ection 3.4Development of a Laboratory
ection 3.4
Development of a Laboratory
This Section suggests another area, highway traffic safety and design, which has technology problems that you could develop into your own Mathematics Journeys labs. It also gives an example of such a problem and offers a point of departure for creating a lab. The following partially formed lab is intended to guide the process of creating a lab by addressing a selected technology problem within the Mathematical Journeys framework. The problem, selected from the area of highway traffic safety and design, involves the design of a highway curve subject to safety constraints. A goal of Mathematical Journeys I is to provide faculty with a framework for creating their own labs around technology problems of their own choosing – problems in which they and/or their community are keenly interested.
The five components of a Mathematical Journeys lab – Technology Problem, Bibliotechnology Research, Mathematics Tools, Model Portfolio, Thesis Defense ‚ are closely interrelated and must be incrementally developed in tandem. The scientific method is a widely accepted habit of thought that underlies development of a lab. The five steps of the scientific method are: problem, question, hypothesis ("tentative answer" or "an explanation on trial"1, test of the hypothesis, and decision whether to retain, reject, or modify the hypothesis. In this simplified example, components of a lab are in boldface and steps of the scientific method are underlined.
Traffic safety is a major public policy concern. Traffic safety involves a range of interrelated issues such as vehicle characteristics, driver characteristics, highway laws, and highway design. Each of theses areas is a continuously evolving field of study that makes use of mathematics and statistics. This framework for developing a Mathematical Journeys lab addresses the highway design element of traffic safety and more specifically, geometric design of curves.
1Campbell, N.A. Biology, 3rd ed. Redwood City, CA: Benjamin/Cummings, 1993. Chapter 1, P. 15Technology Problem: The first two steps (problem, question) of the scientific method depend on exploring the situation and developing a better understanding of it. The purpose of this initial exploration is to obtain background information about the technology problem, to better understand the context of the problem, to give a sharper definition of the technology problem, and to develop useful questions whose answers further illuminate the technology problem. More information is needed. This leads to bibliotechnology research.
Bibliotechnology Research: There are a multitude of sources relative to the geometric design of highways. Key words or phrases for accessing pertinent information on the web include: highway safety, research on highway safety, traffic safety, safety and roadway design, Federal Highway Administration (FHA), National Highway Traffic Safety Agency (NHTSA), Interactive Highway Safety Design Module (IHSDM), Transportation Association of Canada (TAC), and American Association of State Highway and Transportation Officials (AASHTO). Each state in the United States has a department of transportation or bureau of highways that can provide data and information.
After visiting a few of the above sites, you should discover the need to learn more about the geometric design of roads. As you take this direction, you will probably encounter unfamiliar terminology or terminology used in unfamiliar ways, and want to start creating a glossary of terms and diagrams as needed. An example of some terms that you might include in the glossary are: horizontal alignment, horizontal curve, vertical curve, tangent, spiral, grade, superelevation, centrifugal force, compound curve, reverse curve, etc. Four web sources that would introduce you to road design are:
In your initial explorations, you will discover that curves can be categorized as horizontal or vertical. Let's assume that you now turn to a study of horizontal curves. This should lead you to research the layout of a simple, non-banked curve and prove asserted relationships between its parts. Layouts of simple curves appear in most publications that address geometric roadway design. Demonstration of the formulas may require reference to plane geometry and trigonometry texts. In the process of analyzing a simple curve, you should feel free to replace the given symbolism with your own if it makes the problem clearer and links it more closely with your own mathematical background. In any event, before proceeding to work on the layout of a horizontal curve, it would be helpful to study a number of pertinent mathematics tools.
Mathematics Tools: Basic algebra, formula development, unit conversions, scale drawing, geometry, basic analytic geometry, concept of slope, and trigonometry are needed to describe and interrelate characteristics pertaining to the design of horizontal curves. Also, to describe, mathematize, and communicate the characteristics, require variables and a glossary of terms. A working understanding of linear, quadratic, and piecewise functions together with a graphing calculator or pertinent software are needed to graph functions and produce tables of values. After considering these mathematics topics, you will be prepared not only to continue studying the layout of simple, non-banked curves and prove asserted relationships between their parts, but also to address problems that involve banked curves.
