Phone physics and the Gateway Arch: Fun with friends and physics at the AAPT Winter Meeting in St. Louis

Bree Barnett Dreyfuss; David Marasco

Phone physics and the Gateway Arch: Fun with friends and physics at the AAPT Winter Meeting in St. Louis

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Phone barometer data give a better height estimate for the Gateway Arch tram ride than double-integrated accelerometer data, matching the known 192-meter height within a few meters.

desk verdict A modest but honest classroom-resource paper whose central barometer-vs-accelerometer comparison holds up, and the supplied dataset is the real value. read the letter →

arxiv 2506.22746 v1 pith:56UUQQ3N submitted 2025-06-28 physics.ed-ph physics.pop-ph

David Marasco , Bree Barnett Dreyfuss This is my paper · ORCID

classification physics.ed-phphysics.pop-ph

keywords phonesensorsbarometeraccelerometerGatewayArchkinematicsphysicseducationpressurealtitudeEulerintegration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reports a physics measurement made on the Gateway Arch tram in St. Louis: ride the tram while a phone records pressure and acceleration, then compare two ways of reconstructing how high the tram goes. The barometer, using the pressure-to-height relation, gives a peak displacement of about 198 meters, within a few meters of the Arch's known height of 192 meters. Double-integrating the phone's Z-axis acceleration overshoots to about 262 meters, and even after resetting the velocity at the midpoint stop it still reads 251 meters. The authors argue that for this slow-moving, tilting tram the barometer is the stronger representation, and they provide raw data, spreadsheets, and a teacher's guide so students can work through the same comparison.

What carries the argument

The central objects are two complementary sensor pipelines. The barometer pipeline converts pressure to height using the hydrostatic relation $\Delta P = \rho g \Delta h$, or PhyPhox's barometric formula, giving a direct vertical displacement. The accelerometer pipeline uses Euler's method to integrate Z-axis acceleration twice, subtracting an offset tuned so that the round trip returns to its starting height; a variant splits the ride at the midpoint rest and resets velocity there, following the practice recommended in the speed-estimation literature. The comparison of these two pipelines on the same ride is what carries the claim that pressure wins.

What would settle it

Repeat the Gateway Arch tram ride with a phone that also logs gyroscope orientation; if the gyroscope-corrected double integration gives a peak height within a few meters of 192 meters, or beats the barometer's 198 meters, the claim that the barometer is the stronger representation for this ride would fail. Alternatively, a pressure-sensor ride on an elevator of known height that produced a height error larger than the accelerometer's would also undercut it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is comparative: a phone's pressure sensor is a more reliable vertical ruler than its accelerometer for a ride that is slow and whose orientation changes. Pressure-derived altitude peaks at roughly 198 meters versus the Gateway Arch's known height of 192 meters, while two numerical integrations of the Z-acceleration produce 262 meters, or 251 meters when the data are split at the rest stop and each segment receives its own accelerometer offset. The paper also explains why: the tram rides like a Ferris wheel, tilting the phone relative to Earth's vertical, so even one-degree orientation changes can corrupt the acceleration signal, and integration compounds the error quadratically in time. The barometer path also preserves qualitative details the accelerometer misses, such as the tram gently slowing to a stop on the descent.

Load-bearing premise

The argument relies on the accelerometer branch being a fair but imperfect method: if the phone's Z-axis does not stay close enough to Earth's vertical during the ride, then the accelerometer heights are measuring something other than vertical displacement and the comparison with the barometer is fixed in advance.

