61 Ways to Measure the Height of a Building with a Smartphone
Frederic Bouquet, Giovanni Organtini, Amel Kolli, Julien Bobroff
661 Ways to Measure the Height of a Building with a Smartphone
F. Bouquet, A. Kolli, and J. Bobroff
Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.
G. Organtini
Sapienza Universit`a di Roma & INFN Sez. di Roma (Dated: October 23, 2020)We imagined and tested 61 methods to measure the height of a building using a smartphoneand everyday low-cost equipment. This open question forces students to explore various fields ofphysics and confront them with experimental questions such as the validity of a model, the notionof uncertainty, and precision. It allows them to compare different experiments, and can be shapedinto various pedagogical scenarios, engaging students in a concrete task, outside of the lab, easilyset up at almost zero cost.
I. INTRODUCTION
In the last years, it has been demonstrated that smartphones can be used to perform physics experiments:[1–4]they contain many sensors[5] whose data can be accessed with easy-to-use applications. Smartphones can be usedfor various reasons: to reduce the cost of an experiment, for distance learning, to engage students, and even, morerecently, to allow experimental teaching during lockdown.[6] But smartphones have their disadvantages as well: theirready-to-use “black box” interfaces could prevent students from learning good experimental practices, including thenotions of precision, uncertainty, and how to build an experimental setup. We developed a specific teaching to getundergraduate students face three basic questions: • What physics experiment should be carried out to test a model? • What is the precision of the measure, and why it is a key issue? • How is it possible that an experimental result does sometimes not follow the model’s predictions?We based our teaching on the famous urban legend of Niels Bohr and the barometer.[7] When asked how to measurethe height of a building with a barometer, a student — young Niels Bohr — invents a dozen or so experiments thatdo respond to the question but avoid the solution expected by the teacher.We revisited the barometer question as: “How many different ways are there to measure the height of a buildingwith a smartphone?” Unlike Bohr’s legend, we further asked our students to carry out the experiments they thoughtof and to evaluate how the results compare with one another. It turns out that smartphones allow to perform manymore experiments; some give surprisingly good results, others very bad ones. The physics span involved in thesemethods covers many fields: mechanics, magnetism, optics, ...The aim of this article is to show how, using only low-cost equipment, this challenge lets students face directly thequestions of good experimental practices. We first present and compare the 61 methods and discuss how to carrythem from a practical point of view. We then discuss ways to use these experiments in various teaching contexts andhow this could renew the usual approach to labs in undergraduate curricula.
II. HOW TO CONVERT BOHR’S LEGEND INTO REAL LABS
We translated the Bohr’s legend into simple constraints: measuring the height of a building several meters highusing only every-day objects and a smartphone (with its internal sensors: accelerometer, gyroscope, light sensor,magnetometer, GPS, barometer, microphone and camera). We mainly use the “phyphox” app[4] to access the sensorsmeasurements because it is easy to use and is available in many languages, but other apps would also work well.We found 61 different methods. Our sample of investigation was the 15-m high Laboratoire de Physique des Solides(see Figure 1). Each method is labelled by an arbitrary number.[8] We carried out 46 of the 61 methods. Some wedid not carried out (labelled with a star) because of lack of material or technical limitations, see Table I.In order to accurately compare the different methods, a reference altitude was taken at the top of the building. Wechose to quote uncertainties by just deriving them from the device sensitivity. No statistical analysis of data nor thestudy of systematic errors have been done, but some other error sources are discussed. a r X i v : . [ phy s i c s . e d - ph ] O c t FIG. 1. Measurement of the height of the building using method
TABLE I: List of the 61 methods. The methods are described in the text. A resultgiven as 0 means that the experiment was carried out but the height could not becalculated. When the result is not given and the number is starred (e.g. e.g. . ± .
12 Free fall of an object, using a stopwatch 16 ±
23 Free fall of an object, using video analysis 17 . ± .
64 Free fall of an object, using audio analysis 18 . ± .
35 End velocity of the free fall of an object, video analysis 13 . ± .
76* End velocity of the free fall of a speaker, Doppler analysis ∗
7* Distance of landing for an object thrown horizontally ∗ . ± .
