Developing a Quantitative Teaching Tool to Understand Ballistics Accelerated by Explosive Volcanic Eruptions
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V V V V V V O O O O O RESEARCH ARTICLE VOL O NI Trashcano: Developing a quantitative teaching tool to understand ballistics accelerated by explosive volcanic eruptions Fabian B. Wadsworth*α, Holly E. Unwinβ, Jérémie Vasseurγ , Ben M. Kennedyδ, Julia Holzmuellerγ , Bettina Scheuγ , Taylor Witcherγ , Janina Adolfγ , Francisco Cáceresγ , Ana S. Casasγ , Valeria Cigalaγ , Alexandra M. Clement", Mathieu Colombierγ , Shane Croninζ, Marcel Croninη, Donald B. Dingwellγ , Leticia Freitas Guimarãesγ, θ, Laura Höltgenγ , Ulrich Kueppersγ , Gilles Seropianδ, Sönke Sternγ , Adrien Teissierδ, Caron E. J. Vossenγ , Natalie Weichselgartnerγ αDepartment of Earth Sciences, Durham University, Durham, DH1 3LE, United Kingdom βDepartment of Earth Sciences, University of Oxford, South Parks Road, Oxford, OX1 3AN, United Kingdom γ Earth and Environmental Sciences, Ludwig-Maximilians-Universiät, Theresienstr. 41, 80333 Munich, Germany δGeological Sciences, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand " Elly und Stoffl, Röngenstrasse 15, 81679 Munich, Germany ζSchool of Environment, University of Auckland, 3A Symonds St, Auckland 1010, New Zealand ηBruguera Academy, Carrer de les Orenetes, 08690 santa Coloma de Cervelló, Barcelona, Spain θInstituto de Geociências, Universidade de São Paulo. Rua do Lago, 562, Cidade Universitária, São Paulo, SP CEP: 05508-080, Brazil Abstract Accurate predictions of volcanological phenomena, such as the trajectory of blocks accelerated by volcanic explo- sions, require quantitative skills training. Large outdoor experiments can be useful to convey concepts of volcanic processes to students in an exciting way. Beyond the fun aspects, these experiments provide an opportunity to engage with the physics of projectile flight and help promote mathematical learning within the Earth Sciences. We present a quantitative framework required to interpret ballistic trajectories and the outdoor experiment known commonly as “trashcano”, taking a step-by-step approach to the physics of this problem, and deriving a range of mathematical solutions involving different levels of complexity. Our solutions are consistent with the predic- tions from established computer programs for volcanic ballistic trajectory modelling, but we additionally provide a nested set of simplified solutions, useful for a range of teaching scenarios as well as downloadable simulated datasets for use where the full experiment may not be possible. Keywords: Geoscience education; Vulcanian eruption; Calculus; Science outreach 1 Introduction of “math” and develop quantitative skills necessary for their future work [Macdonald et al. 2000]. In this way Teachers, instructors, and lecturers in geoscience at all the geosciences may be a mechanism for understand- levels face a problem: that geoscience graduates are not ing physical science in the same way that the ecological necessarily meeting the needs of an academic or indus- or social sciences often lead into the statistical branch try workforce in terms of quantitative skill sets [Loudin of mathematics. To this end, we aim to provide an ex- 2004; Macdonald et al. 2000; Manduca et al. 2008; ample suite of exercises that help students from high Wenner et al. 2009; Wenner et al. 2011]. This problem school (secondary education), up to undergraduate and has the associated effect that many incoming graduate postgraduate levels, build quantitative skills applied to students see the geosciences as a more tractable and a specific problem in the geosciences. less-quantitative study choice compared with other nat- ural sciences [Manduca et al. 2008; Wenner et al. 2009]. Volcanology is a challenging geoscience topic to teach Conversely, a positive aspect of this problem, is that at any level, from high school (secondary education) undergraduate geoscience courses can offer a mecha- to post graduate levels in a university. The modern nism for students with less developed physical science subject is methodologically broad, incorporating as- backgrounds to break through some perceived hurdles pects of the natural and social sciences (see Sigurds- son et al. [2015] for an overview of the breadth of sub- *Corresponding author: [email protected] disciplines in volcanology). Even when concentrating Trashcano: a quantitative teaching tool Wadsworth et al., 2018 on physical or thermodynamic problems within vol- acceleration of particles such as ping pong balls (first canology, such as the dynamics of lava flow emplace- described by Harpp et al. [2005]). This can be used as ment or large eruption plumes, going beyond superfi- a simple demonstration of the phenomenon of projec- cial explanations brings in a raft of physical and