Editorial extensions

If this is right

A classroom can use the supplied dataset to see that pressure-based height tracks the known 192-meter Arch height while double integration overshoots by about 36 percent.
For short phone-accelerometer experiments, drift is tolerable: the paper notes that the 60-meter overshoot over 450 seconds would be about 3 centimeters in a 10-second experiment.
Qualitative motion features survive integration: the mid-ride slowdown appears in both methods, so accelerometer data still support discussions of velocity and acceleration.
The split-and-reset method improves the accelerometer estimate from 262 meters to 251 meters, showing that using known rest positions is a cheap fix for integration drift.
Horizontal motion is less reliable: the X-displacement is roughly plausible at about 90 meters across the leg-to-center distance, while the Y-displacement is meaningless, warning students about integrating noise-dominated signals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural follow-up is to record gyroscope data alongside pressure and acceleration and rotate the measured acceleration into Earth's frame before integrating; the paper's one-degree-tilt estimate suggests this could eliminate much of the 60-meter overshoot.
The comparison should transfer to elevators, stair climbs, or drone ascents: pressure should remain the better height estimate whenever the device's orientation drifts by more than a degree during the motion.
The roughly 90-meter horizontal excursion from the accelerometer nearly reproduces the Arch's leg-to-center distance of 96 meters, so even imperfect horizontal data may be usable as an order-of-magnitude exercise.
Because air density is the main environmental parameter in the pressure method, temperature-logged repeat runs could turn the 198-meter versus 192-meter difference into a lesson about uncertainty and assumptions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this is a small, transparent physics-education paper, not a research contribution. The value is in the public dataset and the teaching package. The central claim—phone barometer gives a better height estimate than double-integrated accelerometer data for the Gateway Arch tram—is supported by the numbers: barometer reaches 198 m against the arch’s 192 m, while integrated acceleration gives 262 m (or 251 m with a segment reset). The paper is honest about the accelerometer’s offsets, the tilt problem, and the noise, and it credits Yu’s 2025 zeroing method that would have improved their analysis.

The authors do several things well. They make raw data, spreadsheets, and a teacher guide available. They present four methods and show where each fails. Their discussion of student misconceptions—like reading a velocity graph while the tram is descending—is practical. They also correctly treat the barometric altitude as independent of the fitted accelerometer offsets, so the key comparison is not circular. That is the paper’s main strength.

The soft spots are real but not fatal. The accelerometer heights are not independent: the offsets are fitted to match the round-trip boundary condition, so the large overshoot is partly an artifact of that constraint. The paper acknowledges this. The pixel-counting comparison mixes geometry: the 203 m from Figure 1 is a path length along the curved track, not a vertical height, so comparing it to the barometric 198 m vertical rise is apples-to-oranges. That does not change the relative ranking of barometer vs accelerometer, so it is a minor imprecision. The one-degree tilt worry is addressed in the text and, if anything, reinforces why barometers win. The horizontal-axis integration is messy, and the paper says so; the Y-axis giving over a kilometer of noise is actually a useful teachable moment about integrating noise. The authors also rely on a single phone and a tour guide’s speed claim, so formal error bars are absent. For a classroom resource that is acceptable; for a measurements paper it would be thin.

Who is this for? High school or first-year college physics teachers who want a real, mildly messy dataset from a famous landmark with limitations to discuss. It deserves a serious referee because it is clearly written and ships reproducible data. I’d accept it for review and suggest the editor ask the authors to fix the path-length vs vertical-height comparison and to state explicitly that the accelerometer offsets are fitting parameters, not measured quantities.

Referee Report

0 major / 6 minor

Summary. The paper reports a smartphone-based measurement of the vertical motion of the Gateway Arch tram during a round trip to the observation deck. Using the PhyPhox app on a Samsung Galaxy S23, the authors collected simultaneous barometer and accelerometer data. They analyze the data with four methods: (1) pressure change with constant air density, (2) PhyPhox's built-in barometric formula, (3) double numerical integration of the Z-axis accelerometer with a single fitted offset, and (4) double integration with separate offsets for the ascent and descent segments, including a velocity reset at a known stop. The barometric methods give a maximum vertical displacement of 198 m, the accelerometer methods give 262 m (Method 3) and 251 m (Method 4), while the known height of the Gateway Arch is 192 m and a pixel count of the tram path in Figure 1 yields about 203 m. The authors conclude that the barometer provides a more reliable height estimate than double-integrated acceleration, and they attribute the accelerometer errors to sensor offset drift, noise, and the tram's tilting orientation. Raw data, spreadsheets, and a teacher's guide are made available.

