Conference Session

Communication and Literacy

Collection

2015 ASEE Annual Conference & Exposition

Authors

Kai Jun Chew, Stanford University Designing Education Lab; Autumn Turpin, Stanford University

Tagged Divisions

Educational Research and Methods

result for aninter-rater reliability test that includes more than one code, while Cohen’s kappa provides aresult for only one code.With the coding done, the next step is to evaluate the quality of the abstracts written, which wasevaluated through the process of Excerpt Grading. Excerpt GradingThe excerpt grading process was completed to assess the quality of the abstracts analyzed. Fromthe coding process, the sentences or section of an abstract that are categorized to a code areknown as excerpts. This means each code contains excerpts coded from the abstracts. This isexplained in the sample abstract shown below

Conference Session

Curriculum and New Course Development in ET

Collection

2015 ASEE Annual Conference & Exposition

Authors

Fabian Hadipriono Tan P.E., The Ohio State University; Adrian Hadipriono Tan, The Ohio State University

Tagged Topics

Diversity

Tagged Divisions

Engineering Technology

Paper ID #12431A Course in History of Ancient EngineeringProf. Fabian Hadipriono Tan, The Ohio State University Fabian Hadipriono Tan has worked in the areas of construction of infrastructures and buildings, failure assessment of buildings and bridges, construction accident investigations, forensic engineering, ancient buildings, ancient bridges, and the ancient history of science and engineering for over 40 years. The tools he uses include fault tree analysis, fuzzy logic, artificial intelligence, and virtual reality.Adrian Hadipriono Tan, The Ohio State University Adrian H. Tan is a graduate student at the Ohio State

Conference Session

Novel Teaching Methods in a Multidisciplinary Context

Collection

2015 ASEE Annual Conference & Exposition

Authors

Willard W. Neel, Virginia Military Institute; Jon-michael Hardin P.E., Virginia Military Institute

Tagged Topics

Diversity

Tagged Divisions

Multidisciplinary Engineering

fits the description given by Ammianus. This design project can serve as a starting point for Page 26.70.19students to determine the right modern rope to use.Pedagogical ConclusionsAs noted earlier, this historically-based multidisciplinary design and fabrication project wasimplemented to foster the targeted engineering skill sets/learning outcomes of critical thinkingand the ability to determine and apply applicable engineering analysis techniques, and provide aninteresting way to increase student interest in the engineering design process. To assess theimpact of the Onager project on the reinforcement and enhancement of the targeted skillsets

Conference Session

First-year Programs Division Technical Session 5: Using Video, Games, and More in the First Year

Collection

2015 ASEE Annual Conference & Exposition

Authors

Krista M Kecskemety, Ohio State University; Gregory Warren Bucks, University of Cincinnati; Kerry L. Meyers, Youngstown State University

Tagged Divisions

First-Year Programs

Society of Engineering Education, 20124. Anderson, L.W. et al. A Taxonomy for Learning, Teaching, and Assessing. Addison Wesley Longman, Inc., Illinois, 2001.5. Bland, L. “Applying Flip/Inverted Classroom Model in Electrical Engineering to Establish Life-Long Learning,” in Proceedings of the Annual Conference of the American Society of Engineering Education, 2006.6. Redekopp, M.W. and Ragusa, G. “Evaluating Flipped Classroom Strategies and Tools for Computer Engineering,” in Proceedings of the Annual Conference of the American Society of Engineering Education, 2013.7. Stickel, M., Hari, S., and Liu, Q. “The Effect of the Inverted Classroom Teaching Approach on Student/Faculty Interaction and Students’ Self

Conference Session

Circuits and Systems Education 2

Collection

2015 ASEE Annual Conference & Exposition

Authors

Brian J Skromme, Arizona State University; Dan Robinson

Tagged Divisions

Electrical and Computer

sources. We use the DIRECT 1.0 conceptinventory of Engelhardt & Beichner to assess conceptual understanding, administering it as botha pre- and post-test in 20 sections of a linear circuits class involving over 1100 students over twoyears. Pre-test scores are around 50% as found by others. Post-test scores typically rise to only59% (averaged over many instructors), showing that conventional instruction does not addressqualitative misunderstandings very effectively. By introducing targeted instruction in onesection to address misconceptions, the post-test score rose to 69% in Spring 2013 (higher thanany other section) and with further refinement reached 79% in Fall 2013.1. IntroductionIntroductory linear circuit analysis is one of the most

