What is the speed of a geosynchronous satellite orbiting mars?

Mars rotates on its axis once every 24.8 hours. Part A
What is the speed of a geosynchronous satellite orbiting
Mars? Part B
What is the altitude of a geosynchronous satellite orbiting
Mars?

This question is difficult and kind of confusing. I will show you
the solutions:
a) Mass of mars: 6.4191×10^23 kg
F_centripetal = F_gravitational
m_s * a_g = m_s * a_c
Note: m_s is the mass of satellite, a_g is gravitational
acceleration, and a_c is centripetal acceleration
Mass of satellite cancels out from the above equation and why the
mass of satellite is not even given to you in the first
place.
Therefore, a_g=a_c
a_c = (omega)^2*r
a_g = GM/r^2 <-----M is the mass of Mars (omega)^2*r = GM/r^2 r^3=GM/(omega)^2 r= cubic root {GM/(omega)^2} speed of a geosynchronous satellite = (omega)*r = (omega)*cubic root {GM/(omega)^2} = cubic root (GM*omega) Note: simplified using the fact that omega can also be written as cubic root (omega^3) Also know that T=2pi/omega or omega = 2pi/T speed of a geosynchronous satellite =cubic root (GM*omega)= cubic root (GM2pi/T) = cubic root [(6.673 x 10^-11 x 6.4191x10^23 kg x 2 x pi) / (24.8hours x 3600 seconds/hr)] = 1440 m/s = 1.44km/s b) v=omega*r r=v/omega omega = 2pi/T=2*3.14/(24.8hours x 3600 seconds/hr)] v= 1440m/s from part a r=2.05*10^7 But radius of mars is 3397000m Altitude = 2.05*10^7-3397000=1.72*10^7m

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