Question

DERIVE INVERSE SQUARE LAW.

Answer 1

I read Feynman's Lost lecture a while ago. In that spirit, I'm trying to do a more simplified version of coming up with the inverse square relation.

Using Kepler's 2nd,

A1=12r21θ1A1=12r12θ1

A2=12r22θ2A2=12r22θ2

A1=12r21ω1tA1=12r12ω1t

A2=12r22ω2tA2=12r22ω2t

mω1A1=12mr21ω21mω1A1=12mr12ω12

mω2A2=12mr22ω22mω2A2=12mr22ω22

mv1A1=12r21F1mv1A1=12r12F1

mv2A2=12r22F2mv2A2=12r22F2

Dividing one by the other, I get

F1F2=v1v2r22r21F1F2=v1v2r22r12

which is almost there, except for the v1v2v1v2

Any suggestions on how to proceed from here?

Answer 2

It is a law that the intensity of an effect such as illumination or gravitational force changes in reverse proportion to the square of the distance from the source.
Light intensity is proportional to 1/distance squared