Conference Session

Course Development / Curriculum Development

Collection

2015 ASEE Annual Conference & Exposition

Authors

Alisa Gilmore P.E., University of Nebraska, Lincoln

Tagged Divisions

Computers in Education

into Computer and Electronics Engineering Programs. A final evaluation report for the National Science Foundation. 3. Gilmore, Chen, and Grandgenett, “Using Robotics to Equip K-12 Teachers: The Silicon Prairie Initiative for Robotics in Information Technology”, ASEE 2009 4. Gilmore, Detloff, “Assessing Senior Student Experiences with a Novel Mobile Robotics Learning Platform in a Computer and Electronics Engineering Program”, ASEE 2010 5. Gilmore, Santos, and Mills, “Computer Interface Innovations for an ECE Mobile Robotics Platform Applicable to K-12 and University Students”, ASEE 2011-2200 6. Chen, B., Grandgenett, N., Ostler, E., “Silicon Prairie Initiative for Robotics in Information

Conference Session

Microcontrollers, Programming, and Data Acquisition

Collection

2015 ASEE Annual Conference & Exposition

Authors

Sean Patrick Kirby, University Of Maryland, Eastern Shore; Julius Omokiniovo Ejenavi, Morgan State University; Payam Matin, University of Maryland, Eastern Shore

Tagged Divisions

Mechanical Engineering

Conference Session

Building Pathways that Promote Pursuit/Persistence in Engineering

Collection

2015 ASEE Annual Conference & Exposition

Authors

Ebony Omotola McGee, Vanderbilt University; William H. Robinson, Vanderbilt University; Lydia C. Bentley, Vanderbilt University; Stacey Houston II, Vanderbilt University

Tagged Topics

Diversity

Tagged Divisions

Minorities in Engineering

five years, despite intervention programs that aim tobroaden the participation of minorities in engineering. This three-year study explores the barriersand opportunities facing a cohort of: (1) African American engineering PhD students,candidates, and postdocs pursuing engineering faculty careers; (2) African American engineeringtenure-track and tenured faculty; and (3) Minority/Diversity Engineering Program Directors.This study examines factors that impact the production of African American PhDs inengineering, as well as those factors that affect the pathway to tenured faculty positions inengineering. This research includes an assessment of the current engineering faculty climatethrough surveying and interviewing African Americans in

Conference Session

Fundamental: K-12 Students and Engineering Design Practices (Part 2)

Collection

2015 ASEE Annual Conference & Exposition

Authors

Lija Andrija Yang, Tufts Center for Engineering Education and Outreach; Aaron W. Johnson, Tufts University; Merredith D Portsmore, Tufts University

Tagged Divisions

K-12 & Pre-College Engineering

of Understanding the Challenge) and idea generation practices (GeneratingIdeas) when engaged in a more open-ended problem that required them to identify materials,define constraints, and develop their own criteria for success.11 M. McCormick and Hynes alsoshowed that students have the ability to expertly navigate an ill-defined problem space by relyingon and using their own “lived experiences.”12 This lack of consensus suggests that more work isneeded to understand the dynamics of students’ engineering practice in these areas, and how theyare influenced by the nature of the design problem presented to them. This in turn will require us,and others to look at instruction and assessment for engineering curricula.The study described in this

Conference Session

Engineering Physics & Physics Division Technical Session 3

Collection

2015 ASEE Annual Conference & Exposition

Authors

Yu Gong, Purdue University; Tugba Yuksel, Purdue University; Alejandra J. Magana, Purdue University; Lynn A. Bryan, Purdue University

Tagged Divisions

Engineering Physics & Physics

simulation tools for quantum mechanics learning?Method and Research DesignThis work is part of Quantum Learning in Engineering And Physics (Quantum LEAP)project. The Quantum LEAP project aims to develop an integrated framework for the designand assessment of effective simulation-based learning environments for quantum educationbased on studies about engineering and physics students’ non-normative conceptions andmetacognitive learning strategies.Research Design. The presented study is guided by interpretive research design. Interpretiveresearch design enables the researcher to presume that knowledge and understanding areresults of interpretation and based on individual’s subjective experiences12. Interpretiveresearches consider that knowledge and

