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Using Everyday Examples in Engineering (E3)
Using Tangent Lines and Working with Parameters Instead of Numbers: Distance to
Horizon
Bernd Schroeder
Louisiana Tech University
http://www2.latech.edu/~schroder/
Photo Credits: “Rural Montana” by Jimmy Emerson, available under a Creative Commons Attribution
NonCommercial NoDerivs 2.0 Generic License (Left), “Horizon” by Dave Scriven, available under a Creative
Commons Attribution NonCommercial 2.0 Generic License (Right)
How far away is the horizon? Can you see the Rockies from Kansas? How many low Earth
orbit satellites are needed to have a continuous relay network around the Earth?
Where it Fits
This is an application that students can understand once the chain rule has been introduced. The
approximations also set the stage for estimates using differentials.
Setting the Stage
The question about the horizon is natural wherever land is flat as well as for people who live
along coastlines. The view from a tall tower nearby could be used, too. Anywhere else, the
question could be couched into scenes from the latest blockbuster pirate movie (which hopefully
will have a reasonable plot - the last one I saw was all special effects and make-up, loosely held
together by an inane storyline) when heroes or villains spot a ship on the horizon.
The question about the Rockies can be couched into a quick geography lesson as well as remarks
that, although Eastern Kansas and Western Colorado are important and productive agricultural
regions, they are a bit tedious to drive through.
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of NSF.
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The question on the satellites is motivated by considerations for a satellite phone network, which,
because of the weak signal from a satellite phone, needs low orbiting satellites.
Literary connection: I ran across the approximate formula in A. C. Clarke (1997), 3001 - The
Final Odyssey, Random House, Inc., New York. Once I saw it, I stopped reading and derived the
formula.
Presentation
Can be done in straight lecture or in small groups with the teacher leading the class to the next
stage every few minutes (or having a group that is on the right track put the next few steps on the
board).
Exact Formula
1. Model the cross section of the earth as a circle of radius r, centered at the origin. State the
equation of the upper half of this circle as a function
.
2. Compute the equation of a tangent to this function at an arbitrary point
3. Assume you stand on the surface of the Earth, right where the positive y-axis intersects with
the Earth. If the height of your eyes above the ground is h, then your eyes are at (0, r +h).
Because your line-of-sight is a straight line, the farthest away point that you can see is the point
from which a tangent to the circle goes through (0, r+h). Use the result of part 2 and the fact that
the tangent must go through (0, r + h) to obtain the x-coordinate of this point.
2
x=0
y=r+h
Approximate Formula
The formula in part 3 is quite unwieldy. As long as we talk about people, ships, towers, even
mountains, we have
Hence
That's the x-coordinate of the point you can see, not the distance on the circle. However, for
, there is not much difference.
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How far away is the horizon? (For a human, who is not in a spaceborne vehicle.)
Use
as the radius of the Earth (the distance from the center of the Earth to the
surface varies between 6,353km and 6,384km) and h = 1.7m (estimated eye level for a male) to
compute the distance to the horizon when you stand on a flat plain on Earth or on the shore of an
ocean.
Can you see the Rockies from Kansas?
Western Kansas and Eastern Colorado are not completely flat, but the terrain comes close. On
the other hand, even some of the easternmost peaks of the Rocky Mountains rise about 4,000m
above sea level, whereas the plain is about 1,200m above sea level. Compute from how far away
you can see these easternmost peaks of the Rockies when you approach them from the eastern
direction, that is, from Kansas. Then answer the question, assuming that the Kansas border is
about 250km away from these first peaks.
The peaks can be seen from
by eyes that are at ground level.
So, standing up, it's necessary to add the 4.6km from above (why?). Still, you can't see the
Rockies from Kansas. But you see them about 2 hours before you get near them on I-70. (Done
it.) My daughter says she has seen the Rockies from Kansas. How large of a hill would she need
to stand on to do so?
How many satellites at height h do you need to have a direct relay network around the
Earth?
For this one, we need the exact formula. (Why?)
Find the angle defined by the y-axis and the ray from the origin through the point where the
tangent hits the semicircle.
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So the angle is
Twice that angle is the angular “distance" across which two satellites at equal height can see
each other. (Why twice?)
With
(geosynchronous orbit), just looking at the picture gives you the fact that
you need 3 satellites. (Computation verifies that satellites as far apart as 162° can see each other.)
With
(low Earth orbit, say, for satellite phones) we would need at least 10 satellites.
(Satellites that are 39.5° apart can see each other, may need to move them closer to each other
because phone signals traveling tangentially may be dampened too much by the atmosphere,
etc.) Relaying to a geostationary satellite is possible. For full coverage, all satellites are still
needed, because no satellite will stay stationary above its “target."
Note that the above only provides a “belt" around the Earth that covers a little less than 40°. So,
to have overall coverage, at least 10 such belts (and likely more, because you don't want the
phone signal to travel through too much atmosphere) are needed.
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