59 Multiple rebounds of a ball, audio analysis 6 . ± .
510 Giant pendulum, using a stopwatch 14 . ± .
411 Giant pendulum, video analysis 14 . ± .
412 Giant pendulum, using the accelerometer 013 Giant pendulum, using the gyroscope 14 . ± .
414 Giant pendulum, using the magnetometer 14 . ± .
415 Giant pendulum, using the light sensor 14 . ± .
416 Giant pendulum, using the proximity sensor 14 . ± . ∗
18 Giant torsional pendulum, any sensor 10 . ± . ∗
20* Relation between angular velocity and velocity (on a giant pendulum) ∗
21 Thales’ method with shadows 16 . ± .
322 Shadow length and sun elevation from GPS data 14 . ± .
323 Shadow length and sun elevation at the equinox 13 . ± .
224 Measuring the angle from eye level to the top with the accelerometer 15 . ± .
425 Measuring the angle from the bottom to the top with the accelerometer 40 ± . ± .
027 Measuring the angle of view of an object on the ground from a picture 13 . ± .
828 Picture with a scale of the building 14 . ± .
129 Picture of the building knowing the specifications of the camera 14 . ± .
230 Picture of an object on the ground from the top knowing the specifications of the camera 14 . ± .
231 Length of a rope along the facade 14 . ± .
132 Length of a rope along the fa¸cade, using a pulley and the gyroscope 15 . ± . ∗
34 Piling up smartphones along the facade 14 . ± .
35 Number of stairs to the top 15 . ± .
236 Variation of atmospheric pressure 15 . ± . ∗
38 Altitude difference from the GPS 8 ± . ± .
340 Sound time of flight, using two synchronized audio recordings 14 . ± .
441 Sound time of flight, using two audio recordings synchronized by a phone call 10 . ± .
442 Sound time of flight, from the echo ∗
43 Sound time of flight, using slow-motion movie 15 . ± .
044 Audio phase shift along the facade of a single frequency 14 . ± .
345 Audio phase difference from the top and the bottom when changing the frequency 13 . ± . ∗
47* Acoustic interferences at the top created by two speakers at the bottom ∗
48* Resonance of a tube along the facade ∗
49* Decrease of sound intensity with distance ∗
50 Decrease of light intensity with distance 15 . ± .
551 Decrease of Wifi intensity with distance 23 ± ∗
53 Decrease of radioactive intensity with distance ∗
54 Decrease of the surface on a picture with distance 14 . ±
155 Projection of a hair diffraction pattern from the top to the ground 16 . ± .
756 Projection of a smartphone screen diffraction pattern from the top to the ground 15 . ± .
257 Variation of gravity between the top and the ground, determined using small pendulums 0 ± ± E ± E
560 Variation of gravity between the top and the ground, determined by general relativity timedilatation 0 ± E ∗ The results are all presented in Table I; the precision of the methods varies greatly. Some methods give the correctheight with a reasonable precision given the tools at hands, for example, methods using picture analysis (
III. THE TEACHER’S SOLUTION
In Bohr’s legend, the teacher expects a specific solution: the barometer is used to measure the variation of at-mospheric pressure ∆ P between the top and bottom of the building. The altitude H of the building follows by H = ρg ∆ P , with ρ , the density of air, close to 1.2 kg/m . This method can be performed with smartphones that areequipped with a barometer sensor.[9] For a better measurement, a calibration of both the barometer and ρ can beperformed by measuring the difference of pressure between two points of known altitude, such as the distance betweenone’s head and feet. As seen on Table I, this method can be quite precise. IV. METHODS USING FREE FALL
When the air friction is neglected, the fall time t of an object with zero initial velocity gives the height[1, 10] through H = gt . Caveat: throwing objects from the top of a building is potentially hazardous; we used a tennis ball toattenuate the risks. The fall time of the tennis ball can be measured by different means: timed with a stopwatch app( v f of the falling object at impact, using H = v f / (2 g ). The easiestway to measure v f is to film the end of the fall and analyze the video frame by frame ( H = 18 . ± . H = 13 . ± . -100 0 1 2 3 a cc e l e r a t i on ( m / s ²) time (s) FIG. 2. Acceleration recorded by a smartphone during its fall from the top of the building (see Figure 1). Gravity was subtractedfrom the accelerometer raw data. The huge spikes after 2s correspond to the landing and oscillations of the smartphone in thelife net afterwards. The beginning of the fall defines t = 0. The fall time 1 . ± .
