To determine the velocity of a satellite in orbit at a different radius, we can use the formula for the orbital velocity:
\[ v = \sqrt{\frac{GM}{r}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth, and
- \( r \) is the orbital radius.
Given that the original orbital radius is \( r \) and the initial velocity is \( v \), we can express the initial velocity as:
\[ v = \sqrt{\frac{GM}{r}} \]
Now, if the radius is changed to \( 6r \), we can find the new velocity \( v' \):
\[ v' = \sqrt{\frac{GM}{6r}} \]
We can express this in terms of the initial velocity \( v \):
\[ v' = \sqrt{\frac{GM}{6}} \sqrt{\frac{1}{r}} = \frac{1}{\sqrt{6}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{6}} v \]
Therefore, the new velocity of the satellite when it is moved to an orbital radius of \( 6r \) would be
\[ v' = \frac{1}{\sqrt{6}} v \]
The correct response is:
the fraction with numerator 1 and denominator square root of 6 v.
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