A satellite orbiting Earth at an orbital radius r has a velocity v. Which represents the velocity if the satellite is moved to an orbital radius of 5r?

(1 point)
Responses

15–√v
15–√v

15v
15v

5v−−√
5v−−√

5v

1 answer

The velocity of a satellite in orbit is given by the formula:

\[ v = \sqrt{\frac{GM}{r}} \]

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 \( v' \) can be calculated as:

\[ v' = \sqrt{\frac{GM}{5r}} = \frac{1}{\sqrt{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v \]

Since \( \frac{1}{\sqrt{5}} \) is approximately \( 0.447 \), the new velocity is less than the original velocity \( v \).

So, the final answer should represent the new velocity as:

\( \frac{v}{\sqrt{5}} \) which is not among the options provided.

However, examining the multiple-choice options again, it appears none accurately represent this conclusion. Based on the expected use of standard notation, the best response for \( v' \) at an orbital radius 5r is \( 0.447v \) or \( \frac{v}{\sqrt{5}} \). Please verify the answer choices or see if a different question format aligns with typical orbital mechanics.