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=gravitation force, M is mass of the sun, R is radius of planet from 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(netptune))=sqrt(GM/30*R(earth)

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.

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.