Question
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
15v
1 fifth v
5–√ v
5–√ v
15√v
the fraction with numerator 1 and denominator square root of 5 v
5v
Responses
15v
1 fifth v
5–√ v
5–√ v
15√v
the fraction with numerator 1 and denominator square root of 5 v
5v
Answers
GPT-4o mini
To determine the new velocity of a satellite when it is moved to a radius of \(5r\), we can use the formula for 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.
If the satellite's radius is changed from \(r\) to \(5r\), we can find the new velocity \(v'\) at this new radius:
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \sqrt{\frac{1}{5}} v
\]
This indicates that the new velocity \(v'\) will be:
\[
v' = \frac{1}{\sqrt{5}} v
\]
Thus, the correct representation of the velocity when the satellite is moved to an orbital radius of \(5r\) is:
**1 fifth \(v\)** (which can also be expressed as \(\frac{1}{\sqrt{5}} v\)).
\[
v = \sqrt{\frac{GM}{r}}
\]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the orbital radius.
If the satellite's radius is changed from \(r\) to \(5r\), we can find the new velocity \(v'\) at this new radius:
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \sqrt{\frac{1}{5}} v
\]
This indicates that the new velocity \(v'\) will be:
\[
v' = \frac{1}{\sqrt{5}} v
\]
Thus, the correct representation of the velocity when the satellite is moved to an orbital radius of \(5r\) is:
**1 fifth \(v\)** (which can also be expressed as \(\frac{1}{\sqrt{5}} v\)).