If the orbital radius r is doubled in v = sqrt(GM/r) with M fixed, how does v change?

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Multiple Choice

If the orbital radius r is doubled in v = sqrt(GM/r) with M fixed, how does v change?

Explanation:
For a circular orbit around a fixed mass, the orbital speed satisfies v^2 = GM/r, so v ∝ 1/√r. If the radius doubles, v becomes sqrt(GM/(2r)) = [sqrt(GM/r)]/√2, which is the original speed divided by √2. So the speed decreases by a factor of √2. This reflects that moving farther from the mass weakens gravity, requiring a slower orbital motion to provide the same centripetal acceleration.

For a circular orbit around a fixed mass, the orbital speed satisfies v^2 = GM/r, so v ∝ 1/√r. If the radius doubles, v becomes sqrt(GM/(2r)) = [sqrt(GM/r)]/√2, which is the original speed divided by √2. So the speed decreases by a factor of √2. This reflects that moving farther from the mass weakens gravity, requiring a slower orbital motion to provide the same centripetal acceleration.

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