2140) lies outside the region of uncertainty. In (a), the calculated Hugoniot for iron (dashed) based on the original SESAME EOS (material no. Illustration of the analysis technique for the iron(a) and quartz(b) samples.įigures 4 shows a similar comparison for the iron and the quartz samples. 2981, and the results reflect the P-u errors corresponding to the uncertainty in Δ.įigure 4. The values given for the molybdenum (two cases) were obtained in this manner using the SESAME EOS for material no. įor the case in which a material served as a standard without being treated as an upper level sample, the Hugoniot data in Table I were derived from its theoretical EOS using the measured shock velocity and its associated uncertainty. These comparisons are given as percent differences between the calculated (cal) and experimental (exp) results. The calculated shock velocities based on theoretical EOSs are compared with the measured values in columns 5 and 6. The derived P-u Hugoniot points are given for the indicated materials in the last two columns of Table I, with percent errors shown in parentheses. The location of the calculated Hugoniot of the upper material on this graph provided a comparison with the measured point and gave a direct check on the consistency of the theoretical treatments used for the two EOSs. Similar plots using shock velocities that differed by one standard deviation for both materials defined a region of uncertainty in this plane and provided error bars for the Hugoniot point. Each graph consisted of plots in the pressure - particle-velocity (P-u) plane of the experimentally determined Rayleigh line (P = ρ oΔu) for the upper material and the calculated release isentrope (RI) or reflected shock (RS) Hugoniot for the lower standard material, whose initial state was defined by its measured interface shock velocity. Robinson, in Shock Waves in Condensed Matter 1983, 1984 4 IMPEDANCE-MATCHING RESULTSĪ graphical analysis based on the impedance-matching technique and the measured interface shock velocities was used to determine a Hugoniot point for each lower layer sample relative to the molybdenum standard and for each upper layer sample relative to the adjacent lower level material. (The numbers in these examples are hypothetical and only for illustration.) Such bivariate generalized Lorenz curves can, of course, also be usefully compared across populations.Ĭ.E. Or, treating the variables in the other direction, it might be found that the 20% least-consuming families account for 15% of the wealth, or a certain value per family. Thus, it is possible to consider, for example, both wealth and consumption for families, and to draw a curve from which it may be read that the least wealthy 50% of the families consume 27% of total consumption, or a certain quantity per family on the average. One of Mahalanobis' contributions in this domain was to stress the extension of the Lorenz curve idea to two variables. (If wealth were equally distributed, the Lorenz curve would be a straight line.) The comparison of Lorenz curves for two or more populations is a graphical way to compare their distributions of wealth, income, numbers of acres owned, frequency of use of library books, and so on. A Lorenz curve for wealth in a population tells, for example, that the least wealthy 50% of the population owns 10% of the wealth. Radhakrishna Rao, in Encyclopedia of Social Measurement, 2005 Fractile Graphical Analysisįractile graphical analysis is an important generalization of the method and use of concentration (or Lorenz) curves. Find the accns for each of the segments by finding the slope.C. At the end, you can sum up all the distances, with proper signs, to find the total dist travelled. Then the speed is zero => dist covered is 0 and accn =0. The dist is the area of the triangle, giving the formula d= ½ v0*t2. Then the speed decreases linearly from 5 to 0, and the accn is uniform because the slope of the graph is constant. Typically, this represents an impulsive force acting for a very short time (what we call collisions). Next, at a single instant, the speed becomes +5 from -5, showing that accn is infinitely large and not defined at that instant. So, the particle is moving left (say), and the dist covered will be the product of -5*t1 (in metres). Look at the start: the v is -5 m/s for some time t1 (not specified). The accn is dv/dt, and so the slope of the tangent at a point represents the accn. time graph, the area under the curve gives you the distance, since the distance is integral vdt.
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