Satellite Motion Homework: Find 2nd Satellite Speed

[ThePiGeek314]

Homework Statement

A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.70 × 10⁴ m/s, and the radius of the orbit is 5.25 × 10⁶ m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.60 × 10⁶ m. What is the speed of the second satellite?

Homework Equations

Kepler's Laws of Planetary Motion -- T²/r³ = 4pi² / GM

v = sqrt (GM / r)

The Attempt at a Solution

There really isn't much conceptual work involved in this, so I might just be making an arithmetic mistake somewhere. But I've worked the problem out three times and gotten the same (wrong) answer every time...

I used the v = sqrt (GM / r) -- plugged the values in for the first satellite and solved for the planet's mass which must be 2.274 x 10²⁵ kg.

Then I used that mass value for M in the same equation, switching out the radius value for the second satellite. I get that v = 1.3 x 10⁴.

But the answer is 1.3 x 10^-7.

Why is my answer so far off? Is the problem with the planet mass value or just in the calculations for the second satellite's speed? I think my process is right.

 

[Bandersnatch]

But the answer is 1.3 x 10-7.

This is clearly a typo in the answer key. There's no way orbital speed just barely higher would be 11 orders of magnitude lower.
The numerical factor of 1.3 matches, so you're good there, and by eyeballing the distances involved the order of magnitude should be the same for both satellites - i.e. 10E4.

 

[ThePiGeek314]

Thanks! I did wonder about the small difference in the orbit radii, but wasn't sure if I was missing something.