03 s leads to H = 18 . ± . g and tends to zero during the fall, even though theerratic rotation of the falling smartphone makes the curve non-monotonic. The dashed lines are the bracketing curves used todetermine H by integrating twice, leading to H = 14 . ± . effect of air friction, which reduces the impact velocity, making the building appear smaller in the latter method, orincreases the fall time, making it appear larger in the former.Letting the smartphone free fall itself is a way to take air friction into account ( H . One can use a bedsheet hold by two people tocatch the smartphone safely, like fireman life net. However, one should worry especially of unexpected winds. Whenwe threw our smartphone from the 15-m roof of the building, it rolled and looped during the fall, which explains thenot-monotonous acceleration curve we obtained (see Figure 2). Bracketing the value of acceleration by linear curvesgives H = 14 . ± . v i can be added to the ball when throwingit from the top of the building, and measuring v i using video analysis and the horizontal distance d the ball reachedat impact leads to H ( V. METHODS USING A GIANT PENDULUM
Since Galileo, pendulum of known length have been used to measure time. Building a giant pendulum the size ofthe building and timing its period T leads to H = g ( T / π ) . Some care should be given to the construction of thependulum, so that it swings nicely and does not rotate in every directions. The swing construction, with two wiresseparated by a gap helps a lot.[2]To measure the period, the simplest way is to use the smartphone stopwatch ( a c ( v of the pendulum ( ω through simple equations.[20] The accelerometer and gyroscope of a well oriented smartphone can measure a c and ω ,[16] and the velocity could be measured by either Doppler effect[11, 12] or beats between two speakers,[21]one swinging with the pendulum, one motionless on the ground. In practice however these methods are difficult toimplement on a setup the size of a building.A variant ( H = 10 . ± . VI. METHODS USING TRIGONOMETRY
In surveying, triangulation is a well-known technique to measure distances.[22] Since the smartphone can measureangles with the vertical using its accelerometer, different setups can be imagined using this principle.Facing the building, it is either possible to determine the angle to the top of the building knowing the distance tothe building ( H = 15 . ± . H = 40 ±
20 m). Toimprove precision, it is best to attach the smartphone to a tube, and use the latter as homemade theodolite sight.The former method yields better results if one is standing not too close and not too far from the building (a distancecorresponding roughly to the height of the building is good). The latter method, if performed from one’s height,standing on the ground, leads to a large uncertainty on H because the result will be highly sensitive to the measureof the angle to the bottom.A variant is to measure the apparent angle of an object of known size lying on the ground below from the top ofthe building ( VII. METHODS USING PHOTOGRAPHY
The most straightforward method is to take a picture of the building with an object of known size on the image,playing the role of a scale (
VIII. METHODS USING SPEED OF SOUND
A direct method is to record the burst of a balloon at the top of the building, and waiting for the ground echo( H = 15 . ± . FIG. 3. Video analysis of a slow motion movie taken from the top of the building, at 340 frames per second. Frames 81 and96 are represented in the two top panels; the balloon is burst around frame 81, and the sound reaches the camera on frame 96(the sound track is represented in the bottom panel, the vertical red lines correspond to the frames represented above).