Significance. If the result holds, the paper offers a useful, low-cost classroom activity for comparing sensor types and for teaching the pitfalls of numerical integration. The central comparison is supported by the reported data: barometer-derived displacement is much closer to the known height of the Arch than either accelerometer integration approach. A particular strength is that the barometric measurement is independent of the fitted accelerometer offsets, so the main conclusion is not circular. The manuscript is transparent about limitations, including the tilt of the phone's Z-axis, the need to fit offsets, and the failure of the horizontal Y-axis channel. The authors also cite relevant prior work, including Monteiro and Martí and the recent offset-compensation method of Yu. The availability of raw data and a teacher's guide increases the paper's value for instructors.

minor comments (6)

[Eq. (1), barometric pressure analysis] The sentence following Eq. (1), 'the value in the numerator of the ratio is a change in weight,' is confusing because the ratio has already been reduced to ΔP; please rewrite it to say that the numerator ρgAΔh is the weight of the air column of height Δh and that dividing by the area A gives the pressure change.
[Method 3, discussion of offset error] The statement that the error grows quadratically with time is correct and useful, but it would be clearer to state explicitly that the 60 m overshoot corresponds to the fitted offset shift from 9.8234 to 9.8263 m/s², so readers can scale the effect to experiments of different duration.
[Figures 5 and 6] The figures are captioned 'Z-direction Velocity & Height via Method 4,' but the text refers to 'the following two height plots'; please clarify in the captions which panel is velocity and which is height, or split them into separate figures with unambiguous captions.
[Horizontal directions analysis] The phrase 'accelerator sensor offsets' appears to be a typo for 'accelerometer sensor offsets'; please correct it for consistency with the rest of the text.
[Comparison of barometer height to arch height] The paper compares the barometric 198 m result to the pixel-counted 203±2 m path length from Figure 1 and calls the difference reasonable, but the pixel-counted value is a path length along the curved track, not a vertical displacement; the more direct comparison is to the 192 m arch height, which yields a 6 m difference, and the paper should clarify whether the tram's underground loading level explains part of that difference.
[Data availability] The supplemental material is mentioned only through reference [1]; for a teaching-oriented paper, it would be helpful to include a direct URL or repository identifier in the main text so readers can easily locate the raw data and teacher's guide.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: barometer and accelerometer comparisons are independent, with only a legitimate endpoint calibration.

full rationale

The paper's central comparative claim is that barometer-derived height (198 m via Method 1) matches the known Gateway Arch height better than double-integrated accelerometer height (262 m via Method 3, 251 m via Method 4). This is an empirical comparison between two independent sensor channels. The barometric height uses the pressure-altitude relation with an assumed air density and is not derived from the accelerometer data. The accelerometer analyses use a physically motivated calibration: the offset is varied until the round-trip displacement returns to its starting value, or the velocity is reset at a known rest point. These are boundary-condition calibrations, not fits to the quantity being compared; the maximum displacement remains an output of the integration. The paper also discloses key limitations (tilt, offset drift, air-density assumptions) that affect the accelerometer branch more than the barometer branch, which strengthens rather than undermines the comparison. No self-citation is load-bearing, and no known result is merely renamed. The only minor issue is that the pixel-counted 203 m path length is not the same geometric quantity as vertical height, but this does not make the derivation circular and does not affect the relative ranking of methods.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central result rests on standard pressure-height physics and Euler integration, plus two fitted accelerometer offsets and assumptions about air density, rest states, and phone orientation. No new entities are introduced.

free parameters (3)

Acceleration offset (ascent, single trip) = 9.8263 m/s^2
Chosen so the integrated final position matches the initial position over the full round trip (Method 3).
Acceleration offset (ascent, split) = 9.826 m/s^2
Assigned to the upward segment in Method 4 using the rest condition at data restart.
Acceleration offset (descent, split) = 9.812 m/s^2
Assigned to the downward segment in Method 4 to make velocity reset to zero at the rest stop.

