<center>GRAVITY</center>

Derivation of Newton's Universal Law of Gravitation

For a circle (or an ellipse):

P² ~ r³

where r is the radius (semimajor axis) between the planet and the Sun.

So P ~ r^3/2 (Equation 1)

v = ds/dt = circumference / period = 2(pi)r / P

So v ~ r / P (Equation 2)

Putting Equation 1 and 2 together, we get: v ~ r / (r^3/2)

v ~ r^-½

(Equation 3)

We also know, for circular acceleration:a ~ v² / r

Substituting Equation 3 into this, we get: a ~ 1 / r²

So a = k / r², where k is a constant.

Newton's Third Law gives: |F(ps)| = |F(sp)|

where s is for sun and p is for planet.

m(s)a(p) = m(p)a(s)

Here, m(p)a(s) represents the mass of the planet times the acceleration due to the sun. Substituting for a, we get:

[m(s)k(p) / r(ps)²] = [m(p)k(s) / r(ps)²]

Cancelling out the r² and rearranging the equation, we get:

[k(p) / k(s)] = [m(p) / m(s)]

We can see from this equation that: k ~ m

Or written in an equality: k(p) = Gm(p), where G is a constant.

F(ps) = m(s)a(p) = m(s)k(p) / r² = m(s)Gm(p) / r²

=Gm(s)m(p) / r²

Which is Newton's Universal Law of Gravitation.