Two planets revolve around a star in circular orbits. If the ratio of their orbital radii is 1:4 , what is the ratio of their average orbital speeds?

Step-by-step solution

The orbital speed v of a planet in a circular orbit of radius r is given by v = GM r , where M is the mass of the star. This implies that the orbital speed is inversely proportional to the square root of the orbital radius: v 1 r . Alternatively, using Kepler's Third Law ( T^2 r^3 ), the speed is v = 2 r T r r^ 3/2 = r^ -1/2 . Taking the ratio of the speeds of the two planets: v₁ v₂ = r₂ r₁ Given the ratio of radii r₁ r₂ = 1 4 , we have r₂ r₁ = 4 1 . v₁ v₂ = 4 1 = 2 1 Thus, the ratio of their average orbital speeds