assumptions (6)

standard math Euler integration with fixed time steps yields position from acceleration
Used in Methods 3 and 4.
domain assumption Barometric formula relating pressure to altitude as implemented in PhyPhox
Method 2; standard atmospheric model with assumptions about temperature and density.
domain assumption Air density is 1.225 kg/m^3
Used in Method 1; the paper notes a 20 C value would shift the height by 3.5 m.
domain assumption The phone is at rest when data collection is restarted on the descent
Used to reset velocity to zero in Method 4, following Ustin and Cetin.
domain assumption The tram's path is a weighted catenary
Taken from Osserman's reference; used only for qualitative comparison in Figure 9.
domain assumption The phone's Z-axis is approximately aligned with the Earth's vertical throughout the ride
Needed for Methods 3 and 4; the authors acknowledge it is violated by tram rocking.

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Cite this review

Pith. "Pith review of Phone physics and the Gateway Arch: Fun with friends and physics at the AAPT Winter Meeting in St. Louis." pith.science (2026). https://pith.science/paper/56UUQQ3N

@misc{pith2026250622746,
  author       = {Pith},
  title        = {Pith review of: Phone physics and the Gateway Arch: Fun with friends and physics at the AAPT Winter Meeting in St. Louis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56UUQQ3N}},
  note         = {Machine review of arXiv:2506.22746}
}

read the original abstract

As a famous landmark and feat of engineering, the Gateway Arch was a popular destination at the 2025 AAPT Winter Meeting in St. Louis. The visit to the observation deck of the Gateway Arch is unique, climbing the steps after exiting the small tram capsules and seeing a floor that continues to slope upward assures that you are in fact at the very top. Everyone in our group excitedly took pictures, pointing out local features like the Dred Scott Courthouse. There were many selfies at the pinnacle, and we discussed how to work them into future questions for our students. During our tram ride to the top observation deck of the arch, we lamented that we should have brought pendula to measure the acceleration due to gravity. You can take physics teachers out of the physics conference, but you apparently can't get us to stop talking about physics teaching. Recognizing that we had accelerometers on our phones we collected data on the descent. The authors wanted to collect more complete measurements and returned two days later to repeat the journey, the results of which we present here. For readers wishing to repeat with their students, or who want to apply more advanced data analysis techniques, the authors have made the raw data, our spreadsheets, and a teacher's guide available.

Figures

Figures reproduced from arXiv: 2506.22746 by the authors.

**Figure 2.** Height vs time calculations: Method 1 (pressure) results in red, Method 3 (Euler) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 9.** Blue curve is from integrating both directions from accelerometer data (Method 3), red [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

Figures from the paper (1 more)

**Figure 10.** Figure 10: AAPT attendees Bryn Bishop and Jay Kurima with authors Bree Barnett Dreyfuss [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Readers can access the Appendix as supplemental material

work page
[2]

Gateway Arch - Tram System, https://www.nps.gov/jeff/planyourvisit/tram-system.htm

work page
[3]

https://www.nps.gov/parkhistory/online_books/jeff/adhi2-4b.htm

Bob Moore, Urban Innovation and Practical Partnerships: An Administrative History of Jefferson National Expansion Memorial, 1980-1991, 1994. https://www.nps.gov/parkhistory/online_books/jeff/adhi2-4b.htm

work page 1980
[4]

Rhyner, Studying the motion of an elevator, Phys

Charles R. Rhyner, Studying the motion of an elevator, Phys. Teach. 36, 111–113 (1998) https://doi.org/10.1119/1.880007

work page doi:10.1119/1.880007 1998
[5]

Kinser; Relating Time-Dependent Acceleration and Height Using an Elevator

Jason M. Kinser; Relating Time-Dependent Acceleration and Height Using an Elevator. Phys. Teach. 1 April 2015; 53 (4): 220–221. https://doi.org/10.1119/1.4914561

work page doi:10.1119/1.4914561 2015
[6]

Using smartphone pressure sensors to measure vertical velocities of elevators, stairways, and drones,

M. Monteiro and A. Martí, “Using smartphone pressure sensors to measure vertical velocities of elevators, stairways, and drones,” Phys. Educ. 52, 15010 (2016).https://doi.org/10.1088/1361-6552/52/1/015010

work page doi:10.1088/1361-6552/52/1/015010 2016
[7]

