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A satellite is in an elliptic orbit around the earth with aphelion of 6R and perihelion of 2 R where R= 6400 km is the radius of the earth. Find eccentricity of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius 6R ? [G = 6.67 × 10–11 SI units and M = 6 × 1024 kg].

0 2 Views | Posted 4 months ago
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    Answered by

    alok kumar singh | Contributor-Level 10

    4 months ago

    This is a Long Type Questions as classified in NCERT Exemplar

    Explanation- rp= radius of perihelion =2R

    Ra = radius of aphelion =6R

    So ra=a(1+e)=6R

    And rp= a(1-e)=2R

    Solving above eqns

     We get e = ½

    By conservation of angular momentum angular momentum at perigee= angular momentum at apogee

    So mvprp=mvara

    Va/vp=1/3

    Where m is mass of satellite .

    By conservation of energy , energy at perigee =energy at apogee

    1 2 m v p 2 - G M m r p = 1 2 m v a 2 - G M m r a

    So v p 2 1 - 1 9 = - 2 G M 1 r a - 1 r p = 2 G M 1 r p - 1 r a

    Vp= [ 2 G M R ( 1 2 - 1 6 ) ] 1 / 2 [ [ 1 - v a v p ] 2 ] 1 / 2 = 3 4 G M R = 6.85 k m / s

    Vp=6.85km/s , va=2.28km/s

    So orbital velocity vc= G M r

    R=6R ,vc= 3.23km/s

    Hence to transfer to a circular orbit at apogee , we have to boost the velocity by 3.23-2.28=0.95km/s.

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