Planet A and planet B are in circular orbits around a distant star. Planet A is 4.5 times farther from the star than is planet B.
What is the ratio of their speeds vA/vB?
mv²/R=GmM/R ²
v= sqrt (GM/R)
v₁/v₂= sqrt(R₂/R₁) =sqrt(1/4.5)=0.47
Oh, it's the perfect setup for a cosmic joke! The ratio of their speeds is the same as the ratio of their distances from the star. So, planet A goes "4.5 times" faster than planet B! Hope you find that as astronomical as I do!
To find the ratio of the speeds vA/vB, we can use Kepler's Third Law, which states that the square of the orbital period (T) of a planet is proportional to the cube of its average distance (r) from the star.
Since planet A is 4.5 times farther from the star than planet B, we can write the ratio of their distances as rA/rB = 4.5.
Using Kepler's Third Law, we know that T^2 ∝ r^3. Therefore, we can write (TA/TB)^2 = (rA/rB)^3.
Substituting the known value, we have (TA/TB)^2 = (4.5)^3.
Taking the square root of both sides, we get TA/TB = √(4.5^3).
Calculating, we find TA/TB ≈ 2.68.
Since the orbital period is inversely proportional to the speed of the planet, the ratio of their speeds vA/vB would be the inverse of the ratio of their orbital periods.
Thus, the ratio of their speeds vA/vB ≈ 1/(TA/TB) ≈ 1/2.68 ≈ 0.373.
To find the ratio of speeds vA/vB, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is proportional to the cube of the average distance between the planet and its star.
Let's assume that the orbital period of planet A is TA and the orbital period of planet B is TB. Since planet A is 4.5 times farther from the star than planet B, we can express their relative distances as follows:
Distance of planet A = 4.5 × Distance of planet B
Using Kepler's Third Law, we know that the ratio of the square of the orbital periods is equal to the ratio of the cubes of the distances:
(TA^2 / TB^2) = (4.5 × Distance of planet B)^3 / (Distance of planet B)^3
Simplifying this equation, we can cancel out the cubes of the distance:
(TA^2 / TB^2) = (4.5^3)
Taking the square root of both sides of the equation, we find:
((TA / TB) * (1 / (TA / TB))) = sqrt(4.5^3)
Simplifying further, we get:
vA / vB = sqrt(4.5^3)
Calculating the right-hand side of the equation, we find:
vA / vB ≈ 3.59
Therefore, the ratio of their speeds vA/vB is approximately 3.59.