After working with the layout of a non-banked, horizontal curve, it may be worthwhile to construct a scaled model of a curve. Standard dimensions for such curves can be obtained from several sources. You should start with two lines that represent the centerlines of straight-of-ways that are to be connected by a curve. At equal distances from the point of intersection of the centerlines, construct perpendiculars to find the center of a circular arc that is tangent to the centerlines. Construct the arc and record its radius. From the centerline, you can locate the inside and outside edges of the curve. Be sure to state the scaling factor that you use.
Now that you have become somewhat familiar with horizontal curves, you are prepared to make some educated guesses about them. You could make hypotheses about the relationship between: the sharpness of a circular arc and its radius; the sharpness of a circular arc and the reciprocal of its radius; and the measure of the angle subtended by a 100 foot circular arc and the reciprocal of the radius of that arc. The next step is to devise ways to test your hypotheses and to carry out the tests. The last step in the scientific method is to make decisions about rejecting, retaining, or modifying your hypotheses. In this process of formulating and testing hypotheses, you should discover that the sharpness of a curve can be measured using either radius of curvature or degree of curvature. You will need to show how either measure can be expressed in terms of the other.
From the very beginning of a Mathematical Journeys lab, it is important to keep a record of all your sources, as well as the thought processes, arguments, and steps you use to solve the problem. These, as well as any physical models, contribute to the Model Portfolio. Your conclusion or thesis, together with all supporting materials or Thesis Defense is included in the model portfolio. Do not confuse the thesis defense with the oral presentation of the thesis defense to an audience, even though the latter often is referred to simply as "the thesis defense".
The model portfolio is an organized record of all your work on the lab, including any physical models. The record should be sufficiently detailed to allow a reader to follow your work step by step.
The next logical step is to consider the forces acting on a vehicle as it travels along a circular arc. As a vehicle travels along a curve in the road, it experiences an outward force known as centrifugal force. The friction force between the tires and the road counteract the centrifugal force. Another way to counteract the centrifugal force is to bank (or superelevate) the road so that the outside edge is higher than the inside. By doing this, part of the vehicle's weight counteracts the centrifugal force. You will need to make a diagram of a superelevated curve and analyze the forces acting on a vehicle that is traveling at a given velocity. This analysis should lead to the following formula.
where,
e = the superelevation in rate (m/m)
f = side friction factor (ratio of side friction force to normal force)
v = velocity (km/h)
R = radius of curvature (m)
By converting the units for V and R in Equation (1) to miles/hour and feet respectively, you should prove that
where,
e = the superelevation in rate (ft/ft)
f = side friction factor
V = velocity (miles/h)
R = radius of curvature (ft)
You will need to justify why it is reasonable to simplify Equation (2) to obtain Equation (3) below
where,
e = the superelevation in rate (ft/ft)
f = side friction factor
V = velocity (miles/h)
R = radius of curvature (ft)
This is a good place to explore friction, define the coefficient of static friction and kinetic friction, and perform experiments to calculate the coefficients of static friction for various pairs of surfaces. Your investigation of friction should apply the scientific method ‚ problem; question; hypothesis; test of the hypothesis; decision whether to retain, reject, or modify the hypothesis ‚ to questions involving friction. Such questions should address whether friction force depends on surface area, how static and kinetic friction forces are related to normal force, and how static and kinetic friction forces are related.
You are now ready to turn to the problem of developing a way of relating (for a given vehicle speed), the side friction factor, f, and superelevation, e, to the degree of curvature, x. There are several ways to accomplish this. Your final goal in this regard is to develop a function that enables you to find the superelevation for each possible degree of curvature.
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A Guide for Writing Research Papers based on Modern Language Association (MLA) Documentation, Prepared by the Humanities Department and the Arthur C. Banks, Jr., Library,
Capital Community College, Hartford, Connecticut, http://www.ccc.commnet.edu/mla.htm
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