Planet A and planet B are in circular orbits around a distant star. Planet A is 8.0 times farther...

Question:

Planet A and planet B are in circular orbits around a distant star. Planet A is 8.0 times farther from the star than is planet B .What is the ratio of their speeds ?

Orbital speed in a circular orbit

If a body moves in a circular orbit with radius {eq}r {/eq} around a more massive body with mass {eq}M {/eq}, its orbital speed is

{eq}v = \sqrt{\dfrac{G M}{r}}, {/eq}

where {eq}G = 6.67 \times 10^{-11} \ \text{N} \ \text{m}^2/\text{kg}^2 {/eq} is the gravitation constant.

Let the mass of the distant star be {eq}M {/eq}. The radius of the circular orbit of planet B be {eq}r_B {/eq}, while for planet A it is {eq}r_A = 8r_B {/eq}. The orbital speed of planet A and planet B respectively are:

{eq}v_A = \sqrt{\dfrac{GM}{r_A}}, \quad v_B = \sqrt{\dfrac{G M}{r_B}}. {/eq}

Hence, the ratio of their speeds is

{eq}\dfrac{v_A}{v_B} = \dfrac{\sqrt{\dfrac{GM}{r_A}}}{\sqrt{\dfrac{GM}{r_B}}} = \sqrt{\dfrac{r_B}{r_A}} = \sqrt{\dfrac{r_B}{8r_B}} = \sqrt{\dfrac{1}{8}} = 0.35. {/eq}