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Question:

It is known that radius of Neptune in a circular orbit around the sun is 30 times greater than that of Earth's. Does Neptune travel faster than earth and by what factor?

Attempt:

For a circular orbit, $v=\sqrt{GM/R}$ where $G$ is the gravitational constant, $M$ is the mass of the sun, $R$ is the radius of the circular orbit centered around the sun. It is assumed the masses of the planets compared to the sun is negligible.

$v_{earth}=\sqrt{GM/R_{earth}}$

$v_{neptune}=\sqrt{GM/R_{neptune}}$

It can obviously be seen that Earth travels at greater speed, but by finding the factor without inputting the actual numerical values, am i correct in just equating the two equations? Doesn't seem right. Not sure about determining the factor.

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    $\begingroup$ Have you heard of Kepler's laws of planetary motion? $\endgroup$
    – suresh
    Commented Sep 23, 2014 at 9:08
  • $\begingroup$ None of the planets have actually a perfect circular orbit. They are usually ellipses with low eccentricities. However your equation does gives an approximation of the mean motion. $\endgroup$
    – fibonatic
    Commented Sep 23, 2014 at 11:46

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You want to know the ratio of the speeds, i.e. $v_{neptune}/v_{earth}$. You can just divide the two equations you already have. Now, what is the ratio of the radii of the orbits? Your equation should tell you how to relate this to the orbital speeds.

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  • $\begingroup$ with speed of Neptune in terms or R(earth)? 1/sqrt(30)? $\endgroup$
    – Jesse
    Commented Sep 23, 2014 at 9:13
  • $\begingroup$ @Jesse: Bingo! :-) $\endgroup$
    – John Rennie
    Commented Sep 23, 2014 at 9:14

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