To determine the velocity of a satellite at a new orbital radius, we can use the formula for orbital velocity, which is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth, and
- \( r \) is the orbital radius.
If the satellite is moved to an orbital radius of \( 5r \), the new velocity \( v' \) at that radius can be expressed as:
\[ v' = \sqrt{\frac{GM}{5r}} \]
This can be simplified to:
\[ v' = \frac{1}{\sqrt{5}} \sqrt{\frac{GM}{r}} \]
Substituting \( v = \sqrt{\frac{GM}{r}} \) (the original velocity at radius \( r \)), we get:
\[ v' = \frac{1}{\sqrt{5}} v \]
Thus, the new orbital velocity when the satellite is at an orbital radius of \( 5r \) is:
The fraction with numerator 1 and denominator square root of 5 v
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