Physics

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timer Asked: Apr 8th, 2015

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EARTHSC_5642_Ex2-1_08Feb15.pdf 

Exercise 1.1 EarthSc 5642.doc 


 here is the doc you might need them .. you might need to use the data form homework 1.1 named file document 1 YOU have top do file named 

EARTHSC 5642 –

The others u l need they r Notes And previous Assignment u ll need it And be take Care u can do it I got a negative reputation bcuz i didnt paid to a tutor but it was due to tht ihre gve me incompelete wrk i should hve withdrawed but i let him kep the dp fr the incoompelete wrk dont do same this time i m not gonna do like the same got it?

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GEOMATHEMATICAL ANALYSIS Exercise 1.1 You have taken a job at the Johnson Space Flight Center in Houston (TX). In the desk that you were assigned, you find papers with a list of raw travel-time data for the free falls of a feather and a rock hammer. The intriguing thing about the two lists of numbers is that they are exactly the same➔ i ti(s) zi(ft) 1 0.0 25.0 2 0.5 25.7 3 1.0 27.7 4 1.5 31.0 5 2.0 35.6 6 2.5 41.6 7 3.0 48.9 8 3.5 57.5 9 4.0 67.5 10 4.5 78.8 11 5.0 91.4 12 5.5 105.3 13 6.0 120.6 14 6.5 137.2 15 7.0 155.1 16 7.5 174.4 17 8.0 194.6 Explore the inverse properties of numerical differentiation and integration for the above profile of travel-time data – i.e., A) Plot the travel-time data profile using appropriate units. B) Compute and list the 15 horizontal derivative values that may be defined from the successive 3-point data sequences. Taking i=1, 1=2 and i=3 as successive points From the 3 points a square function may be generated of the form ax2+ bx+ c= 0 In this case we wish to generate the equation of the curve of the form Z(t)=at2+bt+c We form the following equations a.02 + b.0+ c=25 a.0.52+b.0.5+c= 25.7 a.12+b.1+c=27.7 Solving the equations using Matrices on Matlab gives a=2.2 b=0.3 c=25 Z(t)=2.2t2+0.3t+25 From the eqation the following values will be generated. T(s) 0 0.5 1 1.5 2 2.5 3 Z(t) 25 25.7 27.5 30.4 36 39.5 45.6 T(s) 4.5 5 5.5 6 6.5 7 7.5 Z(t) 70.9 81.5 93.2 106 119.9 134.9 151 3.5 4 53 61.4 8 168.2 C) Find the derivative values for i = 1 and 17 using the 2nd Fundamental Theorem of Calculus (i.e., a function can be determined from the integral of its derivative) given by equation (1.5) in the GeomathBook.pdf (p. 18/153) and equation (4.ii) in the 5642Lectures_1.pdf (p. 16/21). Z(t)=2.2t2+0.3t+25 ∆Zi(t)/ ∆t=4.4t+0.3 For i=1: t=0.0 Derivative= 4.4(0.0)+ 0.3 =0.3 For i=17: t=8 Derivative= 4.4(8)+ 0.3 35.5 D) Plot the complete derivative profile using appropriate units. Derivative Data i 1 2 3 4 5 6 7 t 0.0 0.5 1 1.5 2 2.5 3 ∆Zi(t)/ 0.3 2.5 4.7 6.9 9.1 11.3 13.5 ∆t i 10 11 12 13 14 15 16 t 4.5 5 5.5 6 6.5 7 7.5 ∆Zi(t)/ 20.1 22.3 24.5 26.7 28.9 31.1 33.3 ∆t 8 3.5 15.7 9 4 17.9 17 8 35.5 E) Numerically integrate the derivative profile and compare to the original data profile. The derivative profile follows the equation ∆Zi(t)/ ∆t=4.4t+0.3 Integrating this profile gives: ∫∆Zi(t)/ ∆t= ∫4.4t+0.3 Zi(t)=2.2t2+0.3t. Profile data i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 t 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Zi(t) 0.7 2.5 5.4 9.4 14.5 20.7 28 39.5 45.9 56.5 68.2 81 94.9 109.9 126 Comparing with the previous data profile which is given by; i ti(s) zi(ft) 1 0.0 25.0 2 0.5 25.7 3 1.0 27.7 4 1.5 31.0 5 2.0 35.6 6 2.5 41.6 7 3.0 48.9 8 3.5 57.5 