Question

Two planets in circular orbits around a star have speeds of v and 2v. What is the ratio of the orbital radii of the planets? What is the ratio of their periods?

a) 1:2, 1:1
b) 1:2, 1:4
c) 1:1, 1:2
d) 1:1, 1:4

Answer 1

The ratio of the orbital radii of the planets is approximately 1:√2, and the ratio of their periods is approximately 1:2.

Step-by-step explanation:

The ratio of the orbital radii of the planets can be determined using Kepler's third law, which states that the square of the period of revolution of a planet is directly proportional to the cube of its average distance from the Sun.

Since the ratio of their speeds is v and 2v, the ratio of their orbital radii would be the square root of the ratio of their speeds, which is √2. Therefore, the ratio of the orbital radii of the planets is approximately 1:√2.

The ratio of their periods can be found using the same law. The ratio of their periods would be the inverse square root of the ratio of their speeds, which is 1/√2. Therefore, the ratio of their periods is approximately 1:1/√2, which simplifies to 1:2.