Question

Mars has surface gravity 0.39 g or about 3.9 m/s2, radius r = 3332 km and a rotation period of 24 hr. 37.38 min.
(a)What is the orbital velocity at distance r?
(b)What is the escape velocity from the surface?
(c) For communication, astronauts on Mars may use a synchronous satellite. At what distance R (in Mars radii) would it orbit?

Answer 1

Thes are quite simple. Rotational velocity is in radians per second. There are 2pi radians in a circumference. So you can easilty figure out the circumference. \[2\pi \times3.332\times10^{6} m\]
But the answer doesn't need this as you now there are 2 pi radians in one rotation. You must turn your hours into seconds.
\[24 \times3600 + (37.38\times60) = 886423 s\] which is the amont of time for one rotation. So how long does it take for one radian to be covered?
Divide last answer by 2 pi,
\[886423\div2\pi=14108 s\] to go one radian. so take the inverse to get,\[14108^{-1 } =7.088 \times10^{-5} rad s ^{-1}\]
Escape velocity is a simple formula\[v _{ e}=\sqrt{2GM/r}\]
Remember to conert mass into kg and radius into metres.
Im having troulble with keyboard so 'Google' the synchronous orbuation. .