Question
A satellite orbiting Earth at an orbital radius r has a velocity b with represents the velocity if the satellite is moved to an orbit radius of 5r
Answers
GPT-4o mini
To determine the relationship between the orbital velocity of a satellite at different radii, we can use the formula for orbital velocity, which is derived from Newton's law of gravitation and the centripetal force needed to keep a satellite in orbit.
The orbital velocity \( v \) of a satellite in circular 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.
Now, if we have a satellite orbiting at an orbital radius \( r \) with velocity \( v = b \):
\[
b = \sqrt{\frac{GM}{r}}
\]
If we move the satellite to an orbital radius of \( 5r \), we can calculate the new orbital velocity \( v' \):
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \sqrt{\frac{1}{5}} b
\]
Thus, if the satellite is moved to an orbital radius of \( 5r \), its new velocity \( v' \) will be:
\[
v' = \frac{b}{\sqrt{5}}
\]
This means that the velocity decreases as you move to a higher orbit. In summary, when a satellite is moved from an orbit of radius \( r \) to an orbit of radius \( 5r \), its new velocity is \(\frac{b}{\sqrt{5}}\).
The orbital velocity \( v \) of a satellite in circular 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.
Now, if we have a satellite orbiting at an orbital radius \( r \) with velocity \( v = b \):
\[
b = \sqrt{\frac{GM}{r}}
\]
If we move the satellite to an orbital radius of \( 5r \), we can calculate the new orbital velocity \( v' \):
\[
v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \sqrt{\frac{1}{5}} b
\]
Thus, if the satellite is moved to an orbital radius of \( 5r \), its new velocity \( v' \) will be:
\[
v' = \frac{b}{\sqrt{5}}
\]
This means that the velocity decreases as you move to a higher orbit. In summary, when a satellite is moved from an orbit of radius \( r \) to an orbit of radius \( 5r \), its new velocity is \(\frac{b}{\sqrt{5}}\).