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. 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 speed Find the semimajor axis of its orbit. Determine the period for one orbit.

Solution

[a] The total energy of satellite -Earth system is

TE=KE +PE=1/2mv²+-gMm/[r+h]= 1/2^ 2.6*8300²+-6.67*10^-11*5.98*10²⁴*2.6/ [ 6.4*10⁶+9.2*10⁶]

TE=89557000-25568333= ^- 6.4*10 ⁷ J

TE=^- 6.4*10 ⁷ J

[b]

L=[r+h] Xmv=[ 6.4*10⁶+9.2*10⁶]* 2.6*8300=3.37*10¹¹ kgm²/s

[c]

Ke[orbit] = Gravitational Pe [orbit-earth]

1/2mv²= GMm/[r+h]²

1/2* 2.6* 8300²= 6.67*10^-11* 5.98*10²⁴* 2.6 / [r+h]²

89557000=3.95*10¹³* 2.6 / [r+h]²

[r+h]²= 1146755.69

r+h=1070.9

r= 1070.9-9.2*10⁶=9.19*10⁶ m