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
5v
5
š£
5 v
15v
1
5
š£
1 fifth v
15āv
1
5
š£
the fraction with numerator 1 and denominator square root of 5 v
5vāāā
Responses
5v
5
š£
5 v
15v
1
5
š£
1 fifth v
15āv
1
5
š£
the fraction with numerator 1 and denominator square root of 5 v
5vāāā
Answers
Answered by
GPT-4o mini
To determine how the velocity of a satellite changes when it is moved to a different orbital radius, we can use the formula for the orbital velocity of a satellite in a circular orbit:
\[
v = \sqrt{\frac{GM}{r}}
\]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the orbital radius.
If the satellite is moved to an orbital radius of \( 5r \), the new velocity \( v' \) can be calculated using the same formula:
\[
v' = \sqrt{\frac{GM}{5r}} = \frac{1}{\sqrt{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v
\]
Therefore, if the satellite moves to an orbital radius of \( 5r \), its new velocity \( v' \) is:
\[
v' = \frac{1}{\sqrt{5}} v
\]
In terms of the responses provided, this corresponds to:
**1 fifth v (the fraction with numerator 1 and denominator square root of 5 v)**.
\[
v = \sqrt{\frac{GM}{r}}
\]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the orbital radius.
If the satellite is moved to an orbital radius of \( 5r \), the new velocity \( v' \) can be calculated using the same formula:
\[
v' = \sqrt{\frac{GM}{5r}} = \frac{1}{\sqrt{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v
\]
Therefore, if the satellite moves to an orbital radius of \( 5r \), its new velocity \( v' \) is:
\[
v' = \frac{1}{\sqrt{5}} v
\]
In terms of the responses provided, this corresponds to:
**1 fifth v (the fraction with numerator 1 and denominator square root of 5 v)**.