Conference Session

Identity and Engineering: ERM Roundtable

Collection

2015 ASEE Annual Conference & Exposition

Authors

Renata A Revelo, University of Illinois, Urbana-Champaign

Tagged Topics

Diversity

Tagged Divisions

Educational Research and Methods

unexplored3,4,5. This paper addresses this literature gap and aims to broaden theconceptualization of engineering identity by studying the development of engineering identity ofLatina/o undergraduates from their perspective and experiences. The forecasted growth of Latinas/os in the United States has encouraged a range ofinstitutions to assess how this shift in population will affect various programs of study especiallythose in science, technology, engineering, and mathematics (STEM) where Latinas/os arecurrently underrepresented. In engineering, the number of Latina/o students enrolling hasincreased since the 1990s and it is projected to continue to increase, though not at the same rateas the Latina/o population growth. Engineering is one

Conference Session

Technical Session: Pedagogical Strategies and Classroom Techniques for Teaching Assistants

Collection

2015 ASEE Annual Conference & Exposition

Authors

Charles Algeo Wilson IV, Louisiana State University; Adrienne Steele, Louisiana State University; Warren N. Waggenspack Jr., Louisiana State University; Wei-Hsung Wang Ph.D., Louisiana State University; Linda Lee Ramsey, Independent evaluator

Tagged Divisions

Student

Partnership for Transforming the Learning Environment. Proceedings from April 2004 Conference Cosponsored by the National Science Foundation (NSF) Division of Undergraduate Education (DUE) and the American Association for the Advancement of Science (AAAS) Directorate for Education and Human Resources Programs (HER): Invention and Impact: Building Excellence in Undergraduate Science, Technology, Engineering and Mathematics (STEM) Education. Crystal City, VA.[6] Loui, M.C., Robbins, B.A., Johnson, E.C., and Venkatesan, N. (2009). “Assessment of peer-led team learning in an engineering course for freshmen”, Retrieved January 31, 2015 from http://brettarobbins.com/papers/PLTLQuant.pdf.[7] Otero, V., Pollock, S

Conference Session

Engineering Ethics Division Technical Session 2

Collection

2015 ASEE Annual Conference & Exposition

Authors

Marilyn A. Dyrud, Oregon Institute of Technology

Tagged Divisions

Engineering Ethics

previously published papers: An exploratory study of potential self-plagiarism. Psychological Reports, 97(1), 43-49.7. Elbeck, M. (2009, Sumer). Exploring the murky waters of self-plagiarism. Journal for the Advancement of Marketing Education, 14, 41-51.8. Martin, B. (2013). Whither research integrity? Plagiarism, self-plagiarism and coercive citation in an age of research assessment. Research Policy, 42, 1005-1014.9. Scanlon, P. M. (2007). Song from myself: An anatomy of self-plagiarism. Plagiary: Cross-Disciplinary Studies in Plagiarism, Fabrication, and Falsification, 2, 57-66.10. Roig, M. (2011, August). On text re-use and the scholarship of science. European Science Editing, 37(3), 73- 76.11. Butler, D

Conference Session

Systems Engineering Division Technical Session 1

Collection

2015 ASEE Annual Conference & Exposition

Authors

Wolter J. Fabrycky PE, Virginia Tech and AAII

Tagged Divisions

Systems Engineering

new alternatives sought. Alternatives that meet all, or the most important,performance criteria can then be evaluated based on estimations and predictions of DDP values.This should be accompanied by an assessment of risk.Within the context of synthesis, analysis, and evaluation is the opportunity to implement systemsengineering over the life cycle in measured ways that can help ensure its effectiveness inprofessional practice. It is a morphology for linking applied research and technologies (Block 0)to customer needs (Block 1). It also provides a structure for visualizing the technologicalactivities of synthesis, analysis, and evaluation. Each of these activities is summarized in theparagraphs that follow, with reference to relevant blocks

Conference Session

Technical Session: Professional Development Opportunities for Students

Collection

2015 ASEE Annual Conference & Exposition

Authors

Kathleen A. Harper, The Ohio State University; Hannah Christine Zierden, The Ohio State University; Kevin Robert Wegman; Rachel Louis Kajfez, The Ohio State University; Krista M. Kecskemety, The Ohio State University

