1-gps-problems
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School
Stony Brook University *
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Course
103
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
Pages
2
Uploaded by SuperPorcupine3432
Homework 1: GPS
AMS 103, Fall 2023
Due: Monday, September 18, 8:30 am EDT
Quiz: Wednesday, September 20, 9:30 am EDT
Objectives
•
To reinforce concepts relevant to GPS technology and the underlying mathematical concepts.
•
To practice quantitative skills related to systems of equations and coordinate systems.
Instructions
•
List, on the first page, any students with whom you worked and/or any outside resources you used in
preparing your solutions. You may not use solutions to previous semesters’ homework assignments.
•
Clearly identify which problem you are solving at the start of the solution.
•
Solutions must be clearly and logically presented. When applicable,
show your work!
The solution
to each problem must be continuous; that is, parts should not be crossed out and solutions to other
problems should not be interspersed.
•
Your solutions must be submitted through Brightspace as a single PDF file. Scans of legibly written
pages are fine.
Improperly submitted or late homework will not receive credit.
Grading Policy
This assignment will be graded for both completeness and accuracy. Completion of the assignment—did
you attempt every problem?—will be worth 50% of the score.
Select problems will also be graded for
accuracy; that is, did you provide a correct answer and, when applicable, show your work? Accuracy will
determine the remaining 50%.
Remember
, if you use Wolfram Alpha
™
(or another software tool), include a screen shot with both input
and pertinent output to show your work. Also, wikipedia is not a valid source (although it may be a great
starting point to find more acceptable sources!).
Problems
1. (
⋆
, 14 pts) Although GPS was designed for determining locations, it has other applications. Explain
how GPS could be used to synchronize events around the world.
For instance, suppose that the
International Olympic Committee wants all broadcasters to show a particular diving event at precisely
the same time (no spoilers!).
2. (
⋆
, 14 pts) Read
http://www.thegreenwichmeridian.org/tgm/articles.php?article=9
. What
was the Board of Longitude, and what problem did it aim to solve? How did its work lead to the
Prime Meridian passing through Greenwich, England?
3. (
⋆⋆
, 22 pts) Meteorites regularly enter the atmosphere, where they heat up, disintegrate, and finally
explode before hitting the surface of the Earth. Such an explosion generates a shock wave (traveling
at the speed of sound) that can be detected by seismographs installed at various locations on the
Earth’s surface. Each seismograph (equipped with a perfectly synchronized clock) then determines
Homework 1: GPS
Page 2/2
the time it took the shockwave to reach the seismograph. How many seismographs need to detect
the shock wave to determine the location of the meteorite explosion? Explain.
4. (
⋆
, 14 pts) For each of the following systems of equations, (i) determine the number of equations
and the number of unknown variables in the system and (ii) classify the system as consistent or
inconsistent. If the system is consistent, (iii) solve the system.
Hint: items (ii) and (iii) are closely related; Wolfram Alpha
™
may be useful.
(a)
|
x
−
5
|
= 7,
|
x
+ 1
|
= 1,
|
x
−
2
|
= 4.
(b) 3
p
−
2
q
= 5, 3
p
+ 2
q
= 1,
p
−
q
= 3.
5. (
⋆
, 14 pts) Calculate the latitude and longitude angles for the following Cartesian coordinates.
(a) (
x
=
−
2
, y
= 7
, z
=
−
1).
(b) (
x
= 5
, y
=
−
1
, z
= 3).
6. (
⋆⋆
, 22 pts) The following table shows the locations of satellites and their distances from you. All
coordinates and distances are in miles.
Satellite
x
y
z
Distance
1
9430
.
08
−
7912
.
78
2170
.
6
11568
.
6
2
2162
.
34
−
12263
.
3
−
1089
.
45
11447
.
6
3
−
4608
.
6
−
6581
.
76
9575
.
56
8700
.
78
(a) Suppose your location is the point (
x, y, z
). Write an equation that determines if your location
is the correct distance from satellite 2.
(b) How would you determine if your location is on the surface of the Earth? You may assume the
Earth is a perfect sphere with radius 4000 miles.
(c) Using the results of parts (a) and (b), and similar ideas for satellites 1 and 3, determine if point
(i) or point (ii) is your location. Explain your response.
Point
x
y
z
(i)
2264
.
84
−
11859
.
9
10350
.
6
(ii)
−
476
.
824
−
2065
.
35
3392
.
19
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