9 4.0 67.5 10 4.5 78.8 11 5.0 91.4 12 5.5 105.3 13 6.0 120.6 14 6.5 137.2 15 7.0 155.1 16 7.5 174.4 17 8.0 194.6 Observation the constant value is lost in the numerical analyses. F) Compute and list the 16 integral values that may be defined from the successive 2-point data sequences of the travel-time data. i t ∆Zi(t)/ ∆t i t ∆Zi(t)/ ∆t 1 0.0 0.3 2 0.5 2.5 3 1 4.7 4 1.5 6.9 5 2 9.1 6 2.5 11.3 7 3 13.5 8 3.5 15.7 10 4.5 20.1 11 5 22.3 12 5.5 24.5 13 6 26.7 14 6.5 28.9 15 7 31.1 16 7.5 33.3 17 8 35.5 9 4 17.9 G) Using the 1st Fundamental Theorem of Calculus (i.e., a function can be determined from the derivative of its integral), find the integral value for i = 1. Using the theorem Z(t)=2.2t2+0.3t For i=1; then t=0.0 Z(t)=0 H) Plot the complete integral profile using appropriate units. Data I) Numerically differentiate time data profile. i 1 2 3 4 5 t 0.5 1 1.5 2 2.5 Zi(t) 0.7 2.5 5.4 9.4 14.5 the integral profile and compare to the original travel6 7 8 9 10 11 12 13 14 16 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 20.7 28 39.5 45.9 56.5 68.2 81 94.9 109.9 126 J) Compute, list, and plot the derivative of the derivative profile in D. The profile in D follows the equation f(t)= 4.4t+3 The derivative will be given by ∆f(t)= 4.4 This implies that for all i(s); ∆f(t)= 4.4 K) Compute, list, and plot the integral of the profile in J and compare to profile D. The intergral will be given by f(t)=4.4t Comparing with the one in D given by f(t)=4.4t+0.3 L) How can you account for the intercepts of the data sets? Despite of the constant 0.3 in D the graphs are similar. this implies that the mathematical function produces the same results despite external forces that may result in a constant. M) On which planetary body of the solar system might these data have been observed? Why? Pluto because it has a surface gravity of 0.58 therefor both a feather and a rock hammer have the same free fall times EARTHSC 5642 – GEOMATHEMATICAL ANALYSIS Exercise 2.1 Referring to Fig. 2.1 on page 24/100 of the notes=> 5642Lectures_2_4.pdf, consider a set of 5 horizontally infinite cylinders with the following parameters è Cylinder # d (km) z (km) R (km) Δρ (gm/cm3) 1 -34 2 2 3.0 2 -20 3 1 5.0 3 0 5 3 1.0 4 20 10 4 2.0 5 35 10 3.5 1.5 A) Along a bisecting profile extending from d = -64 km through 0 km to +64 km at 1-km intervals, compute the 5 gravity profiles in mgal by gz = [41.93 Δρ(R2/z)]/[(d2/z2) + 1], and plot them superimposed on a single graph using different colors or symbols. Computer software (e.g., IMSL, LINPACK, Matlab, Mathematica, MathCad, Maple, etc.) may be helpful here. B) Compute and 1) plot the total gravity effect of the 5 cylinders by summing their effects at each observation point on the profile. 2) What is the mean value and standard deviation of the total gravity effect? 3) What is the utility of these statistics for graphing the profile? C) Suppose you want to estimate the 5 densities (Δρi) from the total gravity observations in B-above. Determine 1) the [ATA]-matrix and 2) least-squares estimates of Δρi, and 3) compare the estimated densities with those in the above table. D) Determine 1) the Choleski factorization of [ATA] – i.e., determine a lower triangular matrix L such that [LLT = ATA]. 2) Find the coefficients of [P] such that [LP = ATB], and 3) solve the system for the least-squares estimates of Δρi. 4) Compare your density estimates with those you obtained in C-above.
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