A satellite of mass 26 kg is in an elliptical orbit around t

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. For this orbit find the total energy of the satellite-Earth system. Your response differs from the correct answer by more than 100%. J For this orbit find the magnitude of the angular momentum of the satellite. 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 Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully, m speed The correct answer is not zero. m/s Find the semimajor axis of its orbit. Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully, m Determine the period for one orbit. Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully, min

Solution

a) total energy = PE + KE

   = - G M m / d + m v^2 /2

= -[ (6.67 x 10^-11 x 5.98 x 10^24 x 2.6 ) / (9200000) ] + [ 2.6 x 8300^2 / 2 ]

= - 112723000 + 89557000

= - 23166000 J

= - 2.3166 e+7 J

b) p = 198.54 e+9 kg m^2 / s

c) angular momentum will conserved.

m v r = 198.54 x 10^9

v = 76.36e+9 / r

now applying energy conservation,

- G M m / r + m v^2 /2 = - 2.3166 e+7

-[ (6.67 x 10^-11 x 5.98 x 10^24 x 2.6 ) / r ] + [ 2.6 x v^2 / 2 ] = - 2.3166 e+7

- ( 7.97732e+14)/(r) + v^2 =   - 1.782 e+7

- ( 7.97732e+14)/(r) + (76.36e+9 / r )^2 =   - 1.782 e+7

- 7.97732e+14 r + 5.83e+21 = - 1.782 e+7 r^2

1.782 e+7 r^2 - 7.97732e+14 r + 5.83e+21 = 0

r = 3.56 x 10^7 m

v = 76.36e+9 / (3.56 x 10^7) = 2145 m/s

d) semimajor axis = (9.2 x 10^6 + 3.56 x 10^7 )/ 2   = 2.24 x 10^7 m

e) T = 2 pi sqrt [ a^3 / GM ]

T = 2 pi sqrt[ (2.24 x 10^7)^3 / (6.67 x 10^-11 x 5.98 x 10^24 )]

T = 33353.27 sec