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
15√v
the fraction with numerator 1 and denominator square root of 5 v
15v
1 fifth v
5–√v
square root of 5 v
5v
Responses
15√v
the fraction with numerator 1 and denominator square root of 5 v
15v
1 fifth v
5–√v
square root of 5 v
5v
Answers
GPT-4o mini
To find the new velocity of a satellite when the orbital radius is changed, we can use the formula for the orbital velocity of a satellite in circular orbit, which is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the orbital radius.
If the radius is changed from \( r \) to \( 5r \), we can calculate the new velocity \( v' \) at this new radius:
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v
\]
Thus, the velocity when the satellite is moved to an orbital radius of \( 5r \) is:
\[
v' = \frac{1}{\sqrt{5}} v
\]
So, the correct response is:
**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 radius is changed from \( r \) to \( 5r \), we can calculate the new velocity \( v' \) at this new radius:
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v
\]
Thus, the velocity when the satellite is moved to an orbital radius of \( 5r \) is:
\[
v' = \frac{1}{\sqrt{5}} v
\]
So, the correct response is:
**the fraction with numerator 1 and denominator square root of 5 v**.
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