# Why do planets' orbital velocities drop off by less than a third when twice as distant from the sun, rather than three-quarters? If gravity is $1/4$?

Looking at the speeds of the planets in our solar system when traveling around the sun and their distances from it (or, using Kepler's third law,), it seems that when a planets distance is doubled, it's velocity drops to about 70.7% of its previous value...

Why is that, if the gravitational strength between the planet and sun is now only a quarter of its previous value? Shouldn't it be moving at 25% the speed it would have if it was half as distant, not 70.7%?...

All I can think of is that the more distant planet doesn't have to expend as much 'gravitational energy' or whatever changing its direction (which is a form of 'acceleration'), because it is moving in a straighter line (less curvature; less angle in its angular velocity) during its orbit....

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$
Cancelling the $$m$$ and $$r$$ and square rooting gives:
$$\frac{\sqrt{GM}}{\sqrt{r}} = v$$
So you see the velocity is inversely proportional to the square root of distance $$r$$. If the distance doubles, the velocity decreases by a factor of $$1/\sqrt{2}\approx 0.707$$. This is the 70.7% that you observed.