# Satellite A orbits a planet with a speed of 10,000 m/s. Satellite B is twice as massive as...

## Question:

Satellite A orbits a planet with a speed of 10,000 m/s. Satellite B is twice as massive as satellite A and orbits at twice the distance from the center of the planet. What is Satellite B's speed?

## Force balance equations in Satellite Motion:

Assuming that the satellite is moving in a circular orbit, we can easily write the gravitational force acting between the Earth and the Satellite. Since the Satellite is changing its direction of motion, the satellite is accelerating. This makes the Satellite a non-inertial frame of reference. Thus for our convenience of study, we can represent the force on the Satellite as

{eq}F=m\dfrac{v^{2}}{r} {/eq}

where m is the mass of the Satellite, r is the distance between the Satellite and the Earth. And this force is generated by Gravitation. Thus

{eq}G\dfrac{Mm}{r^{2}}=m\dfrac{v^{2}}{r} {/eq}

or more simply, we can write

{eq}G\dfrac{M}{r}=v^{2} {/eq}

A better expression would be

{eq}v^{2}r=\text{constant} {/eq}

This expression is a direct result that comes from Kepler's 2nd Laws of Planetary Motion which states that the areal velocity of a body orbiting a planet remains constant. This result is further extended to show how angular momentum in planetary motion remains conserved.

It should be clear from the above expression, that the orbital velocity of a satellite is independent of its mass.

Let the initial orbital radius be {eq}r_{\circ} \\ v_{\circ}=10000 \ ms^{-1} {/eq}

The orbital radius for the second satellite is

{eq}r'=2r_{\circ} {/eq}

Therefore

{eq}(10000)^{2}r_{\circ}=v^{2}(2r_{\circ})\\ v=\dfrac{10000}{\sqrt{2}}\\ v=7071.06 \ ms^{-1} {/eq}