Using a phone call to time the travel of sound is tempting, since the electromagnetic waves that carry phoneconversations are alike light. A person at the top of the building phones to someone at the bottom, and bursts aballoon. The person at the bottom records two bursts, one having travelled through air and one having being carriedout by the communication cell tower ( H = 10 . ± . IX. METHODS PHYSICS OF WAVES
Using the phase of sound provides additional methods. The measure of a phase difference is easily converted indistance using the speed of sound (the difference between phase velocity and group velocity is not relevant here).Two audio records are needed to determine a phase difference, and such measurements require more time, care, andanalysis skills than previous methods. We performed all sound-phase analysis using Audacity software.[27]A direct setup is to use two smartphones and a speaker emitting a pure continuous tone. At the beginning, they allare at the bottom of the building, at the foot of an external stair. Both smartphones are audio recording, one is lefton the ground, the other is slowly brought up to the top using the external stairs, still recording the continuous tone.Audio analysis of both audio records will show an increase of the phase difference between the two smartphones,[28]related to the distance from the top smartphone to the ground ( ≈ H = 14 . ± . δ Φbetween the two recordings when the frequency f of the tone is changed is related to H and the speed of sound v s through the relation: d( δ Φ)d f = 2 πHv s Plotting δ Φ as a function of f and determining the slope leads to H , though with larger errors than the previousmethod (we obtained H = 13 . ± . d from it ( δ Φd( d ) = πfHv s Another type of classical experiments is the standing wave experiments.[29] If one has a long tube running alongthe building fa¸cade, the determination of resonance frequencies should give H ( l . To ensure the phase coherence, both speakers should be driven by the same sound generator (a smartphone witha split jack connection for example). The sound intensity is measured at the top of the building. The distance d between the first minimums can then be found when moving laterally ( l, d ) (cid:28) H , the equation simplifiesinto H = ldf /v s .Light wave can also be used instead of sound waves: diffraction pattern of hair lighted by a laser is well known,[31]and the resulting pattern depends on the size between the screen (the ground at the bottom of the building) and thehair (at the top of the building). It will also depend on the diameter of the hair which can be determined using a dropof water on the smartphone lens to increase the magnification of the camera[32] ( X. METHODS BASED ON THE INVERSE-SQUARE LAW
The inverse-square law happens when a quantity is freely propagating from a punctual source without any othereffect (diffusion, absorption, reflection, interferences ...). After proper calibration, a measure of this quantity can beused to determine the distance to the source. Having the source at one point of the building and the measure at theother allows to determine H . Several quantities can be used, with varying degree of precision.Light is a prime quantity for this method, and it works well ( H = 15 . ± . H = 14 . ± . B ∝ /r ), and since somesmartphone can measure a magnetic field the same process of calibration/measurement could be done ( XI. DIRECT METHODS
Some methods rely on more direct approach rather than specific physics laws, generally in a simpler (but not alwaysprecise) way.Going up the stairs and counting how many smartphones should be piled up to reach the top is an easy and relativelyprecise way of determining H , if done with care on a convenient stairway ( H , which can then be measured with a meter( XII. METHODS THAT ONLY WORK IN THEORY
Keeping par with the Bohr’s legend, it is interesting to explore methods that would only work in an ideal worldof “spherical cows”. These methods only work in an idealized world where no perturbations are present, and requirevery precise measurements. Smartphones are obviously not the right tool, but some error bars can be calculated byestimating how tall the building would need to be for a signal to be measurable.For example, assuming Earth is a perfect magnetic dipole, using the magnetomer to measure the magnetic fieldat the top and bottom of the building should lead to H . A 15-m change of altitude would correspond to a 0.0008%change of magnetic field, below the standard smartphone sensor resolution. But Earth magnetic field is not exactlythat of a dipole, and more importantly the magnetic field created in a building in activity is not neglectible (especiallyin a physics lab hosting NMR experiments). When we did the experiment, we measured 25 µ T at the bottom of thebuilding and 38 at the top, a 40% difference which an ideal model would translate in a height of several hundreds ofkilometers.The variation with the altitude of the value of g is mentioned in Bohr’s legend as a possible way to determine theheight of the building (in reality, the variation of g also depends on the density of the building and its foundation[40]).Using a smartphone, one could then build two pendulums, and measure g through the period value ( g , assuming a 1-meter pendulum and a 0.1-second resolution in the period measurement givesan uncertainty of 3.2 km in the height value if using the former method. Using the latter