students may use online research only to find physical constants or realistic values requested by the prompt. They may not search for solved examples, use AI to solve the problem, or copy another group's work.

The Situation

A small environmental field station sits on a hillside above a stream. During power outages, the station must fill an elevated emergency water tank using only gravity-fed flow from a spring box located farther uphill. The system is old, and the station director wants your physics class to determine whether the current pipe can fill the tank quickly enough before a forecasted storm.

The spring box feeds water into a pipe that descends to the tank. The pipe narrows at a partially repaired section near the tank. Your job is to decide whether the flow rate is high enough and whether the pressure at the narrow section creates a risk of cavitation or air release.

Data Embedded in the Scenario

Quantity	Value	Context
Vertical height of water surface in spring box above tank inlet	18.0 m	Measured from station map
Length of main pipe	96 m	Marked on old installation plan; ignore viscous losses for the main AP C model
Main pipe inside diameter	5.00 cm	Pipe label from maintenance log
Narrow repaired section inside diameter	2.50 cm	Measured by maintenance crew
Emergency tank volume needed	2.40 m³	Minimum water volume for one day of operation
Required fill time	30.0 min	Director's requirement before storm arrival
Gauge pressure at the spring box water surface	0 Pa	Open to the atmosphere
Gauge pressure at tank inlet/outlet opening	0 Pa	Discharges into open tank

Values Students Must Research Online

Density of fresh water near room temperature, rho.
Atmospheric pressure at sea level, P_atm.
Vapor pressure of water near 20°C, P_vapor.
Optional extension: dynamic viscosity of water near 20°C if you want to discuss whether ignoring viscous losses is reasonable.

Students should record the source and units for each researched value.

Useful Relationships

Density

rho = m/V

Continuity

A₁v₁ = A₂v₂ = Q

Volume Flow Rate

Q = ΔV/Δt = Av

Bernoulli’s Equation

P + ½ρv² + ρgy = const

Part A — Extract the Known Information

Identify the physical quantities given in the narrative and convert all values to SI units. Include pipe radii, cross-sectional areas, required volume, and required fill time.

A1. Written setup and extracted data A2. Main pipe cross-sectional area, A_main, in m² A3. Narrow section cross-sectional area, A_narrow, in m²

Part B — Research the Missing Physical Values

Find reasonable values for water density, atmospheric pressure, and vapor pressure of water near 20°C. Record the source and explain whether each value is appropriate for this situation.

B1. Density of water, rho, in kg/m³ B2. Atmospheric pressure, P_atm, in Pa B3. Vapor pressure of water near 20°C, P_vapor, in Pa B4. Sources and justification

Part C — Required Flow Rate

Determine the minimum volume flow rate needed to fill the tank in the required time.

C1. Required flow rate, Q_required, in m³/s C2. Show your calculation and interpret the result in everyday terms

Part D — Ideal Exit Speed from Bernoulli’s Equation

Use Bernoulli’s equation between the open spring box surface and the open tank inlet. Assume the spring box surface is large enough that the water speed at the surface is approximately zero. Ignore viscosity and frictional losses for this first model.

D1. Ideal speed at the tank inlet, v_exit, in m/s D2. Derive the result from Bernoulli’s equation

Part E — Predicted Flow Rate Through the Narrow Section

Assume the tank inlet is controlled by the narrow repaired section. Use the ideal exit speed from Part D and the narrow-section area to estimate the actual delivered flow rate.

E1. Predicted ideal flow rate, Q_predicted, in m³/s E2. Predicted fill time in minutes E3. Explain whether the tank can fill within 30.0 minutes

Part F — Speed in the Main Pipe

Use continuity to determine the water speed in the 5.00 cm main pipe when the water speed in the 2.50 cm repaired section equals your ideal exit speed.

F1. Speed in the main pipe, v_main, in m/s F2. Explain why the speed changes when the pipe narrows

Part G — Pressure at the Narrow Section

The repaired narrow section is located 1.20 m above the tank inlet. Estimate the absolute pressure in the narrow section using Bernoulli’s equation between the narrow section and the open tank inlet. Then compare this pressure with the vapor pressure of water.

G1. Absolute pressure in the narrow section, P_narrow, in Pa G2. Show your Bernoulli setup and discuss cavitation risk

Part H — Engineering Recommendation

Write a final recommendation to the station director. You must address the fill-time requirement, the pressure/cavitation concern, and at least one limitation of the ideal-fluid model.

H1. Final recommendation

Submit and Check

When every section is complete, submit your work. After submission, the answer checks and scoring feedback will unlock.

Answer Checks and Scoring Guide

These checks use reasonable reference values: rho = 998 kg/m³, P_atm = 101325 Pa, P_vapor ≈ 2300 Pa, and g = 9.8 m/s². Small differences are acceptable when researched constants differ slightly.

Part A Check

Main pipe radius = 0.0250 m, so A_main ≈ 1.96 × 10&supthree; m². Narrow radius = 0.0125 m, so A_narrow ≈ 4.91 × 10⁻&sup4; m².

Part C Check

Q_required = 2.40 m³ / 1800 s ≈ 1.33 × 10⁻³ m³/s.

Part D Check

With both endpoints open to atmosphere and the spring surface speed approximately zero: rho g h = ½ rho v², so v = sqrt(2gh) = sqrt(2 · 9.8 · 18.0) ≈ 18.8 m/s.

Part E Check

Q_predicted = A_narrow v_exit ≈ (4.91 × 10⁻&sup4;)(18.8) ≈ 9.23 × 10⁻³ m³/s. Fill time ≈ 2.40 / 9.23 × 10⁻³ ≈ 260 s ≈ 4.33 min. In the ideal model, the tank easily fills within 30.0 min.

Part F Check

v_main = Q/A_main ≈ 9.23 × 10⁻³ / 1.96 × 10⁻³ ≈ 4.70 m/s. Since the narrow area is one-fourth the main area, the narrow-section speed is four times the main-pipe speed.

Part G Check

If the narrow section and outlet have nearly the same diameter, their speeds are approximately the same. Bernoulli between the narrow section 1.20 m above the outlet and the open outlet gives P_narrow ≈ P_atm − rho g(1.20) ≈ 101325 − 11700 ≈ 8.96 × 10&sup4; Pa. This is far above the vapor pressure of water near 20°C, so this simplified model does not predict cavitation at that location.

Model Limitation Check

The ideal calculation likely overestimates the flow rate because it ignores pipe friction, turbulence, entrance/exit losses, bends, valves, roughness, and losses at the narrowed repair. The system may still meet the requirement, but the final recommendation should mention that a real flow test or head-loss calculation is needed before relying on the system.