PART I Determine the velocity and period of a satellite whose height above groun

Question

PART I

Determine the velocity and period of a satellite whose height above ground is 2.5 times the radius of Earth and whose orbit is circular.

PART II

We wish to place a satellite in a geosynchronous orbit about the planet Earth. Assume that the Earth rotates once about its axis in 24.0 hours. Determine the theoretical volocity and height above the planet's surface required.

PART III

What is the acceleration due to gravity at an altitude of 500km above the surface of Mars? With what velocity must an object leave the planet Mars with, assume vertical and no air resistance, in order to obtain this height as its maximum height?

F= G(m1m2/r^2)

G=6.67X10^-11 (N-m^2)/kg^2

Mmars= 6.42X10^23 kg

Rmars= 3.37X10^3 km

Explanation / Answer

Part I

d = 3.5 R from center.

Force of attraction = centripetal force

G*M*m/d^2 = m*v*v/d

v = sqrt(GM/d) = 4.225 km/s

Period = 2*pi*d/v = 9.24 hr.

Part II

Period, T = 24 hr

v = 2*pi*d/Period ; v*v = 4*pi*pi*d*d/T^2 = GM/d

d^3 = GMT*T/4*pi*pi => d = 42318 km

Velocity = sqrt(GM/d) = 3.076 km/s

Part III

g = GMmars/d^2 = 2.86 m/s2

PE + KE at surface = KE + PE at 500 km

-GMm/R + m*v*v/2 = -GMm/2d (total Energy at 500 km)

v = 1.812 km/s

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