method, our smartphoneaccelerometer had a slow drift of roughly 0.01 ms − , which corresponds to 3 km of uncertainty. We can safely concludethat for our building H = 0 ± g with altitude, general relativity tells us that time is not thesame at the top and at the bottom of the building:[41] if two smartphone stopwatches are started at the same timeat the bottom of the building and one of them is brought up to the top for a given time t , say 1 hour, then broughtdown, a delay δt should exist between the two stopwatches: δtt = (cid:104) g (cid:105) Hc with (cid:104) g (cid:105) the averaged value of g , and c the speed of light. Assuming that we are able to detect a 1 ms differencebetween the two stopwatches on a 1-hour experiment, this would result in an uncertainty of 3000000 km on H ! Thismethod seems farfetched, but atomic clocks do have the resolution to measure this effect on Earth (with an altitudedifference much higher than a building height). Special relativity tells us that when the clock is brought up and down,the speed of displacement v will also change the local time: the correction is given by v / c compared to the effect ofgeneral relativity gH/c . Back to the envelope calculations shows that the effect of velocity is neglectible. XIII. USING BOHR’S LEGEND TO CREATE ENGAGING TEACHINGS
Undergraduate or high-school labs are usually devoted to measure a specific law with a given setup. Here, the varietyof available methods allows a new approach by comparing different experiments and models. Table I clearly showsthat some methods are better than others in terms of precision. Letting students compare different measurements theycarried out themselves forces them to address the issue of uncertainty and the question of the underlying hypothesisin a more meaningful way for them than during a more traditional students’ lab or theoretical course. It also forcesthem to question the quality of their experiment design, not just taking it for granted.We tested this approach in various teachings over the past two years. In a first course, we left it as a complete openquestion where students had to invent and build their own way to measure the building’s height. They clearly enjoyedthe opportunity of getting out of the lab and doing measurements that had a direct and concrete meaning to theirdaily life. However, the proposed solutions were always the same (the barometer, a variation using the shadow of thebuilding, a picture of the building, and sometimes the elevator method). To force variety, we developed another typeof course where we proposed a subset of the 61 methods in “DIY” sheets[8] depending on our pedagogical objectivesand the available time and material. We found out that students were still engaged by the challenge and by thecreativity required to design the measurement (such as how to build a giant pendulum) even though it was no longer“their” idea.For example, in a 1.5-hour session, we asked 20 students to compare 10 different methods that cover different fieldsof physics and could be performed rapidly (
XIV. CONCLUSION
Open-ended activities in students’ laboratory activities have been shown to be more efficient than guided activitiesto develop a more expert-like behavior toward experimental physics.[42] We propose here the building’s height questionas an effective way to challenge students to invent their own ways to measure a quantity and confront their results.New activities could be developed and adapted from this question, challenging students to invent their own ways tomeasure a quantity and then to confront their results. This classroom activity can easily be adapted to other publics.0For example, we used it with high school physics teachers in the context of yearlong formation or in science museumoutreach activities. Last but not least, this “smartphone physics challenge” can be implemented for teaching at adistance, everyone carrying out experiments at home. This could serve for homework activities or as distant labssessions during lock-down periods such as the brutal ones experienced by many of us recently.
ACKNOWLEDGMENTS
We gratefully acknowledge Ulysse Delabre and Jo¨el Chevrier for discussions and inspiration, and Anna Kazhinafor her work on the graphic presentation of the methods. We thank the students who participated to these projects,K.V. Pham for welcoming this new teaching in the physics curiculum of Paris-Saclay University, and the phyphoxteam for their helpfulness. We thanks the Institut Villebon -
Georges Charpak for welcoming us and helping us todevelop new teachings. This work has been partially supported by the Erasmus+ teaching mobility programme. Italso benefited from the support of the Chair “Physics Reimagined” led by Paris-Saclay University and sponsored byAIR LIQUIDE, and also from a grant “p´edagogie innovante” from IDEX Paris-Saclay. [1] P. Vogt and J. Kuhn, “Analyzing free fall with a smartphone acceleration sensor,” The Physics Teacher, (3), 182–183(2012).[2] P. Vogt and J. Kuhn, “ Analyzing simple pendulum phenomena with a smartphone acceleration sensor,” The PhysicsTeacher, (7), 439–440 (2012).[3] J. Chevrier et al. , “Teaching classical mechanics using smartphones,” The Physics Teacher, (6), 376–377 (2013).[4] S. Staacks et al. , “Advanced tools for smartphone-based experiments: phyphox,” Physics Education, (4), 045009-1–6(2018).[5] A database of sensors installed in smartphones can be found on the phyphox Web Site