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/