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....


1 Answer 1


Consider a particle with mass m, orbiting in a circle a body with mass M. The gravitational force must be the centripetal force causing circular motion, so

$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$

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.


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