A satellite of mass 2.6 kg is in an elliptical orbit around the Earth (mass 5.98

Question

A satellite of mass 2.6 kg is in an elliptical orbit around the Earth (mass 5.98e+24 kg). The satellite's minimum distance from the center of the Earth is 9200000 m, at which point it has a speed of 8300 m/s. (a) For this orbit find the total energy of the satellite–Earth system. J (b) For this orbit find the magnitude of the angular momentum of the satellite. kg · m2/s (c) Use conservation of energy and conservation of angular momentum to find the farthest distance from the center of the Earth that is reached by the satellite and its speed at that point. distance from the center of the Earth m speed m/s (d) Find the semimajor axis of its orbit. m (e) Determine the period for one orbit. min

Explanation / Answer

m = mass of satellite = 2.6 kg

M = mass of earth = 5.98 x 1024 kg

rmin = minimum distance = 9.2 x 106 m

v = speed = 8300 m/s

a)

Total energy = gravitational potential energy + KE = - GMm/rmin + (0.5) m v2

Total energy = - (6.67 x 10-11) (5.98 x 1024 ) (2.6)/(9.2 x 106) + (0.5) (2.6) (8300)2 = - 2.32 x 106 J

b)

angular momentum = mv rmin = 2.6 x 8300 x 9.2 x 106 = 1.986 x 1011

c)

V = speed at maximum distance

R = maximum distance

using conservation of momentum

mVR = 1.986 x 1011

VR = 0.764 x 1011                    eq-1

using conservation of energy

- GMm/R + (0.5) m V2 = - 2.32 x 106

- (6.67 x 10-11) (5.98 x 1024 ) (2.6)/R + (0.5) (2.6) (0.764 x 1011 /R)2 = - 2.32 x 106

R = 4.4 x 108 m

V = 0.764 x 1011 /(4.4 x 108 )

V = 173.64 m/s

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