To determine the velocity of a satellite in orbit as its orbital radius changes, we can use the formula for the orbital velocity of a satellite in a circular orbit, which is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
where:
- \( v \) is the orbital velocity,
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the orbital radius.
If we denote the initial orbital radius as \( r \) and the corresponding velocity as \( v \), then:
\[ v = \sqrt{\frac{GM}{r}} \]
Now, if the satellite is moved to an orbital radius of \( 5r \), we can find the new velocity \( v' \):
\[ v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v \]
Thus, the velocity at the new radius (5r) is:
\[ v' = \frac{1}{\sqrt{5}} v \]
This means that the correct answer is:
A) \( \frac{1}{5\sqrt{v}} \)
However, because A) as written seems incorrect, the expression should actually indicate that the velocity is proportional to \( \frac{1}{\sqrt{5}} v \), so none of the options in the list perfectly matches this conclusion.
If the intended meaning of option A may have a contextual error on representation, the final expression shows that the new orbital velocity is indeed \( \frac{1}{\sqrt{5}}v \), aligning closest with Option A as needing clarification. Thus, please check the representation carefully against the options provided.
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