Tagged Divisions

Student

at Virginia Tech. Paper presented at the 36th Annual Frontiers in Education Conference, San Diego, CA.5 Nicklow, J. W., Marikunte, S. S., & Chevalier, L. R. (2007). Balancing pedagogical and professional practice skills in the training of graduate teaching assistants. Journal of Professional Issues in Engineering Education and Practice, 133(2), 89-94.6 Kajfez, R. L. & McNair, L. D. (2014). Graduate student identity: A balancing act between roles. Paper presented at the 121th American Society for Engineering Education Annual Conference & Exposition, Indianapolis, IN.7 Guba, E. G. (1981). Criteria for assessing the trustworthiness of naturalistic inquiries, Educational Communication

Conference Session

Topics Related To Engineering Design Graphics Division

Collection

2015 ASEE Annual Conference & Exposition

Authors

Edward E. Osakue, Texas Southern University

Tagged Divisions

Engineering Design Graphics

for revolution operation. Some assessment must be done for each segment or feature to determine whether it should be constructed by extrusion or revolution. However, extrusion is more versatile and should be used when revolution is not suitable. Virtually any solid form can be created by a series of extrusion processes. Cross-sectional view profiles are used mainly in extrusion operation while longitudinal section profiles are used in revolution operations. Use extrusion for simple profiles of polygons, non-symmetrical, and non- rounded segments. Use revolution for externally rounded segments that have complicated internal contours.5. Assemble solid segments and features into part: After all solid segments

Conference Session

Multidisciplinary Approaches for Enhancing Non-technical Skills

Collection

2015 ASEE Annual Conference & Exposition

Authors

Marissa H. Forbes, University of Colorado, Boulder; Angela R. Bielefeldt, University of Colorado, Boulder; Jacquelyn F. Sullivan, University of Colorado, Boulder

Tagged Topics

Diversity

Tagged Divisions

Multidisciplinary Engineering

Technology (DSST), where she also created and taught a year-long, design-based DSST engineering course for seniors. Before beginning her PhD work, Marissa returned to CU, where she supervised K-12 fellows and worked with educators engaged in the creation of K-12 engineering curriculum for the TeachEngineering digital library.Dr. Angela R Bielefeldt, University of Colorado, Boulder Angela Bielefeldt is a professor at the University of Colorado Boulder in the Department of Civil, Envi- ronmental, and Architectural Engineering (CEAE). She serves as the Associate Chair for Undergraduate Education in the CEAE Department, as well as the ABET assessment coordinator. Professor Bielefeldt is the faculty director of the Sustainable

Conference Session

Focus on the Classroom: Novel Approaches to Course Delivery

Collection

2015 ASEE Annual Conference & Exposition

Authors

Gary J. Mullett, Springfield Technical Community College

Tagged Topics

Diversity

Tagged Divisions

Two Year College Division

. Noncontact vital sign detection has beendemonstrated by researchers all over the world using radar systems. Radar sensing can beutilized for indoor detection of human falls which allows for timely and accurate reporting of fallincidents which can become severe if not dealt with in a timely (immediate) manner. Accuraterespiration measurements and monitoring also lends itself to many other important applicationssuch as: lie detection, psychotherapy, assessment of human physical states, and speechmonitoring. Respiration is a natural human activity; however, it is not uniform but changes inassociation with a human’s physical state. Conventional medical monitoring techniques aremostly based on contact methods and therefore are not as reliable or

Conference Session

Identity and Engineering: ERM Roundtable

Collection

2015 ASEE Annual Conference & Exposition

Authors

Allison Godwin, Purdue University, West Lafayette; Gerhard Sonnert, Science Education Department, Harvard-Smithsonian Center for Astrophysics; Philip Michael Sadler, Harvard Smithsonian Center for Astrophysics

Tagged Topics

Diversity

Tagged Divisions

Educational Research and Methods

interests include assessment of students’ misconceptions and how they change with instruction, K-12 curriculum development, the transition to college of students who wish to purse STEM careers, and the professional development teachers. Dr. Sadler won the Journal of Research in Science Teaching Award, the AIP’s Computers in Physics Prize, the American Astronomical Society Education Prize, and the AAPT’s Millikan Medal. He holds five patents. Materials and curricula developed by Dr. Sadler are used by an estimated fifteen million students every year. Page 26.1552.1 c American Society

Conference Session

Online Teaching

Collection

2015 ASEE Annual Conference & Exposition

Authors

Larry Lagerstrom, Stanford University; Petr Johanes, Stanford University; Umnouy Ponsukcharoen, Stanford University

