geomath

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 sameè

 

i

1

2

3

4

5

6

7

8

ti(s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

zi(ft)

25.0

25.7

27.7

31.0

35.6

41.6

48.9

57.5

 

9

10

11

12

13

14

15

16

17

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

67.5

78.8

91.4

105.3

120.6

137.2

155.1

174.4

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.

 

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).

 

D)   Plot the complete derivative profile using appropriate units.

 

E)    Numerically integrate the derivative profile and compare to the original data profile.

 

F)    Compute and list the 16 integral values that may be defined from the successive 2-point data sequences of the travel-time data.

 

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. 

 

H)   Plot the complete integral profile using appropriate units.

 

I)     Numerically differentiate the integral profile and compare to the original travel-time data profile.

 

J)    Compute, list, and plot the derivative of the derivative profile in D.

 

K)   Compute, list, and plot the integral of the profile in J and compare to profile D. 

 

L)    How can you account for the intercepts of the data sets?

 

M)  On which planetary body of the solar system might these data have been observed?  Why?