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### Earth's Curvature

```Date: 07/07/2003 at 22:30:20
From: James
Subject: Earth's Curvature

My son and I wondered what would happen if there were no gravity.  I
jokingly said that you'd take a step and simply float in a straight
line forever. That got me to thinking: How far would you have to
"walk" this way to get 1 foot off the ground? Scientifically, how far
would you have to follow a tangent of the earth to be one foot above
the surface of the curvature?

Could you use triangulation somehow?  Could you use the formula for
a circle to calculate the slope?

In another discussion:

Earth's Curvature
http://mathforum.org/library/drmath/view/54904.html

you say that the formula for a circle is x^2 + (y+R)^2 = R^2 and that,
"if 2 points are separated by a distance L, they might be point A at
x = -L/2 and point B at x = +L/2.  At both A and B we have:

y = - L^2 / 8R"

That leads me off in the general direction, but I'm not finishing
the connection.
```

```
Date: 07/08/2003 at 02:23:31
From: Doctor Jeremiah
Subject: Re: Earth's Curvature

Hi James,

Imagine a perfectly spherical world with a tangent:

A           B
+++++ -------+
+++    |    +++  /
+++        |        +++
+             |       /     +
+               |      /        +
+                |     /          +
+                 R   R+1           +
+                  |   /              +
+                  |  /               +
+                   | /                 +
+                   |/                  +
+                   +                   +

You travel from A to B. Your distance from the center of the Earth at
A is R, and your distance from the center of the Earth at B is R+1.
This is a simple right angle triangle and can be solved for the
distance from A to B with the Pythagorean formula:

(R+1)^2 = R^2 + d^2   where  d = the distance from A to B
d^2 = (R+1)^2 - R^2
d^2 = R^2+2R+1 - R^2
d^2 = 2R+1
d = sqrt(2R+1)

The radius of the Earth (R) is 3963.19 statute miles or, more
importantly for us, 20925643.2 feet. That means:

d = sqrt(2R+1)   where  R = 20925643.2 feet
d = sqrt(2*20925643.2+1)
d = sqrt(41851287.4)
d = 6469.26 feet

So you would have to travel along the tangent for over a mile to get
one foot off the ground.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles

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