Tagged Divisions

Computers in Education

, L., D.E. Pritchard, J. DeBoer, G.S. Stump, A.D. Ho, and D. T. Seaton. 2013. Studying learning in the Page 26.1558.16worldwide classroom: Research into edX’s first MOOC. Research & Practice in Assessment, 8(1), 13-25.2. Guo, P. J., J. Kim, and R. Rubin. 2014. How video production affects student engagement: An empirical study ofMOOC videos. Paper presented at L@S 2014, March 4–5, 2014, Atlanta, Georgia, USA.3. Kim, J., P.J. Guo, D.T. Seaton, P. Mitros, K.Z. Gajos, and R.C. Miller. 2014. Understanding in-video dropoutsand interaction peaks in online lecture videos. Paper presented at L@S 2014, March 4–5, 2014, Atlanta

Conference Session

College Industry Partnerships Division Technical Session 2

Collection

2015 ASEE Annual Conference & Exposition

Authors

Mark Angolia, East Carolina University; Leslie Pagliari, East Carolina University; James Kirby Easterling, Eastern Kentucky University School of Business

Tagged Divisions

College Industry Partnerships

) Department withinsix months of hire. The current Executive-in-Residence is assessing options to pursue a PhD orDoctor of Business Administration (DBA) program through an AACSB-accredited BusinessSchool which offers a flexible format as a means to advance into a tenure-track position.5.1 Earning a Terminal Degree Post Career ChangeCritical to obtaining an assistant professor position is completion of a terminal degree. If afixed-term faculty is fortunate enough to work at a University offering a relevant PhD, the task ismade easier, although still daunting. Table 5 is a summary of online PhD or DBA programsfrom traditional universities that include a residency requirement accommodating enough tocontinue working in a full-time position. While most

Conference Session

Promoting Engineering and Technological Literacy

Collection

2015 ASEE Annual Conference & Exposition

Authors

John Heywood, Trinity College Dublin, The University of Dublin; Mani Mina, Iowa State University

Tagged Divisions

Technological and Engineering Literacy/Philosophy of Engineering

the United Kingdom, South Hampton, Department of mechanical engineering university of Southhapton 1982.[24] Vinck, D (ed) (2003). Everyday Engineering. An Ethnography of Design and Innovation. Cambridge, MA. MIT Press.[25] Trevelyan, J (2014). The Making of an Expert Engineer. London, Taylor and Francis (CRC Press).[26] SCANS (1992). Learning a Living. A Blueprint for Higher Performance. Washington DC. US Department of Labour.[27] Heywood, J (1994) Enterprise Learning and its Assessment. Technical report No 20. Learning Methods Branch. Sheffield. Employment Department Page 26.1572.14

Conference Session

Women in Engineering Division: Retention of Undergraduate Students

Collection

2015 ASEE Annual Conference & Exposition

Authors

Kimberly Grau Talley P.E., Texas State University; Araceli Martinez Ortiz, Texas State University

Tagged Topics

Diversity

Tagged Divisions

Women in Engineering

Engineering degrees from The University of Michigan and Kettering University. She holds a Masters degree in Educa- tion from Michigan State and a PhD in Engineering Education from Tufts University. Her research inter- ests include studying the role of engineering as a curricular context for mathematics and science learning in K-20 and developing research-based active-learning instructional models and assessment instruments to enhance engineering students’ learning experiences and STEM Teacher professional development. She works with teachers, families, and students from underrepresented communities. Page

Conference Session

Multidisciplinary Approaches to Sustainability and Grand Challenges

Collection

2015 ASEE Annual Conference & Exposition

Authors

Richard M. Goff, Virginia Tech

Tagged Topics

Diversity

Tagged Divisions

Multidisciplinary Engineering

to choosewhat aspect of the streamlined motorcycle fuel economy competition they would pursue andwhat they would design and fabricate to move the project forward. To assess their interest in andengagement in as well as their general impression of the project, they were asked to answerquestions on their interest in and engagement in the project at three different points in the projectas well as their overall reflection on the experience.Student Survey Response DataThe Four Survey Questions asked Students at the A. beginning, M. the middle, and F. finally atthe end of the REU experience:1. When you first saw the VT Student Engineering Education Collaboratory (SEEC) Project “Critical Engineering Challenges”, what did you think this project

Conference Session

Mechanics of Materials

Collection

2015 ASEE Annual Conference & Exposition

Authors

William E. Howard, East Carolina University; Rick Williams, Auburn University; Sarah Christine Gurganus, NAVAIR Fleet Readiness Center East

