Say a planet (mass $m$) is orbiting a star (mass $M$) in a perfect circle, so it is in circular motion.
$F=ma$ and the gravitational force between two masses $F=\frac{GMm}{r^2}$ so
$\frac{GMm}{r^2}=ma$
$\frac{GM}{r^2}=a$
And in circular motion $a=\frac{v^2}{r}$ so
$\frac{GM}{r^2}=\frac{v^2}{r}$
$\frac{GM}{r}=v^2$
And gravitational potential $V=-\frac{GM}{r}$
So $v^2=-V$
Is there a (qualitative/less math/wordy) reason why this is the case? (or have I got this wrong?) and is this limited to the specific case of perfect circular motion?
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