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?
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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 is proportional to the cube of the semi-major axis of the orbit.
1. Let's assume that the orbital period of planet B is TB.
2. According to the given information, the semi-major axis of planet A's orbit is 4.5 times the semi-major axis of planet B's orbit.
3. Therefore, the semi-major axis of planet A's orbit is 4.5 times larger than that of planet B's orbit.
4. Since the two planets are in circular orbits around the same star, we can assume that their orbital radii are equal to their semi-major axes.
5. So, the orbital radius of planet A is 4.5 times larger than the orbital radius of planet B.
6. According to Kepler's third law, the square of the orbital period is proportional to the cube of the semi-major axis.
Therefore, (TA)^2 = (TB)^2 * (4.5)^3.
7. Rearranging the equation, we get TA/TB = (4.5)^(3/2).
8. Taking the square root of both sides of the equation, we have TA/TB = 4.5^(3/4).
9. Finally, to find the ratio of their speeds vA/vB, we can take the inverse value of the square root of the above ratio.
So, vA/vB = 1/(4.5^(3/4)).
Therefore, the ratio of their speeds vA/vB is approximately 0.5986.
To find the ratio of their speeds, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is directly proportional to the cube of its average distance from the star.
Let's assume that the orbital period of planet A is TA and the orbital period of planet B is TB. According to Kepler's Third Law, we have:
TA^2 / TB^2 = (DA / DB)^3
where DA is the average distance of planet A from the star, and DB is the average distance of planet B. Given that planet A is 4.5 times farther from the star than planet B, we can substitute these values into the equation:
(4.5 * DB)^2 / DB^2 = (4.5)^3
Simplifying the equation, we get:
20.25 = 91.125
Now, we need to find the ratio of their speeds. The speed of a planet in a circular orbit is given by:
v = 2πr / T
where v is the speed, r is the distance from the star, and T is the orbital period. Since we know that the distance from the star is directly proportional to the orbital period, we can write:
v ∝ r / T
So, the ratio of their speeds is simply the ratio of their distances from the star:
vA / vB = DA / DB
Since planet A is 4.5 times farther from the star than planet B, the ratio of their speeds is:
vA / vB = 4.5
Therefore, the ratio of their speeds is 4.5.