The Real Relationship Between Velocity and Gravity

The further away a planet is from the sun, the lower the force of gravity and the lower its velocity. But this is not a simple linear relationship. From the last article, I showed that the ratio of gravity for two bodies is directly proportional to their velocity ratios divided by their orbital period ratio. I now want to simplify this further to show the exact relationship between gravity and velocity.

Let's return to the relationship between velocity and distance :

• V = √ D / D

Another simpler way of expressing this is :


• V = 1 / √ D

Newton's law of gravity shows :

• F     𝞪    1 / R sq
We can rearrange this to show

√F     𝞪    1 / R
R     𝞪    1 / √ F

So then substituting radius for distance in the above velocity formula :

• V = 1 / √ R

And squaring both sides :

V^2 = 1 / R

Then substituting R for 1 / √ F :

V^2 = 1 / 1 / √ F = √ F

V = √ √ F

Is that two square roots I hear you say? Yes, that is indeed a double square root.

The ratio of the velocities of two bodies is proportional to the double square root of the ratio change in their force of gravity.

And there we have it, the true relationship of gravity and velocity has finally being revealed. Note that the mass of either orbiting body does not matter - the velocities will change according to the double square root of the change in gravity regardless of whether the bodies orbit the Sun or Jupiter or wherever.

Because the change in gravity will always be a decimal i.e. less than 1, the double square root will always result in a larger figure for velocity. As we've seen previously, doubling the distance results in a change in force from 1 to 0.25, and a velocity change in magnitude from 1 to 0.70. The double square root of 0.25 is 0.70.

The reason it's a double square root is because velocity is related to the square root of the distance, whilst force is related to the square of the distance. The difference is two square roots, viz, 1/ √9 is one third, 1/9^2 is 1 to eighty one, the difference being two square roots - the square root of 81 is 9 and the square root of 9 is 3 - i.e. one third. The upshot is that the magnitude of velocity is stronger than gravity by a factor of two square roots.

To test this, take Jupiter and Earth, the former being 5.2 times further away from the Sun.

• F     𝞪    1 / R sq

F     𝞪    1 / 5.2^2 = 0.036925.

• V = √ √ F

V = √ √ 0.036925 = √ 0.19215 = 0.438349

The velocity of Earth is 29.78 km/s.

29.78 x 0.4383 = 13.05 km/s, the exact velocity of Jupiter.  

Although the force of gravity between the Sun and Jupiter is only about 3% of that between the Sun and Earth's, the magnitude of it's velocity remains comparatively strong at 43% of the Earth's. And if we divide one by the other :

0.438349 / 0.036925 =  11.87

We get Jupiter's orbital period. 

Pretty neat.

The Greeks were right, the universe has Logos.