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)

Question

Responses

A) 1/√5 v
the fraction with numerator 1 and denominator square root of 5 v

B) 1/5v
1 fifth v

C) √5v
square root of 5 v

D) 5v

Answer 1

To determine the velocity of a satellite when it is moved to an orbital radius of \(5r\), 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, and \(r\) is the orbital radius.

1. At radius \(r\), the velocity is: \[ v = \sqrt{\frac{GM}{r}} \]

2. At radius \(5r\), the new velocity \(v'\) can be calculated as: \[ v' = \sqrt{\frac{GM}{5r}} = \sqrt{\frac{1}{5}} \sqrt{\frac{GM}{r}} = \frac{1}{\sqrt{5}} v \]

Therefore, if the satellite is moved to an orbital radius of \(5r\), the new velocity becomes:

\[ v' = \frac{1}{\sqrt{5}} v \]

The correct response is: A) \( \frac{1}{\sqrt{5}} v \).