Littleton, Richard Secco; Smartphones and Gravitational Acceleration II: Applications

Meryem Berrada, Joshua A.H. Littleton, Richard Secco; Smartphones and Gravitational Acceleration II: Applications. Phys. Teach. 1 October 2020; 58 (7): 473–476. https://doi.org/10.1119/10.0002064

work page doi:10.1119/10.0002064 2020
[9]

Compensating smartphone accelerometers for more precise classroom experiments

Hoon Yu, “Compensating smartphone accelerometers for more precise classroom experiments.” Phys. Teach., 63, 189-192. (2025) https://doi.org/10.1119/5.0202891

work page doi:10.1119/5.0202891 2025

Show all 14 references

[10]

https://github.com/phyphox/phyphox-experiments/blob/master/elevator.phyphox
[11]

A Quick Derivation relating altitude to air pressure,

Portland State Aerospace Society, “A Quick Derivation relating altitude to air pressure,” https://archive.psas.pdx.edu/RocketScience/PressureAltitude_Derived.pdf
[12]

https://www.engineeringtoolbox.com/air-density-specific-weight-d_600.html
[13]

https://phyphox.org/wiki/index.php/Integrated_acceleration
[14]

Ustun, I., & Cetin, M. (2019). Speed Estimation using Smartphone Accelerometer Data. Transportation Research Record, 2673(3), 65-73. https://doi.org/10.1177/0361198119836977

2019 doi
[15]

Mathematics of the gateway arch

Osserman, Robert. "Mathematics of the gateway arch." Notices of the AMS 57.2 (2010): 220-229. https://www.ams.org/journals/notices/201002/rtx100200220p.pdf

2010

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Readers can access the Appendix as supplemental material

work page

[2] [2]

Gateway Arch - Tram System, https://www.nps.gov/jeff/planyourvisit/tram-system.htm

work page

[3] [3]

https://www.nps.gov/parkhistory/online_books/jeff/adhi2-4b.htm

Bob Moore, Urban Innovation and Practical Partnerships: An Administrative History of Jefferson National Expansion Memorial, 1980-1991, 1994. https://www.nps.gov/parkhistory/online_books/jeff/adhi2-4b.htm

work page 1980

[4] [4]

Rhyner, Studying the motion of an elevator, Phys

Charles R. Rhyner, Studying the motion of an elevator, Phys. Teach. 36, 111–113 (1998) https://doi.org/10.1119/1.880007

work page doi:10.1119/1.880007 1998

[5] [5]

Kinser; Relating Time-Dependent Acceleration and Height Using an Elevator

Jason M. Kinser; Relating Time-Dependent Acceleration and Height Using an Elevator. Phys. Teach. 1 April 2015; 53 (4): 220–221. https://doi.org/10.1119/1.4914561

work page doi:10.1119/1.4914561 2015

[6] [6]

Using smartphone pressure sensors to measure vertical velocities of elevators, stairways, and drones,

M. Monteiro and A. Martí, “Using smartphone pressure sensors to measure vertical velocities of elevators, stairways, and drones,” Phys. Educ. 52, 15010 (2016).https://doi.org/10.1088/1361-6552/52/1/015010

work page doi:10.1088/1361-6552/52/1/015010 2016

[7] [7]

Littleton, Richard Secco; Smartphones and Gravitational Acceleration II: Applications

Meryem Berrada, Joshua A.H. Littleton, Richard Secco; Smartphones and Gravitational Acceleration II: Applications. Phys. Teach. 1 October 2020; 58 (7): 473–476. https://doi.org/10.1119/10.0002064

work page doi:10.1119/10.0002064 2020

[8] [9]

Compensating smartphone accelerometers for more precise classroom experiments

Hoon Yu, “Compensating smartphone accelerometers for more precise classroom experiments.” Phys. Teach., 63, 189-192. (2025) https://doi.org/10.1119/5.0202891

work page doi:10.1119/5.0202891 2025