A satellite orbiting the earth has a speed of 5.5 km/s when its distance from the center of the earth is 11,000 km. Is the satellite in a bound orbit?

Question

A satellite orbiting the earth has a speed of 5.5 km/s when its distance from the center of the earth is 11,000 km. Is the satellite in a bound orbit?

 

I have already calculated that the satellite is not in a circular orbit, but I know elliptical orbits can be bound so that is not very helpful. I also know that anything in a bound orbit has negative total mechanical energy, so is that how you would approach solving this problem?

 

which would be E mechanical = K + U = 1/2 m2v^2 - (Gm1m2)/r ? And we have all the values except the mass of the satellite itself? Or is there a better way to approach this problem?

 

Thanks!

Answer 1

Your approach is correct, but you do not need to know the mass of the satellite.  It is sufficient to show that the ratio of orbital energy to the mass of the satellite is negative.  If m is the mass of the satellite and M is the mass of the earth,

 

E = mv²/2 - GMm/r < 0 

 

in order to be in a bound orbit.  Then

 

E/m = v²/2 - GM/r < 0

 

You know v, G, M, and r.  Evaluate v²/2 - GM/r and examine the sign of the result.  Be careful with units.