Tagged Divisions

Mechanics

disciplineduring the final phase. During the year for which the assessment results are reported in thispaper the first three weeks were spent in the rotations (two hours in each of the three disciplineseach day) and the final week was devoted to a focused project in one discipline. The bracketproject was included in the first phase of the program, and so all 31 students with engineering asone of their rotation disciplines completed the project.The engineering portion of the program began with instruction in the use of SolidWorks solidmodeling software. Very few of the high school students had used a solid modeling program, sothe first two-hour session of the summer provided basic instruction in creating parts. During thesecond session, students were given

Conference Session

New Engineering Educators Division Technical Session 2

Collection

2015 ASEE Annual Conference & Exposition

Authors

Ted Eschenbach P.E., University of Alaska Anchorage; Neal A Lewis, University of Bridgeport; Gillian M. Nicholls, Southeast Missouri State University; William J. Schell IV P.E., Montana State University

Tagged Divisions

New Engineering Educators

enhanced by using agile to remain flexible in how courses are designed, delivered, and assessed. Rather than sticking to a rigid plan for delivering X amount of content in Y period of time, deciding upon the critical content areas and structuring delivery to suit learning by the students can be more efficient and effective.  The selection of a set of service opportunities to meet career, departmental, college, and/or university requirements is an essential part of success in academia. A specific service commitment that resembles a project may also benefit from an agile approach when the scope is unclear or adjustable as customer requirements become more apparent.  Reading just a few reference sources

Conference Session

First-year Programs Division Technical Session 4: The Best of the All: FPD Best Papers

Collection

2015 ASEE Annual Conference & Exposition

Authors

Mary A. Fraley, Michigan Technological University; A. J. Hamlin, Michigan Technological University; Amber Kemppainen, Michigan Technological University; Gretchen L. Hein, Michigan Technological University

Tagged Divisions

First-Year Programs

student readiness and performance. The responses on the studentpreparedness surveys for the different groups will be compared to see if the developed materialsimproves student readiness. The three groups were:  No change to the current course format  Students could watch the videos  Students could watch the videos and they had to complete a short, on-line quiz based on the video content.Additionally, student learning will be assessed by comparing performance on exam scorespertaining to topics for the fall 2013 traditional reading group and the fall 2014 experimentalgroup.Introduction and BackgroundAs engineering education evolves, a new way of student learning has developed called blended

Conference Session

College Industry Partnerships Division Technical Session 2

Collection

2015 ASEE Annual Conference & Exposition

Authors

Waddah Akili, Iowa State University

Tagged Topics

Diversity

Tagged Divisions

College Industry Partnerships

stressful for somecandidates, is an invaluable means for assessing candidate’s teaching skills. A scheduledpresentation is also an opportunity for regular faculty to meet adjunct candidates, before andafter the seminar, to get to know the candidate and discuss matters of mutual interest, includingpotential future collaboration. (1)Some of the adjunct faculty-particularly those who are seniors in specific industries- could offerimportant linkages for the development of industrial affiliate programs, co-op activities, summertraining opportunities, and employment opportunities for new graduates. They may also providenew ideas for senior design projects, topics for graduate theses, or render help in theestablishment of collaborative research

Conference Session

Mobile Devices and Apps

Collection

2015 ASEE Annual Conference & Exposition

Authors

Mohammadjafar Esmaeili, University of Dayton; Ali Eydgahi, Eastern Michigan University; Ilkhomjon Amanov, Eastern Michigan University

Tagged Topics

Diversity

Tagged Divisions

Computers in Education

Conference Session

Engineering Physics & Physics Division Technical Session 4

Collection

2015 ASEE Annual Conference & Exposition

Authors

Jeffrey C. Hayen, Oregon Institute of Technology

Tagged Topics

Diversity

Tagged Divisions

Engineering Physics & Physics

problem may be solved by the method of Lagrange multipliers [9], but such aneffort is beyond the scope of this article. The values of o and i required to satisfy Eqs. (45)and (46) have been numerically determined [7] for various values of  . With these values, themaximum range R  can be evaluated via the relation for R ( o , i ) in Eqs. (46). These resultsare utilized in the next section to assess/judge the accuracy of the analogous results generated viathe cubic law.Maximum Projectile Range: Approximate AnalysisAs described above, y  0 at x  R (with R  0 ). The cubic law, Eq. (30), then requires that a R 2  b R  o  0 (47)where the expressions for a and b were