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 4r?

Answer 1

The  velocity of the satellite if it is moved to an orbital radius of 4r is v/2

Which represents the velocity if the satellite is moved to an orbital radius of 4r?

Let F = gravitational force on satellite = GMm/r² where

• G = universal gravitational constant,
• M = mass of earth,
• m = mass of satellite and
• r = radius of orbit.

Also, let F' = centripetal force on satellite = mv²/r where

• m = mass of satellite,
• v = velocity of satellite and
• r = radius of orbit

Now, at the orbit of the satellite, the gravitational force equals the centripetal force

So, F = F'

GMm/r² =  mv²/r

The velocity of the satellite

So, making v subject of the formula, we have the velocity of satellite is

v = √(GM/r)

Since G and M are constant, we see that v ∝ 1/√r

Now, given that satellite orbiting Earth at an orbital radius r has a velocity v. We need to find its velocity at an orbit of R = 4r

Let

• v = velocity of satellite at orbit r,
• r = radius of orbit r,
• v' = velocity of satellite at orbit R = 4r and
• R = radius of orbit T = 4r

Since v ∝ 1/√r, we have that

v//v = √(r/R)

So, v' = [√(r/R)]v

Substituting R = 4r into the equation, we have

v' = [√(r/R)]v

v' = [√(r/4r)]v

v' = [√(1/4)]v

v' = [1/2]v

v' = v/2

So, the  velocity of the satellite is v/2

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