Question

complete this assessment to review what youve learned. it will not count toward your grade.

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 ( 4r )? (1 point)

( \bigcirc ) ( 4v )

( \bigcirc ) ( \frac{1}{2}v )

( \bigcirc ) ( \frac{1}{4}v )

( \bigcirc ) ( 2v )

Explanation:

Step1: Recall orbital velocity formula

The orbital velocity \( v \) of a satellite around a planet (here, Earth) 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.

Step2: Find velocity for new radius \( 4r \)

Let the new velocity be \( v' \). Substituting \( r' = 4r \) into the formula, we get \( v' = \sqrt{\frac{GM}{4r}} \).

Step3: Simplify the new velocity expression

We can rewrite \( \sqrt{\frac{GM}{4r}} \) as \( \frac{1}{2}\sqrt{\frac{GM}{r}} \). But from the original formula, \( \sqrt{\frac{GM}{r}} = v \). So, \( v' = \frac{1}{2}v \).

Answer:

\(\frac{1}{2}v\) (corresponding to the option \(\boldsymbol{\frac{1}{2}v}\))