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SECOND PROOF for GRAVITY EQUATION

Hypothesize two spheres (funnel cross-sections), one above the other, and use the earth's surface as the lower sphere (called "e") with an upper sphere (called "a"). We want to find the average gravity between the two spheres.

The difference between "v_e" and "v_a" is the increase in nether velocity between the two spheres. Gravity is an increase in nether velocity which is normally given as increase per second.

[(v_e - v_a) / H], where "H" is the distance in feet between the two spheres, is the increase in nether velocity per foot. To find the increase in nether velocity per second, we must multiply
[(v_e - v_a) / H] by the average number of feet in one second.

v_ave = [(v_e + v_a) / 2] gives us the average number of feet per second.

H = (r_a - r_e) which is the difference in the radii of the two spheres.

So:

1.    g_ave = [(v_e - v_a) / H] [(v_e + v_a) / 2] for the two spheres.

Substituting for H:

2.    g_ave = [(v_e - v_a) / (r_a - r_e)] [(v_e + v_a) / 2]

Multiplying the two expressions at the right:

3.    g_ave = [(v_e - v_a) (v_e + v_a)] / [2 (r_a - r_e)]

Multiplying terms at top right:

4.    g_ave = [v_e² - v_a²] / [2 (r_a - r_e)]

Multiplying both sides by [2 (r_a - r_e)]

5.    2g_ave (r_a - r_e) = (v_e² - v_a²)

From the inverse square law:

v_a / v_e = r_e^1/2 / r_a^1/2

v_a² / v_e² = r_e / r_a

v_a² = v_e² (r_e/r_a)

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Substituting for v_a² in equation 5 above:

6.    2g_ave (r_a - r_e) = {v_e² - [v_e² (r_e / r_a)]}

Simplifying:

7.    2g_ave (r_a - r_e) = v_e² [1 - (r_e / r_a)]

Providing a common denominator:

8.    2g_ave (r_a - r_e) = v_e² [(r_a - r_e)/r_a]

Dividing both sides by (r_a - r_e):

9.    2g_ave = v_e² / r_a

Multiplying both sides by r_a

10.    2r_ag_ave = v_e²

Let H approach and become zero, then r_a becomes r_e and g_ave becomes g_e.

11.    2r_eg_e = v_e²

Taking the square root of both sides and reversing the sides:

12.    v_e = (2r_eg_e) ^1/2 and
g_e = v_e² / 2r_e

For the general equation:

13.    v = (2rg) ^1/2 or g = v² / 2r

Again we have the equation for escape velocity being the same as incoming nether velocity at any point in a gravity funnel.
 

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