chem- tile motions, or as a rigorous way to compare observa- ical complexities. Here, we address one such problem: tions with mathematical predictions, just as volcanol- the trajectory taken by a ballistic volcanic block accel- ogists aim to do during explosive eruptions. By using erated from an eruptive vent by an explosion. The un- trashcano, we simplify the problem of projectile trajec- derstanding of the trajectories of ballistics propelled by tories, stripping away many of the complexities asso- an explosion at a volcanic vent is crucial to our predic- ciated with real volcanic scenarios, but while keeping tion of the distribution of hazards around active vol- the relevant scales and physics and exploring the chal- canoes [e.g. Breard et al. 2014; Fitzgerald et al. 2014]. lenges associated with visual monitoring of projectiles. Indeed, the 63 fatalities caused by the September 2014 Some examples of successful and widely used, scaled eruption of Ontake volcano in Japan [Tsunematsu et al. in-class experiments or exercises for understanding 2016], highlight how hazardous ballistics can be. Bal- volcanic systems include fudge factors to explore lava listic analysis of projectile trajectories can also be key flow rheology [Rust et al. 2008], the M&M®magma to extracting important information about the source chamber for Bowen crystallization sequences [Wirth parameters of the initial explosion, such as the ener- 2003], the volcanic hazards simulation for training in gies, subsurface geometries, or explosion mechanisms crisis communication [Dohaney et al. 2015; Harpp and involved [Blackburn et al. 1976; Chouet et al. 1974; Fa- Sweeney 2002], and the example here, trashcano, for gents and Wilson 1993]. learning about explosive eruption dynamics [Harpp et In its essence, predicting the flight path of a vol- al. 2005]. Rust et al. [2008] found that using such prac- canic block is cognate with the problem of predicting tical exercises facilitated discussion and improved com- the trajectory of any other projectile in flight; a topic prehension of volcanological physical concepts, and familiar to students of classical mechanics and calcu- better performance in early education within physics or lus. There exist useful computer programs to solve this biology has also been found with practical hands-on ac- problem at the scales of volcanic environments, such tivities [Penner et al. 1998]. Indeed, many undergradu- as Eject! [Mastin 2001], which could be used as teach- ate volcanology programs involve fun and often large- ing tools. However, these do not necessitate proficiency scale class activities, not only to inspire and motivate with, or understanding of, the underpinning mathe- students but as a serious pedagogic strategy to improve matics of the problem, divorcing numerical skills from engagement and learning. While we do not undertake the output results. To tackle this, we develop a frame- an evidence-based approach to demonstrate the peda- work in which the full solution to the problem is al- gogic efficacy of our framework, we take the evidence most identical in its physical aspects to that given by for the success of active learning techniques as a moti- Eject! [Mastin 2001], but which has nested simplifica- vating starting point for designing our study. tions, developed step-by-step. In doing so we provide In this paper, we describe how to perform the trash- solutions with different levels of predictive accuracy, cano experiment, how to easily collect and process data which are suitable to different levels of mathematical from video observations of the explosion, and a mathe- ability. In their entirety, the nested approach developed matical and numerical framework to interpret the re- here demonstrates the mathematical steps an instructor sults. An illustrative analysis is provided as a guide or student can take from an initially complex problem, for the reader, but also to demonstrate the efficacy of to various examples of simpler, more manageable prob- this exercise. By providing added quantitative value to lems, and how to assess their respective validity. a widely used experiment, we also highlight the fit of Additional challenges facing teachers include the physics and mathematics to a modern volcanologist’s fact that large scale natural phenomena, including vol- toolbox and promote core STEM (science, technology, canic eruptions, are not easily adapted to in-classroom engineering, mathematics) skills that are the corner- experiments or demonstrations. While volcanic phe- stone of education policy in many countries around the nomena are pedagogically explicable from a purely world. theoretical perspective [e.g. Martin 1993], successful learning is thought to be increased when hands-on ex- periments, non-traditional techniques, or active learn- 2 Using Trashcano for teaching in four ing